[ { "index": 1, "ocr_en": "On line AE, select two points A and E. Draw line segment CE from point E such that CE is perpendicular to AE, with the foot of the perpendicular at E. Connect points A and C to form the red triangle ACE. Draw line segment AB from point A such that AB is perpendicular to AC, with the foot of the perpendicular at A, and points B and C are on the same side of line AE. Draw BD from point B such that BD is perpendicular to EA at point D, with the foot of the perpendicular at D and point D is to the left of point A, forming the purple triangle ADB.", "ocr_zh": "在直线AE上取两点A、E,过点E作CE垂直于AE，垂足为E,连接AC,作红色三角形ACE。过点A作AB垂直于AC，垂足为A，且点B、点C在直线AE同侧。过点B作BD垂直于EA于点D，垂足为D且点D在点A左侧，得紫色三角形ADB。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1.jpg" }, { "index": 2, "ocr_en": "Left Figure: In the green triangle ABE, AE is perpendicular to BE, with the foot of the perpendicular at E. Draw line segment AC from point A such that AC is perpendicular to AB, with the foot of the perpendicular at A. Extend EA, and draw CD from point C such that CD is perpendicular to EA at point D, with the foot of the perpendicular at D. Form the blue triangle ACD. Connect points B and C to form triangle ABC. At this point, point E lies on the extension of DA.\nMiddle Figure: In the green triangle ABD, AD is perpendicular to BD, with the foot of the perpendicular at D. Draw line segment AC from point A such that AC is perpendicular to AB, with the foot of the perpendicular at A. Lines AD and BC intersect. Draw CE from point C such that CE is perpendicular to AD at point E, with the foot of the perpendicular at E. Form the blue triangle ACE. Connect points B and C to form triangle ABC. At this point, point E lies on line segment AD.\nRight Figure: In the green triangle ABD, AD is perpendicular to BD, with the foot of the perpendicular at D. Draw AC from point A such that AC is perpendicular to AB, with the foot of the perpendicular at A. Extend BD to intersect AC at point C. Form the blue triangles ACE and ABC. At this point, points D and E coincide.", "ocr_zh": "左图：在绿色三角形ABE中，AE垂直于BE，垂足为E。过点A作线段AC垂直于AB，垂足为A。延长EA,过点C作CD垂直于EA于点D，垂足为D，作蓝色三角形ACD。连接BC作三角形ABC。此时点E在DA延长线上。\n\n中图：在绿色三角形ABD中，AD垂直于BD，垂足为D。过点A作线段AC垂直于AB，垂足为A，AD、BC有交点。过点C作CE垂直AD于点E，垂足为E，作蓝色三角形ACE。连接BC作三角形ABC。此时点E在线段AD上。\n\n右图：在绿色三角形ABD中，AD垂直于BD，垂足为D。过点A作AC垂直于AB，垂足为A。延长BD与AC交于点C，作蓝色三角形ACE，三角形ABC。此时点D、E重合。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2.jpg" }, { "index": 3, "ocr_en": "Top Figure: On line AE, select two points A and E. Draw line segment CE from point E such that CE is perpendicular to AE, with the foot of the perpendicular at E. Connect points A and C to form the red triangle ACE. Draw line segment AB from point A such that AB is perpendicular to AC, with the foot of the perpendicular at A. Draw BD from point B such that BD is perpendicular to AE at point D, forming the purple triangle ABD. Triangles ABD and ACE are on opposite sides of line AE, and points D and E do not coincide.\nBottom Figure: On line AE, select two points A and E. Draw line segment CE from point E such that CE is perpendicular to AE, with the foot of the perpendicular at E. Connect points A and C to form the red triangle ACE. Draw line segment AB from point A such that AB is perpendicular to AC, with the foot of the perpendicular at A. Draw BD from point B such that BD is perpendicular to AE at point D, forming the purple triangle ABD. Triangles ABD and ACE are on opposite sides of line AE, and points D and E coincide.", "ocr_zh": "上图：在直线AE上取两点A、E,过点E作CE垂直于AE，垂足为E，连接AC作红色三角形ACE。过点A作AB垂直于AC,垂足为A，过点B作BD垂直于AE于点D作紫色三角形ABD。三角形ABD与三角形ACE在直线AE两侧且点D、E不重合。\n下图：在直线AE上取两点A、E,过点E作CE垂直于AE，垂足为E，连接AC作红色三角形ACE。过点A作AB垂直于AC,垂足为A，过点B作BD垂直于AE于点D作紫色三角形ABD。三角形ABD与三角形ACE在直线AE两侧且点D、E重合。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3.jpg" }, { "index": 4, "ocr_en": "On line AE, select points A and E. Take a point C outside line AE, and connect AC and EC to form the red triangle ACE, where angle CAE is acute and angle AEC is obtuse. On the other side of line AE, take point B, and connect AB and DB to form the purple triangle ABD, where angle BAC is acute.", "ocr_zh": "在直线AE上取点A、E。在直线AE外取一点C，连接AC，EC作红色三角形ACE且角CAE为锐角，角AEC为钝角。在直线AE另一侧取点B，连接AB、DB作紫色三角形ABD，且角BAC为锐角。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4.jpg" }, { "index": 5, "ocr_en": "On line AE, select points A and E. Take a point C outside line AE, and connect AC and EC to form the red triangle ACE, where angle AEC is acute. Extend EA to point D. On the same side as point C, take point B, and connect AB and DB to form the purple triangle ABD, where angles ADB and BAC are acute.", "ocr_zh": "在直线AE上取点A、E,在直线AE外取一点C，连接AC，EC作红色三角形ACE且角AEC为锐角。延长EA至点D，在点C同侧取一点B，连接AB,DB作紫色三角形ABD且角ADB、角BAC为锐角。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_5.jpg" }, { "index": 6, "ocr_en": "On the line segment ED, take two points E and D, and take the midpoint A of ED. On the same side of the line segment ED, take points C and B. Connect CE and CA to form a red triangle ACE, with angle AEC denoted as angle 3 and angle AEC being an acute angle. Connect BA and BD to form a purple triangle ABD, with angle ADB denoted as angle 1 and angle ADB being an acute angle. Connect BC to form triangle ABC, with angle BAC denoted as angle 2 and angle BAC being an acute angle.", "ocr_zh": "在直线ED上取两点E、D，取ED的中点A。在直线E、D同侧取点C、B，连接CE，CA作红色三角形ACE，角AEC记为角3且角AEC为锐角。连接BA，BD作紫色三角形ABD，角ADB记为角1且角ADB为锐角。连接BC作三角形ABC，角BAC记为角2且为锐角。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_6.jpg" }, { "index": 7, "ocr_en": "Take two points A and E on the line AE. Take one point C outside the line AE. Connect CA and CE to form a red triangle ACE, with angle AEC being an obtuse angle and angle AEC denoted as angle 3. Take one point D on the extension of EA. Take one point B on the same side of the line AE as point C. Connect BA and BD to form a purple triangle ABD, with angle ADB being an obtuse angle and angle ADB denoted as angle 1. Angle BAC is denoted as angle 2.", "ocr_zh": "在直线AE上取两点A、E，在直线AE外取一点C，连接CA，CE作红色三角形ACE且角AEC为钝角，角AEC记为角3。在EA的延长线上取一点D，在直线AE外取与点C同侧的一点B，连接BA，BA作紫色三角形ABD且角ADB为钝角，记角ADB为角1，角BAC为角2。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_7.jpg" }, { "index": 8, "ocr_en": "Upper figure:Take two points A and D on the line AD. Take one point B on one side of the line AD. Connect BA and BD to form a purple triangle ABD. The exterior angle of angle ABD is denoted as angle 3 and angle 3 is an obtuse angle. Take one point E on the line segment AD. Take one point C outside the line AD such that points B and C are on different sides. Connect CA and CE to form a red triangle ACE. Angle CAB is denoted as angle 2 and angle 2 is an obtuse angle. Angle CED is denoted as angle 1 and angle 1 is an obtuse angle.\n\nLower figure:Take two points D and E on the line DE. Take one point A on the line segment DE such that A is the midpoint of DE. Take two points B and C on the same side of the line DE. Connect BD and BA to form a purple triangle ADB. Angle ADB is denoted as angle 1 and angle 1 is an obtuse angle. Connect CA and CE to form a red triangle AEC. Angle AEC is denoted as angle 3 and angle 3 is an obtuse angle. Connect BC to form triangle BAC. Angle BAC is denoted as angle 2 and angle 2 is an obt angleuse.", "ocr_zh": "上图：在直线AD上取两点A，D，在直线AD一侧取一点B，连接BA，BD作紫色三角形ABD，角ABD的外角记为角3且角3为钝角。在线段AD上取一点E，在直线AD外取一点C且点B，C异侧，连接CA，CE作红色三角形ACE。角CAB记为角2且角2为钝角，角CED记为角1且角1为钝角。下图：在直线DE上取两点D，E，在线段DE上取一点A且A为DE的中点。在直线DE同侧取点B、C，连接BD，BA作紫色三角形ADB且角ADB记为角1，角1为钝角。连接CA，CE作红色三角形AEC，角AEC记为角3且角3为钝角。连接BC作三角形BAC，角BAC记为角2且角2为钝角。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_8.jpg" }, { "index": 9, "ocr_en": "In rectangle ABCD, points H, G, and F are the midpoints of sides AD, DC, and BD, respectively. Connect AG to form a purple triangle ADG. Connect GF to form a red triangle GCF. Connect AC, EC, and EH.", "ocr_zh": "在矩形ABCD中，点H、G、F分别为边AD、DC、BD的中点。连接AG作紫色三角形ADG，连接GF作红色三角形GCF。连接AC、EC、EH。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_9.jpg" }, { "index": 10, "ocr_en": "Triangles ABC and DEF are two identical isosceles triangles, with CA = CB and DF = EF. Point D is on side AB, point M is the intersection of DF and CA, and point N is the intersection of DE and BC. Points A, D, and M form a purple triangle ADM, with angle MAD denoted as angle 1. Points D, B, and N form a red triangle DBN, with angle DBN denoted as angle 3. Angle MDN is denoted as angle 2. ", "ocr_zh": "三角形ABC、DEF为形状、大小完全相同的两个等腰三角形，CA等于CB，DF等于EF。点D在AB边上，点M为DF与CA的交点，点N为DE与BC的交点。点A、D、M构成紫色三角形ADM，角MAD记为角1。点D，B，N构成红色三角形DBN，角DBN记为角3。角MDN记为角2。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_10.jpg" }, { "index": 11, "ocr_en": "In the plane rectangular coordinate system xOy, the line AB intersects the x-axis at point A and the y-axis at point B. Point P is the midpoint of line segment OB. Connect AP to form a red triangle ABP. Point D is on the negative half-axis of the y-axis. Connect CD and CP to form a purple triangle DCP.", "ocr_zh": "在平面直角坐标系xOy中，直线AB交x轴于点A，交y轴于点B。点P为线段OB中点，连接AP作红色三角形ABP。D为y轴负半轴上一点，连接CD，CP作紫色三角形DCP。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_11.jpg" }, { "index": 12, "ocr_en": "In triangle ABC, AB is perpendicular to AC at point B. Lines l1, l2, and l3 pass through points A, B, and C, respectively, and are parallel to each other. Draw an auxiliary line BD through point B, perpendicular to l1 at point D, forming a purple triangle ABD. Draw an auxiliary line BE through point B, perpendicular to l3 at point E, forming a red triangle CBE. Draw an auxiliary line CF through point C, perpendicular to l1 at point F, with the foot of the perpendicular being F.", "ocr_zh": "在三角形ABC中，AB垂直于AC，垂足为B。直线l1、l2、l3分别过点A、B、C且l1、l2、l3互相平行。过点B作辅助线BD垂直于l1于点D，垂足为D，作紫色三角形ABD。过点B作辅助线BE垂直l3于点E，垂足为E，作红色三角形CBE。过点C作辅助线CF垂直l1于点F，垂足为F。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_12.jpg" }, { "index": 13, "ocr_en": "In the plane rectangular coordinate system xOy, the vertices B and D of the rhombus ABCD are on the graph of the inverse proportion function, with point B in the second quadrant and point D in the fourth quadrant. Connect AC and BD, and AC intersects BD at the coordinate origin O. Draw an auxiliary line AF through point A, perpendicular to the x-axis at point F, forming a red triangle AFO. Draw an auxiliary line BE through point B, perpendicular to the x-axis at point E, forming a purple triangle BOE.", "ocr_zh": "在平面直角坐标系xOy中，菱形ABCD的顶点B、D在反比例函数的图像上，且点B在第二象限，点D在第四象限。连接AC，BD，AC与BD交于坐标原点O。过点A作辅助线AF垂直x轴于点F，垂足为F，构成红色三角形AFO。过点B作辅助线BE垂直x轴于点E，垂足为E，构成紫色三角形BOE。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_13.jpg" }, { "index": 14, "ocr_en": "**Left figure:** In the plane rectangular coordinate system xOy, the graph of a quadratic function intersects the x-axis at points A and B, and intersects the y-axis at point C, where point A is on the negative half-axis of the x-axis, point B is on the positive half-axis of the x-axis, and point C is on the negative half-axis of the y-axis. Draw a circle C centered at point C. Draw an auxiliary circle passing through points C, O, and B, intersecting circle C at points P3 and P4, with P3 in the third quadrant and P4 in the fourth quadrant. Draw an auxiliary tangent line to the auxiliary circle through point C, intersecting circle C at points P1 and P2, with P1 in the third quadrant and P2 in the fourth quadrant. Draw an auxiliary tangent line to circle BOC through point B, parallel to P1P2.\n\n**Right figure:** In the plane rectangular coordinate system xOy, the graph of a quadratic function intersects the x-axis at points A and B, and intersects the y-axis at point C, where point A is on the negative half-axis of the x-axis, point B is on the positive half-axis of the x-axis, and point C is on the negative half-axis of the y-axis. Draw a circle C centered at point C. Point P3 is on circle C, with P3 in the third quadrant. Draw P3D perpendicular to the x-axis at point D, forming a purple triangle BDP3. Draw P3E parallel to the y-axis, and draw CE parallel to the x-axis intersecting at point E, forming a red triangle CEP3.", "ocr_zh": "左图：在平面直角坐标系xOy中，二次函数的图像于x轴交于A、B两点，与y轴交于点C，其中点A在x轴负半轴，点B在x轴正半轴，点C在y轴负半轴。过点C作圆C，过点C、O、B作辅助圆交圆C于点P3、P4且P3在第三象限，P4在第四象限。过点C作辅助圆的辅助线切线交圆C于点P1、P2且点P1在第三象限，P2在第四象限。过点B作圆BOC的辅助线切线且该切线平行于P1P2。右图：在平面直角坐标系xOy中，二次函数的图像于x轴交于A、B两点，与y轴交于点C，其中点A在x轴负半轴，点B在x轴正半轴，点C在y轴负半轴。过点C作圆C，P3为圆C上一点且点C在第三象限。过点P3作P3D垂直于x轴于点D，垂足为D，作紫色三角形BDP3。过点P3作P3E平行于y轴，过点C作CE平行于x轴交于点E，构成红色三角形CEP3。", "class": "Analytic Geometry", "difficulty": 3, "image_path": "fig_14.jpg" }, { "index": 15, "ocr_en": "In the plane rectangular coordinate system xOy, Point A is on the graph of the inverse proportion function \\(y = \\frac{k}{x}\\). Draw a ray AB through points A and B. Rotate the ray AB counterclockwise by 45 degrees around point A, intersecting the graph of the inverse proportion function at point C. Draw an auxiliary line CD perpendicular to AC at point C, with the foot of the perpendicular being C, and CD intersects the ray AB at point D. Draw an auxiliary line DE parallel to the y-axis through point D, and an auxiliary line CE parallel to the x-axis through point C. DE and CE intersect at point E, with the foot of the perpendicular being E. Draw an auxiliary line AF parallel to the y-axis through point A, and an auxiliary line CF parallel to the x-axis through point C. AF and CF intersect at point F, with AF perpendicular to CF.", "ocr_zh": "在平面直角坐标系xOy中，点A在反比例函数y=k/x的图像上，过点A，B作射线AB，将射线AB绕点A按逆时针方向旋转45度交反比例函数图像于点C。过点C作辅助线CD垂直于AC于点C，垂足为C，CD交射线AB于点D。过点D作辅助线DE平行于y轴，过点C作辅助线CE平行于x轴，DE、CE交于点E且垂足为E。过点A作辅助线AF平行于y轴，过点C作辅助线CF平行于x轴，CF、AF交于点F且AF垂直于CF。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_15.jpg" }, { "index": 16, "ocr_en": "Points A and B are two points on opposite sides of line l. Connect AB, intersecting line l at point P.", "ocr_zh": "点A、B为直线l异侧两点，连接AB，交直线l于点P。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_16.jpg" }, { "index": 17, "ocr_en": "Points A and B are two points on the same side of line l. Point A' is the reflection of point A across line l. Connect A'B to draw an auxiliary line intersecting l at point Q. Point P is a point on line l. Connect AQ and A'P to draw auxiliary lines, and connect AP and BP.", "ocr_zh": "点A、B为直线l同侧两点。点A'为点A关于直线l的对称点，连接A'B作辅助线交l于点Q。点P为直线l上一点，连接AQ、A'P作辅助线，连接AP，BP。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_17.jpg" }, { "index": 18, "ocr_en": "There is a straight line representing a river. Outside the line and on the same side, there is a point A representing a mountain peak and a point B representing a campsite. Point A is to the right of point B and is closer to the river.", "ocr_zh": "有一直线为河流，直线外同侧有一点山峰A和一点营地B。点山峰A在点营地B右侧且距离直线河流更近。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_18.jpg" }, { "index": 19, "ocr_en": "In angle MON, there are two points P and Q. Construct the reflection of point P across ray OM, denoted as P', and the reflection of point Q across ray ON, denoted as Q'. Connect P' and Q' to draw an auxiliary line that intersects OM at point A and ON at point B. Connect points A, P, Q, and B to form a red quadrilateral APQB. Draw auxiliary lines PP' and QQ'.", "ocr_zh": "在角MON中有两点P、Q，作点P关于射线OM的对称点PP'，作点Q关于射线ON的对称点QQ'，连接P'Q'作辅助线交OM于点A，交ON于点B。连接APQB作红色四边形APQB。连接PP'，QQ'作辅助线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_19.jpg" }, { "index": 20, "ocr_en": "Point A is a point outside line l. Line k is a line that is not parallel to line l. Take a point B on line k such that points A and B are on the same side of line l. Construct the reflection of point A across line l, denoted as A'. Connect AP to draw an auxiliary line. Connect A' and B to intersect line l at point P.", "ocr_zh": "点A为直线l外一点。直线k为与直线l不平行的一条直线，在直线k上取一点B且点A、B在直线l同侧。作点A关于直线l的对称点A'，连接AP作辅助线。连接A'B交直线l于点P。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_20.jpg" }, { "index": 21, "ocr_en": "In triangle ABC, points D, E, and F are points on sides AB, AC, and BC, respectively. Connect D, E, and F to form triangle DEF.", "ocr_zh": "在三角形ABC中，点D、E、F分别为边AB、AC、BC上一点。连接DEF作三角形DEF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_21.jpg" }, { "index": 22, "ocr_en": "In angle MON, there is a point P. Construct the reflection of point P across ray OM, denoted as P1, and the reflection of point P across ray ON, denoted as P2. Connect P1 and P2, P1 and P, P2 and P to draw auxiliary lines. Connect points A, B, and P to form a red triangle ABP.", "ocr_zh": "在角MON内有一点P，作点P关于射线OM的对称点P1，关于射线ON的对称点P2。连接P1P2,P1P,P2P作辅助线。连接ABP作红色三角形ABP。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_22.jpg" }, { "index": 23, "ocr_en": "In triangle ABC, draw an auxiliary line CD perpendicular to AB at point D. Construct the reflection of point D across AC, denoted as D', and the reflection of point D across BC, denoted as D''. Connect D' and D''. Draw auxiliary lines CD', CD'', DE, DF, and BE. With C as the center and CD' as the radius, draw an arc D'D'' passing through points D', D, and D''.", "ocr_zh": "在三角形ABC中，过点C作辅助线CD垂直AB于点D。作点D关于AC的对称点D'，关于BC的对称点D''。连接D'D''。连接CD',CD'',DE,DF,BE作辅助线。以C为圆心，CD'为半径，过点D',D,D''作圆弧D'D''。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_23.jpg" }, { "index": 24, "ocr_en": "Line \\( l \\) is a straight line, and points A and B are two points outside line \\( l \\) such that A and B are on opposite sides of \\( l \\).", "ocr_zh": "l为一条直线，点A、B为直线l外两点且A，B异侧。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_24.jpg" }, { "index": 25, "ocr_en": "Line \\( l \\) is a straight line, and points A and B are two points outside line \\( l \\) on the same side. Connect A and B, and extend AB to draw an auxiliary line BA that intersects line \\( l \\) at point P.", "ocr_zh": "l为一条直线，点A,B为直线l外同侧两点。连接AB，延长AB作辅助线BA交直线l于点P。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_25.jpg" }, { "index": 26, "ocr_en": "Line \\( l \\) is a straight line, and points A and B are two points outside line \\( l \\) on the same side. Construct the reflection of point B across line \\( l \\), denoted as B'. Draw an auxiliary line BB'. Connect A and B' to intersect line \\( l \\) at point P, where B'P is a dashed line and AP is a solid line. Connect BP.", "ocr_zh": "l为一条直线，点A,B为直线l外同侧两点。作点B关于直线l的对称点B'，连接BB'作辅助线。连接AB'交直线l于点P，其中B'P为虚线，AP为实线。连接BP。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_26.jpg" }, { "index": 27, "ocr_en": "Line \\( l \\) is a straight line, and points A and B are two points outside line \\( l \\) on opposite sides. Construct the reflection of point B across line \\( l \\), denoted as B'. Draw an auxiliary line BB'. Extend AB' to intersect line \\( l \\) at point P, where B'P is a dashed line and AB' is a solid line.", "ocr_zh": "l为一条直线，点A,B为直线l外异侧两点。作点B关于直线l的对称点B'，连接BB'作辅助线。延长AB'交直线l于点P。其中B'P为虚线，AB'为实线。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_27.jpg" }, { "index": 28, "ocr_en": "Line \\( l \\) is a straight line, and points A and B are two points outside line \\( l \\) on opposite sides. Connect A and B to intersect line \\( l \\) at point P.", "ocr_zh": "l为一条直线，点A,B为直线l外异侧两点。连接AB交直线l于点P。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_28.jpg" }, { "index": 29, "ocr_en": "Square ABCD has an equilateral triangle ABE inside it, with E being a point inside the square. Connect AE and BE to form the equilateral triangle ABE. Connect AC, and let P be a point on AC. Connect DP and EP.", "ocr_zh": "四边形ABCD为正方形,点E为正方形ABCD内一点，连接AE,BE且三角形ABE为等边三角形。连接AC。P为AC上一点，连接DP，EP。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_29.jpg" }, { "index": 30, "ocr_en": "Triangle ABC is an isosceles right triangle with AC perpendicular to BC at point C. Construct angle DCB with BC as a side and C as the vertex, such that ray CD intersects side AB. Point P is on ray CD and inside triangle ABC. Connect AP and BP.", "ocr_zh": "三角形ABC为等腰直角三角形，AC垂直于BC且垂足为C。以BC为边，C为顶点作角DCB，射线CD与边AB有交点。点P在射线CD上且在三角形ABC内，连接AP，BP。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_30.jpg" }, { "index": 31, "ocr_en": "In triangle ABC, AC = BC, and angle ACB is 90 degrees. Point D is the midpoint of side BC, and point E is on side AB. Connect CE and DE.", "ocr_zh": "在三角形ABC中，AC等于BC，角ACB为九十度。点D为边BC的中点，点E在边AB上。连接CE，DE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_31.jpg" }, { "index": 32, "ocr_en": "In the plane rectangular coordinate system xOy, the coordinates of point A are (0, 3), and the coordinates of point B are (2, 0).", "ocr_zh": "在平面直角坐标系xOy中，点A的坐标为（0，3），点B的坐标为（2，0）。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_32.jpg" }, { "index": 33, "ocr_en": "In square ABCD, connect AC, and let point N be a point on AC. Point M is a point on side CD. Connect DN and MN.", "ocr_zh": "在正方形ABCD中，连接AC，点N为AC上一点。点M为边CD上一点，连接DN,MN。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_33.jpg" }, { "index": 34, "ocr_en": "There is an angle AOB, and point P is inside angle AOB. Connect OP.", "ocr_zh": "有一角AOB，点P在角AOB内，连接OP。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_34.jpg" }, { "index": 35, "ocr_en": "In quadrilateral ABCD, angles B and D are both 90 degrees. Point M is on BC, and point N is on AD. Connect AM, AN, and MN to form triangle AMN.", "ocr_zh": "在四边形ABCD中，角B等于角D等于九十度。M为BC上一点，N为CD上一点，连接AM,AN,MN作三角形AMN。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_35.jpg" }, { "index": 36, "ocr_en": "There is an angle MON, and point P is inside angle MON.", "ocr_zh": "有一角MON，点P在角MON内。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_36.jpg" }, { "index": 37, "ocr_en": "There is an angle MON, and points A and B are on rays OM and ON, respectively.", "ocr_zh": "有一角MON，点A，B分别在射线OM，ON上。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_37.jpg" }, { "index": 38, "ocr_en": "In the plane rectangular coordinate system xOy, point B is in the first quadrant, and quadrilateral AOCB is a rectangle. Point D is the midpoint of OC. Points E and F are on line segment OA with point E to the left of point F. Connect DE and BF.", "ocr_zh": "在平面直角坐标系xOy中，点B在第一象限内且四边形AOCB为矩形。点D为OC的中点。点E，F在线段OA上且点E在点F左侧，连接DE,BF。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_38.jpg" }, { "index": 39, "ocr_en": "**Upper left figure:** Line \\( l \\) is a straight line, and segment \\( d \\) is a line segment parallel to \\( l \\) above \\( l \\). Points A and B are two points outside line \\( l \\) and on the same side as segment \\( d \\).\n\n**Upper right figure:** Line \\( l \\) is a straight line, and segment \\( d \\) is a line segment parallel to \\( l \\) above \\( l \\). Points A and B are two points outside line \\( l \\) and on the same side as segment \\( d \\). Translate point A to the right by \\( d \\) units to point A'. Construct the reflection of point A' across line \\( l \\), denoted as A''. Draw auxiliary lines A''A and A'A. Connect A'' and B to intersect line \\( l \\) at point N, where A''N is a dashed line and BN is a solid line. Translate point N to the left by \\( d \\) units to point M, and connect AM.\n\n**Lower left figure:** Lines \\( l_1 \\) and \\( l_2 \\) are two parallel lines with a distance \\( d \\) between them. Point A is a point above line \\( l_1 \\), and point B is a point below line \\( l_2 \\).\n\n**Lower right figure:** Lines \\( l_1 \\) and \\( l_2 \\) are two parallel lines with a distance \\( d \\) between them. Point A is a point above line \\( l_1 \\), and point B is a point below line \\( l_2 \\). Draw an auxiliary line between \\( l_1 \\) and \\( l_2 \\) with a distance \\( d \\). Translate point A downward by \\( d \\) units to point A', and draw an auxiliary line AA'. Connect A' and B to intersect line \\( l_2 \\) at point N, where A'N is a dashed line and BN is a solid line. Translate point N upward by \\( d \\) units to point M, and connect AM and MN.", "ocr_zh": "左上图：l为一条直线，d为直线l上方平行于l的一条线段。点A,B为直线l外与线段d同侧两点。右上图：l为一条直线，d为直线l上方平行于l的一条线段。点A,B为直线l外与线段d同侧两点。将点A向右平移d个单位长度到点A'，作点A'关于直线l的对称点A''，连接A''A,A'A作辅助线。连接A''B交直线l与点N，且A''N为虚线，BN为实线。将点N向左平移d个单位长度到点M，连接AM。左下图：l1,l2为两条平行直线且l1,l2之间距离为d，点A为直线l1上方一点，点B为直线l2下方一点。右下图：l1,l2为两条平行直线且l1,l2之间距离为d，点A为直线l1上方一点，点B为直线l2下方一点。作l1,l2之间的辅助线距离d，将点A向下平移d个单位长度到点A'，连接AA'作辅助线。连接A'B交直线l2于点N，且A'N为虚线，BN为实线。将点N向上平移d个单位长度到点M，连接AM,MN。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_39.jpg" }, { "index": 40, "ocr_en": "In the plane rectangular coordinate system xOy, the vertices of rectangle OACB are the coordinate origin O, vertex A on the positive half-axis of the x-axis, and vertex B on the positive half-axis of the y-axis. Point D is the midpoint of side OB. Connect CD.", "ocr_zh": "在平面直角坐标系xOy中，矩形OACB的顶点为坐标原点O，顶点A在x轴正半轴上，顶点B在y轴正半轴上。点D为边OB的中点，连接CD。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_40.jpg" }, { "index": 41, "ocr_en": "There are two parallel lines \\( l_1 \\) and \\( l_2 \\), and point A is a point above line \\( l_1 \\).", "ocr_zh": "有两条平行直线l1，l2，点A为直线l1上方一点。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_41.jpg" }, { "index": 42, "ocr_en": "In a cone, Q is the vertex of the cone, and point A is a point on the circumference of the base circle, Connect the center of the base circle with point Q and point A, and connect QA.", "ocr_zh": "在圆锥中，Q为圆锥顶点，点A为地面圆上一点，连接底面圆的圆心与点Q，点A，连接QA。", "class": "Solid Geometry", "difficulty": 1, "image_path": "fig_42.jpg" }, { "index": 43, "ocr_en": "There is a rectangular prism，Point A is the lower left vertex of the rectangular prism, and point B is the upper right vertex of the rectangular prism.", "ocr_zh": "有一长方体，点A为长方体左下顶点，点B为长方体右上顶点。", "class": "Solid Geometry", "difficulty": 1, "image_path": "fig_43.jpg" }, { "index": 44, "ocr_en": "There is a cylinder，Rectangle ABCD is a cross-section of the cylinder, with AB and DC being the height of the cylinder. Points A and D are on the same base, and points B and C are on the other base.", "ocr_zh": "有一圆柱体，矩形ABCD为该圆柱体的横截面，且AB,DC为圆柱体的高，点A,点D在同一底面上，点B，点C在另一底面上。", "class": "Solid Geometry", "difficulty": 1, "image_path": "fig_44.jpg" }, { "index": 45, "ocr_en": "There is a cylinder，Rectangle ABCD is a cross-section of the cylinder, with AB and DC being the height of the cylinder. Points A and D are on the same base, and points B and C are on the other base.", "ocr_zh": "有一圆柱体，矩形ABCD为该圆柱体的横截面，且AB,DC为圆柱体的高，点A,点D在同一底面上，点B，点C在另一底面上。", "class": "Solid Geometry", "difficulty": 1, "image_path": "fig_45.jpg" }, { "index": 46, "ocr_en": "There is a cylinder, and points A and B are on the same cross-section passing through the axis of the cylinder.", "ocr_zh": "有一圆柱，点A,B在过圆柱轴的同一横截面上。", "class": "Solid Geometry", "difficulty": 1, "image_path": "fig_46.jpg" }, { "index": 47, "ocr_en": "**Figure A:** The sector OA(A) is the lateral surface development of the cone, with point B being the midpoint of arc A(A). Connect OB as an auxiliary line, and point C is the midpoint of OB. Let point X be a point on OA, and connect AC, BX, and CX.\n\n**Figure B:** The sector OA(A) is the lateral surface development of the cone, with point B being the midpoint of arc A(A). Connect OB as an auxiliary line, and point C is the midpoint of OB, and connect AC. Let point X be a point on OA, and connect XC, extending XC to intersect arc B(A) at a point.\n\n**Figure C:** The sector OA(A) is the lateral surface development of the cone, with point B being the midpoint of arc A(A). Connect OB as an auxiliary line, and point C is the midpoint of OB, and connect AC. Let point X be a point on OA, and let point (X) be the point symmetric to X about OB on O(A). Connect C(X) and BX.\n\n**Figure D:** The sector OA(A) is the lateral surface development of the cone, with point B being the midpoint of arc A(A). Connect OB as an auxiliary line, and point C is the midpoint of OB. Let point X be a point on OA, and let point (X) be the point symmetric to X about OB on O(A). Connect BX, (A)C, and C(X).", "ocr_zh": "下图：有一圆锥，点O为圆锥顶点，OA，OB为圆锥的母线且点A、B在圆锥底面的同一条直径上。点C为母线OB的中点，连接AC。OA上有一点设为X，连接BX，CX。图A：扇形OA(A)为圆锥的侧面展开图，点B为弧A(A)的中点。连接OB作辅助线，点C为OB的中点。设OA上一点为X，连接AC，BX，CX。图B：扇形OA(A)为圆锥的侧面展开图，点B为弧A(A)的中点。连接OB作辅助线，点C为OB的中点，连接AC。设OA上一点为X，连接XC并延长XC与弧B(A)交于一点。图C：扇形OA(A)为圆锥的侧面展开图，点B为弧A(A)的中点。连接OB作辅助线，点C为OB的中点，连接AC。设OA上一点为X，O(A)上与关于OB与点X对称的一点为（X），连接C(X)，BX。图D：扇形OA(A)为圆锥的侧面展开图，点B为弧A(A)的中点。连接OB作辅助线，点C为OB的中点。设OA上一点为X，O(A)上与关于OB与点X对称的一点为（X），连接BX，(A)C,C(X)。", "class": "Solid Geometry", "difficulty": 2, "image_path": "fig_47.jpg" }, { "index": 48, "ocr_en": "There is a cube, and point P is the lower left corner vertex of the bottom face, point A is the lower right corner vertex of the top face, and point Q is on the right edge of the top face.", "ocr_zh": "有一正方体，点P为下底面左下角顶点，点A为上底面右下角顶点，点Q在上底面右侧棱上。", "class": "Solid Geometry", "difficulty": 1, "image_path": "fig_48.jpg" }, { "index": 49, "ocr_en": "There is a cube，Point A is the lower left corner vertex of the bottom face, point C is a point on the right edge of the front face, and point B is the upper left corner vertex of the top face. Let point X be a point on the right edge of the right face, and connect AC, CX, and XB.", "ocr_zh": "有一正方体。点A为下底面左下角顶点，点C为正面右侧棱上一点，点B为上底面左上角顶点。设右侧面右侧棱上一点为X，连接AC,CX,XB。", "class": "Solid Geometry", "difficulty": 1, "image_path": "fig_49.jpg" }, { "index": 50, "ocr_en": "In triangle ABC, point D is the midpoint of side BC, and point F is a point on side AB. Connect DF.", "ocr_zh": "在三角形ABC中，点D为边BC的中点，点F为边AB上一点，连接DF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_50.jpg" }, { "index": 51, "ocr_en": "In triangle ABD, point D is the midpoint of side BC. Connect AD.", "ocr_zh": "在三角形ABD中，点D为BC边中点。连接AD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_51.jpg" }, { "index": 52, "ocr_en": "Above: In triangle ABC, point D is the midpoint of the BC edge, connecting AD. Extend AD as an auxiliary line to point E and DE is equal to AD, and connect BE as an auxiliary line. Bottom: In triangle ABC, point F is on BC and point D is the midpoint of BC, connecting DF. Extend FD to point E as auxiliary line DE and FD is equal to DE, and connect EC as auxiliary line.", "ocr_zh": "上图：在三角形ABC中，点D为BC边中点，连接AD。延长AD作辅助线到点E且DE等于AD，连接BE作辅助线。下图：在三角形ABC中，点F在BC上，点D为BC的中点，连接DF。延长FD到点E作辅助线DE且FD等于DE，连接EC作辅助线。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_52.jpg" }, { "index": 53, "ocr_en": "In triangle ABC, AD is the median on side BC. Point E is a point on AD. Connect BE and extend BE to intersect AC at point F.", "ocr_zh": "在三角形ABC中，AD为BC边上的中线，点E是AD上一点，连接BE并延长BE交AC与点F。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_53.jpg" }, { "index": 54, "ocr_en": "In triangle ABC, point D is the midpoint of side BC. Connect AD.", "ocr_zh": "在三角形ABC中，点D为BC的中点，连接AD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_54.jpg" }, { "index": 55, "ocr_en": "In the acute triangle ABC, point D is the midpoint of BC, connecting AD. Point M is a point on AB close to B, and point N is a point on AC, connecting MD and ND, and MD is perpendicular to ND, and the vertical foot is D.", "ocr_zh": "在锐角三角形ABC中，点D为BC的中点，连接AD。点M为AB上靠近B的一点，点N为AC上一点，连接MD、ND，且MD垂直于ND，垂足为D。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_55.jpg" }, { "index": 56, "ocr_en": "Left diagram: In triangle ABC, AB equals AC, and point D is the midpoint of BC.\nRight diagram: In triangle ABC, AB equals AC. Point D is the midpoint of BC. Connect AD as an auxiliary line such that AD is perpendicular to BC, with the foot of the perpendicular being D.", "ocr_zh": "左图：在三角形ABC中，AB等于AC，点D为BC中点。右图：在三角形ABC中，AB等于AC。点D为BC中点，连接AD作辅助线AD垂直于BC，垂足为D。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_56.jpg" }, { "index": 57, "ocr_en": "In triangle ABC, AB equals AC. Point M is the midpoint of BC, and point N is a point on AC. Connect MN such that MN is perpendicular to AC, with the foot of the perpendicular being N.", "ocr_zh": "在三角形ABC中,AB等于AC。点M为BC中点，N为AC边上一点，连接MN且MN垂直于AC，垂足为N。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_57.jpg" }, { "index": 58, "ocr_en": "In triangle ABC, AB equals AC, and point D is the midpoint of BC. Point E is a point outside triangle ABC on the left side, and point F is a point outside triangle ABC on the right side. Connect AE, AF, DE, and DF such that AE is perpendicular to DE with the foot of the perpendicular being E, and AF is perpendicular to DF with the foot of the perpendicular being F.", "ocr_zh": "在三角形ABC中，AB等于AC，点D为BC中点。点E为三角形ABC外左侧一点，点F为三角形ABC外右侧一点，连接AE、AF、DE、DF，AE垂直于DE且垂足为E，AF垂直于DF且垂足为F。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_58.jpg" }, { "index": 59, "ocr_en": "In triangle ABC, AC equals BC and AC is perpendicular to BC with the foot of the perpendicular being C. Point D is the midpoint of AB, point E is a point on AC, and point F is a point on BC. Connect DE and DF such that DE is perpendicular to DF with the foot of the perpendicular being D, and DE is perpendicular to AC with the foot of the perpendicular being E.", "ocr_zh": "在三角形ABC中，AC等于BC且AC垂直于BC，垂足为C。点D为AB中点，点E为AC上一点，点F为BC上一点，连接DE、DF且DE垂直于DF，垂足为D，DE垂直于AC，垂足为E。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_59.jpg" }, { "index": 60, "ocr_en": "In triangle ABC, AC equals BC and AC is perpendicular to BC with the foot of the perpendicular being C. Point D is the midpoint of AB, point E is a point on AC, and point F is a point on BC. Connect DE and DF such that DE is perpendicular to DF with the foot of the perpendicular being D. Note that DE is not perpendicular to AC.", "ocr_zh": "在三角形ABC中，AC等于BC且AC垂直于BC，垂足为C。点D为AB中点，点E为AC上一点，点F为BC上一点，连接DE、DF且DE垂直于DF，垂足为D。DE不垂直于AC。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_60.jpg" }, { "index": 61, "ocr_en": "In triangle ABC, AC equals BC and AC is perpendicular to BC with the foot of the perpendicular being C. Point D is the midpoint of AB, point E is a point on the extension of AC, and point F is a point on the extension of CB. Connect DE and DF such that DE is perpendicular to DF with the foot of the perpendicular being D.", "ocr_zh": "在三角形ABC中，AC等于BC且AC垂直于BC，垂足为C。点D为AB中点，点E为AC延长线上一点，点F为CB延长线上一点，连接DE、DF且DE垂直于DF，垂足为D。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_61.jpg" }, { "index": 62, "ocr_en": "Left diagram: In triangle ABC, point D is the midpoint of AB.\nRight diagram: In triangle ABC, point D is the midpoint of AB, and point E is the midpoint of AC. Connect DE as an auxiliary line.", "ocr_zh": "左图：在三角形ABC中，点D为AB的中点。右图：在三角形ABC中，点D为AB的中点，点E为AC的中点，连接DE作辅助线。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_62.jpg" }, { "index": 63, "ocr_en": "In quadrilateral ABCD, AB equals CD. Point E is the midpoint of BC, and point F is the midpoint of AD. Connect EF, which intersects the extension of BA at point M and the extension of CD at point N.", "ocr_zh": "在四边形ABCD中，AB等于CD。点E为BC的中点，点F为AD的中点，连接EF，交BA的延长线于点M，交CD的延长线于点N。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_63.jpg" }, { "index": 64, "ocr_en": "On line MN, there are two points B and C with point B to the left of point C. Outside line MN, there is a point A, and connecting AB and AC forms an acute triangle ABC. BD is the angle bisector of angle ABM, and CE is the angle bisector of angle ACN. Draw AD perpendicular to BD at point D, with the foot of the perpendicular being D, and draw AE perpendicular to CE at point E, with the foot of the perpendicular being E. Connect DE.", "ocr_zh": "在直线MN上有两点B、C且点B在点C左侧。在直线MN外有一点A，连接AB、AC作锐角三角形ABC。BD为角ABM的角平分线，CE为角ACN的角平分线，过点A作AD垂直于BD于点D，垂足为D，过点A作AE垂直CE于点E，垂足为E。连接DE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_64.jpg" }, { "index": 65, "ocr_en": "In triangle ABC, BD is the angle bisector of angle ABC intersecting AB at point G, and CE is the angle bisector of angle ACB intersecting AC at point F. Draw AE perpendicular to CE at point E, with the foot of the perpendicular being E, and draw AD perpendicular to BD at point D, with the foot of the perpendicular being D. Connect DE.", "ocr_zh": "在三角形ABC中，BD平分角ABC交AB于点G，CE平分角ACB交AC于点F。过点A作AE垂直CE于点E，垂足为E，过点A作AD垂直BD于点D，垂足为D。连接DE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_65.jpg" }, { "index": 66, "ocr_en": "In triangle ABC, BD bisects angle ABC. Extend BC to point N, and CE bisects angle ACN. Draw AD perpendicular to BD at point D, with the foot of the perpendicular being D, and draw AE perpendicular to CE at point E, with the foot of the perpendicular being E. Connect DE.", "ocr_zh": "在三角形ABC中，BD平分角ABC。延长BC到点N，CE平分角ACN。过点A作AD垂直BD于点D，垂足为D，过点A作AE垂直CE于点E，垂足为E。连接DE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_66.jpg" }, { "index": 67, "ocr_en": "In quadrilateral ABCD, connect AB and CD intersecting at point O, and AB equals CD. Point F is the midpoint of AD, and point E is the midpoint of BC. Connect EF intersecting DC at point M and AB at point N.", "ocr_zh": "在四边形ABCD中，连接AB、CD交于点O且AB等于CD。点F为AD中点，点E为BC中点，连接EF交DC于点M，交AB于点N。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_67.jpg" }, { "index": 68, "ocr_en": "In triangle ABC, AC is longer than AB. Point D is on AC such that AB equals CD. Point E is the midpoint of BC, and point F is the midpoint of AD. Connect EF and extend EF to intersect the extension of BA at point G. Connect GD.", "ocr_zh": "在三角形ABC中，AC长于AB。D为AC上一点且AB等于CD。点E为BC中点，点F为AD中点，连接EF并延长EF交BA的延长线于点G。连接GD。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_68.jpg" }, { "index": 69, "ocr_en": "Left diagram: In triangle ABC, AC is perpendicular to BC with the foot of the perpendicular being C. Point D is the midpoint of AB.\nRight diagram: In triangle ABC, AC is perpendicular to BC with the foot of the perpendicular being C. Point D is the midpoint of AB. Connect CD as an auxiliary line.", "ocr_zh": "左图：在三角形ABC中，AC垂直于BC且垂足为C。点D为AB中点。右图：在三角形ABC中，AC垂直于BC且垂足为C。点D为AB中点，连接CD作辅助线。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_69.jpg" }, { "index": 70, "ocr_en": "In triangle ABC, BE is perpendicular to AC with the foot of the perpendicular being E, and CF is perpendicular to AB with the foot of the perpendicular being F. Connect EF. Point D is the midpoint of BC. Draw DM perpendicular to FE at point M, with the foot of the perpendicular being M.", "ocr_zh": "在三角形ABC中，BE垂直于AC且垂足为E，CF垂直于AB且垂足为F。连接EF。点D为BC中点，过点D作DM垂直FE于点M，垂足为M。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_70.jpg" }, { "index": 71, "ocr_en": "In triangle ABC, AD is perpendicular to BC with the foot of the perpendicular being D. Point M is the midpoint of BC and is located to the right of point D.", "ocr_zh": "在三角形ABC中，AD垂直于BC且垂足为D。点M为BC中点且点M在点D右侧。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_71.jpg" }, { "index": 72, "ocr_en": "In triangle ABD, AB is perpendicular to DB with the foot of the perpendicular being B. Point C is a point on AB. Draw CE perpendicular to AB at point C, with the foot of the perpendicular being C, and connect AE to form triangle ACE. Connect DE, and let M be the midpoint of DE, with M located outside of triangles ABD and ACE. Connect MC and MB.", "ocr_zh": "在三角形ABD中，AB垂直于DB且垂足为B。点C为AB上一点，过点C作CE垂直AB于点C，垂足为C，连接AE作三角形ACE。连接DE，DE中点为M且M在三角形ABD、三角形ACE外，连接MC，MB。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_72.jpg" }, { "index": 73, "ocr_en": "In triangle ABC, CB is not equal to CA. Point M is inside triangle ABC. Connect MB and MA such that angle MAC equals angle MBC. Draw ME perpendicular to BC at point E, with the foot of the perpendicular being E, and draw MF perpendicular to AC at point F, with the foot of the perpendicular being F. Point D is the midpoint of AB. Connect ED and FD.", "ocr_zh": "在三角形ABC中，CB不等于CA。点M在三角形ABC内部，连接MB、MA，且角MAC等于角MBC。过点M作ME垂直于BC，垂足为E，过点M作MF垂直于AC，垂足为F。点D为AB中点，连接ED，FD。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_73.jpg" }, { "index": 74, "ocr_en": "In triangle ABC, point D is the midpoint of AB. AE is perpendicular to BC with the foot of the perpendicular being E, and BF is perpendicular to AC with the foot of the perpendicular being F. AF and AE intersect at point M. Connect DF and EF.", "ocr_zh": "在三角形ABC中，点D为AB中点。AE垂直于BC且垂足为E，BF垂直于AC且垂足为F。AF、AE交于点M。连接DF、EF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_74.jpg" }, { "index": 75, "ocr_en": "In triangle ABC, CB equals CA. Point M is inside triangle ABC. Connect MB and MA such that angle MAC equals angle MBC. Draw ME perpendicular to BC at point E, with the foot of the perpendicular being E, and draw MF perpendicular to AC at point F, with the foot of the perpendicular being F. Point D is the midpoint of AB. Connect ED and FD.", "ocr_zh": "在三角形ABC中，CB等于CA。点M在三角形ABC内部，连接MB、MA，且角MAC等于角MBC。过点M作ME垂直于BC，垂足为E，过点M作MF垂直于AC，垂足为F。点D为AB中点，连接ED，FD。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_75.jpg" }, { "index": 76, "ocr_en": "Quadrilateral ABCD is a square. Point E is on BC, point F is on DC, and point G is on the extension of CD. Connect AE, AF, AG, and EF.", "ocr_zh": "四边形ABCD为正方形，点E在BC上，点F在DC上，点G在CD延长线上。连接AE、AF、AG、EF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_76.jpg" }, { "index": 77, "ocr_en": "In square ABCD, point E is on BC, and point F is on DC. Connect AE, AF, and EF.", "ocr_zh": "在正方形ABCD中，点E在BC上，点F在DC上。连接AE、AF、EF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_77.jpg" }, { "index": 78, "ocr_en": "Upper diagram: In angle AOB, OA equals OB. Points E and F are points inside angle AOB with point E to the left of point F. Connect OE and OF. Angle AOE is denoted as angle 1, angle EOF as angle 2, and angle FOB as angle 3.\nLower diagram: In angle AOB, OA equals OB. Points E and F are points inside angle AOB with point E to the left of point F. Connect OE and OF. Angle AOE is denoted as angle 1, angle EOF as angle 2, and angle FOB as angle 3. Rotate angle FOB clockwise around point O to the position of angle F'OA, and denote angle F'OA as angle 4. Connect F'A, F'E, EF, and FB as auxiliary lines.", "ocr_zh": "上图：在角AOB中，OA等于OB。点E、F为角AOB内一点且点E在点F左侧，连接OE、OF。角AOE记为角1，角EOF记为角2，角FOB记为角3。下图：在角AOB中，OA等于OB。点E、F为角AOB内一点且点E在点F左侧，连接OE、OF。角AOE记为角1，角EOF记为角2，角FOB记为角3。将角FOB绕点O顺时针旋转到角F'OA处，且角F'OA记为角4。连接F'A、F'E、EF、FB作辅助线。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_78.jpg" }, { "index": 79, "ocr_en": "Triangle ABC is an equilateral triangle. Point D is a point outside triangle ABC. Connect BD and CD. Construct angle EDF with vertex at point D such that angle EDF equals 60 degrees, with DE intersecting AB at point E and DF intersecting AC at point F. Connect EF.", "ocr_zh": "三角形ABC为等边三角形。点D为三角形ABC外一点，连接BD，CD。以点D为顶点作角EDF等于六十度，且DE交AB于点E，DF交AC于点F。连接EF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_79.jpg" }, { "index": 80, "ocr_en": "In triangle ABC, points D and E are on side BC with point D to the left of point E. Connect AD and AE such that angle DAE equals 45 degrees.", "ocr_zh": "在三角形ABC中，点D、E为BC边上两点且点D在点E左侧，连接AD、AE，角DAE等于45度。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_80.jpg" }, { "index": 81, "ocr_en": "Triangle ABC is an equilateral triangle. Point D is a point outside triangle ABC. Connect BD and CD. Construct angle EDF with vertex at point D such that angle EDF equals 60 degrees, with DE intersecting AB at point E and DF intersecting AC at point F. Connect EF to form the pink triangle DEF. Extend AC to point G and connect DG to form the orange triangle DFG.", "ocr_zh": "三角形ABC为等边三角形。点D为三角形ABC外一点，连接BD，CD。以点D为顶点作角EDF等于六十度，且DE交AB于点E，DF交AC于点F。连接EF作粉色三角形DEF。延长AC至点G，连接DG作橙色三角形DFG。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_81.jpg" }, { "index": 82, "ocr_en": "In triangle ABC, points D and E are on side BC with point D to the left of point E. Connect AD to form the yellow triangle ABD, and connect AE to form the red triangle AEC, with angle DAE equaling 45 degrees. Construct the yellow triangle ADF symmetric to triangle ABD about AD, and construct the red triangle AEF symmetric to triangle ACE about AE. DE is a dashed line, and EF is perpendicular to DF with the foot of the perpendicular being F.", "ocr_zh": "在三角形ABC中，点D、E为BC边上两点且点D在点E左侧，连接AD作黄色三角形ABD，连接AE作红色三角形AEC，角DAE等于45度。作三角形ABD关于AD对称的黄色三角形ADF，作AE作三角形ACE有关AE对称的红色三角形AEF。DE为虚线，EF垂直于DF且垂足为F。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_82.jpg" }, { "index": 83, "ocr_en": "In triangle ABC, points D and E are on side BC with point D to the right of point E. Connect AD and AE such that angle DAE equals 45 degrees. Triangle ABD is blue. Rotate triangle ABD counterclockwise around point A to the position of triangle ACF, where AB coincides with AC. Draw FC perpendicular to BC at point C, with the foot of the perpendicular being C. Connect EEF to form the red quadrilateral ADEF.", "ocr_zh": "在三角形ABC中，点D、E为边BC上两点且点D在点E右侧。连接AD、AE且角DAE为45度。三角形ABD为蓝色，将三角形ABD绕点A逆时针旋转至三角形ACF处且AB与边AC重合。过点F作FC垂直BC于点C且垂足为C。连接EEF得红色四边形ADEF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_83.jpg" }, { "index": 84, "ocr_en": "In square ABCD, point E is the midpoint of BC, and point F is the midpoint of CD. Connect AE, AF, and EF.", "ocr_zh": "在正方形ABCD中，点E为BC中点，点F为CD中点，连接AE、AF、EF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_84.jpg" }, { "index": 85, "ocr_en": "In square ABCD, connect BD. Point E is a point on BC, and point F is a point on CD. Connect AE, AF, and EF. AE intersects BD at point M, and AF intersects BD at point N. H is a point on EF, and connect AH.", "ocr_zh": "在正方形ABCD中，连接BD。点E为BC上一点，点F为CD上一点，连接AE、AF、EF。AE交BD于点M，AF交BD于点N。H为EF上一点，连接AH。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_85.jpg" }, { "index": 86, "ocr_en": "In square ABCD, point E is a point on BC, and point F is a point on CD. Connect AE, AF, and EF. Extend CD to point G, and connect AG.", "ocr_zh": "在正方形ABCD中，点E为BC上一点，点F为CD上一点。连接AE、AF、EF。延长CD至点G，连接AG。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_86.jpg" }, { "index": 87, "ocr_en": "In square ABCD, point E is a point on BC, and point F is a point on CD. Connect AE, AF, and EF. Extend CD to point E and connect AE as an auxiliary line.", "ocr_zh": "在正方形ABCD中，点E为BC上一点，点F为CD上一点，连接AE、AF、EF。延长CD作辅助线至点E，连接AE作辅助线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_87.jpg" }, { "index": 88, "ocr_en": "In square ABCD, AB is perpendicular to BC at point B, and AD is perpendicular to CD at point D. Point M is a point on BC, and point N is a point on CD. Connect AM, AN, and MN.", "ocr_zh": "在正方形ABCD中，AB垂直BC于点B且垂足为B，AD垂直CD于点D且垂足为D。点M为BC上一点，点N为CD上一点，连接AM、AN、MN。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_88.jpg" }, { "index": 89, "ocr_en": "In square ABCD, point F is a point on BC, and point E is a point on CD. Connect AF, AE, and EF. Draw AD perpendicular to EF at point G, with the foot of the perpendicular being G. Extend CB to point Q and connect AQ.", "ocr_zh": "在正方形ABCD中，点F为BC上一点，点E为CD上一点，连接AF、AE、EF。过点A作AD垂直EF于点G且垂足为G。延长CB至点Q，连接AQ。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_89.jpg" }, { "index": 90, "ocr_en": "In quadrilateral ABCD, angle ABC is an obtuse angle. Point K is a point on AB, and point N is a point on BC. Connect DK, DN, and KN.", "ocr_zh": "在四边形ABCD中，角ABC为钝角。点K为AB上一点，点N为BC上一点，连接DK、DN、KN。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_90.jpg" }, { "index": 91, "ocr_en": "In square ABCD, point M is a point on BC, and point N is a point on CD. Connect AM, AN, and MN such that BM is less than DN.", "ocr_zh": "在正方形ABCCD中，点M为BC上一点，点N为CD上一点，连接AM、AN、MN且BM小于DN。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_91.jpg" }, { "index": 92, "ocr_en": "In square ABCD, point M is a point on BC, and point N is a point on CD. Connect AM, AN, and MN such that BM equals DN.", "ocr_zh": "在正方形ABCCD中，点M为BC上一点，点N为CD上一点，连接AM、AN、MN且BM等于DN。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_92.jpg" }, { "index": 93, "ocr_en": "In square ABCD, point M is a point on the extension of CB, and point N is a point on the extension of DC. Connect AM, AN, and MN.", "ocr_zh": "在正方形ABCD中，点M为CB延长线上一点，点N为DC延长线上一点，连接AM、AN、MN。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_93.jpg" }, { "index": 94, "ocr_en": "In square ABCD, point M is a point on the extension of CB, and point N is a point on the extension of DC. Connect AM, AN, and MN. Take a point Q on DC such that DQ equals BM, and connect AQ as an auxiliary line.", "ocr_zh": "在正方形ABCD中，点M为CB延长线上一点，点N为DC延长线上一点，连接AM、AN、MN。在DC上取一点Q且DQ等于BM，连接AQ作辅助线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_94.jpg" }, { "index": 95, "ocr_en": "In square ABCD, point M is a point on BC, and point N is a point on CD such that BM is not equal to DN. Connect AM, AN, and MN. Point E is a point on the extension of CB, and BM is a dashed line. Connect AE as an auxiliary line.", "ocr_zh": "在正方形ABCD中，点M为BC上一点，点N为CD上一点且BM不等于DN。连接AM，AN，MN。点E为CB延长线上一点，BM为虚线。连接AE作辅助线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_95.jpg" }, { "index": 96, "ocr_en": "In quadrilateral ABCD, AB equals AD. Point M is a point on BC, and connect AM as an auxiliary line. Point F is a point on the extension of CD, and point E is a point on the extension of MC. Connect AF, EF, and AE.", "ocr_zh": "在四边形ABCD中，AB等于AD。点M为BC上一点，连接AM作辅助线。点F为CD延长线上一点，点E为MC延长线上一点，连接AF、EF、AE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_96.jpg" }, { "index": 97, "ocr_en": "In quadrilateral ABCD, AB equals AD. Point E is a point on the extension of BC, and point F is a point on the extension of CD. Connect AE, AF, and EF.", "ocr_zh": "在四边形ABCD中，AB等于AD点E为BC延长线上一点，点F为CD延长线上一点，连接AE、AF、EF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_97.jpg" }, { "index": 98, "ocr_en": "In the quadrilateral ABCD, AB is equal to AD. Point E is a point on BC and point F is a point on CD. Connect AE, AF, EF. Extend the CB to the point M and the BM is the dotted line, and connect the AM as an auxiliary line.", "ocr_zh": "在四边形ABCD中，AB等于AD。点E为BC上一点，点F为CD上一点。连接AE、AF、EF。延长CB至点M且BM为虚线，连接AM作辅助线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_98.jpg" }, { "index": 99, "ocr_en": "In quadrilateral ABCD, AB equals AD. Point E is a point on BC, and point F is a point on CD. Connect AE, AF, and EF.", "ocr_zh": "在四边形ABCD中，AB等于AD。点E为BC上一点，点F为CD上一点。连接AE、AF、EF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_99.jpg" }, { "index": 100, "ocr_en": "In pentagon ABCDE, AB equals AE. Connect AD.", "ocr_zh": "在五边形ABCDE中，AB等于AE。连接AD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_100.jpg" }, { "index": 101, "ocr_en": "In pentagon ABCDE, AB equals AE. Connect AD. Connect AC as an auxiliary line. Extend DE to point F such that EF equals BC, with EF being a dashed line. Connect AF as an auxiliary line.", "ocr_zh": "在五边形ABCDE中，AB等于AE。连接AD。连接AC作辅助线。延长DE到点F且EF等于BC，EF为虚线。连接AF作辅助线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_101.jpg" }, { "index": 102, "ocr_en": "In pentagon ABCDE, AB equals CD equals AE. AB is perpendicular to BC with the foot of the perpendicular being B, and AE is perpendicular to DE with the foot of the perpendicular being E.", "ocr_zh": "在五边形ABCDE中，AB等于CD等于AE。AB垂直于BC且垂足为B，AE垂直于DE且垂足为E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_102.jpg" }, { "index": 103, "ocr_en": "In pentagon ABCDE, AB equals CD equals AE. AB is perpendicular to BC with the foot of the perpendicular being B, and AE is perpendicular to DE with the foot of the perpendicular being E. Extend DE to point F such that EF equals BC. Connect AC, AD, and AF as auxiliary lines.", "ocr_zh": "在五边形ABCDE中，AB等于CD等于AE。AB垂直于BC且垂足为B，AE垂直于DE且垂足为E。延长DE至点F作辅助线EF且EF等于BC。连接AC、AD、AF作辅助线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_103.jpg" }, { "index": 104, "ocr_en": "In triangle ABC, AB equals BC, and point P is a point inside triangle ABC. Connect AP, BP, and CP.", "ocr_zh": "在三角形ABC中，AB等于BC且点P为三角形ABC内一点。连接AP、BP、CP。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_104.jpg" }, { "index": 105, "ocr_en": "In triangle ABC, point P is a point within triangle ABC. Connect AP, BP, CP. Extend BP to point D, connect AD, PD, CD, and AD and PD as dashed lines and CD as solid lines. Angular BCP is denoted as angular 1, angular ACP is denoted as angular 2, angular ACD is denoted as angular 3, angular BDC is denoted as angular 4, and angular ADB is denoted as angular 5.", "ocr_zh": "在三角形ABC中，点P为三角形ABC内一点。连接AP、BP、CP。延长BP至点D，连接AD、PD、CD，且AD、PD为虚线，CD为实线。角BCP记为角1，角ACP记为角2，角ACD记为角3，角BDC记为角4，角ADB记为角5。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_105.jpg" }, { "index": 106, "ocr_en": "In square ABCD, AB is perpendicular to BC with the foot of the perpendicular being B. Point M is a point on BC, and point N is a point on CD. Connect AM, AN, and MN such that angle MAN equals 45 degrees. Draw AP perpendicular to MN at point P, with AP being an auxiliary line.", "ocr_zh": "在正方形ABCD中，AB垂直于BC且垂足为B。点M为BC上一点，点N为CD上一点，连接AM，AN，MN。角MAN等于45度。过点A作辅助线AP垂直于MN于点P且AP垂直于MN。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_106.jpg" }, { "index": 107, "ocr_en": "In square ABCD, point M is a point on BC, and point N is a point on CD. Connect AM, AN, and MN such that angle MAN equals 45 degrees. Extend CB to point F such that BF equals DN, and connect AF as an auxiliary line, forming the red triangle ABF. Extend CD to point E such that DE equals BM, and connect AE as an auxiliary line, forming the yellow triangle ADE.", "ocr_zh": "在正方形ABCD中，点M为BC上一点，点N为CD上一点，连接AM，AN，MN。角MAN等于45度。延长CB至点F作辅助线BF且BF等于DN。连接AF作辅助线，得红色三角形ABF。延长CD至点E作辅助线且DE等于BM，连接AE作辅助线，得黄色三角形ADE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_107.jpg" }, { "index": 108, "ocr_en": "In square ABCD, AD is perpendicular to CD with the foot of the perpendicular being D, and AB is perpendicular to BC with the foot of the perpendicular being B. Point M is a point on the extension of CB, and point N is a point on the extension of DC. Connect AM, AN, and MN such that angle MAN equals 45 degrees. Draw AH perpendicular to the extension of NM at point H, with the extension of CB being a dashed line. Take a point X on CD such that DX equals BM, and connect AX as an auxiliary line. Triangle ABM is green, and triangle ADX is purple.", "ocr_zh": "在正方形ABCD中，AD垂直于CD且垂足为D，AB垂直于BC且垂足为B。点M为CB延长线上一点，点N为DC延长线上一点，连接AM，AN，MN。角MAN等于45度。过点A作AH垂直NM的延长线于点H且CB的延长线为虚线。在CD上取一点设为X且DX等于BM。连接AX作辅助线。三角形ABM为绿色，三角形ADX为紫色。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_108.jpg" }, { "index": 109, "ocr_en": "In square ABCD, AB is perpendicular to BC with the foot of the perpendicular being B, and AD is perpendicular to CD with the foot of the perpendicular being D. Point M is a point on BC, and point N is a point on CD. Connect AM, AN, and MN.", "ocr_zh": "在正方形ABCD中，AB垂直于BC且垂足为B，AD垂直于CD且垂足为D。点M为BC上一点，点N为CD上一点，连接AM、AN、MN。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_109.jpg" }, { "index": 110, "ocr_en": "In square ABCD, AD is perpendicular to CD with the foot of the perpendicular being D, and AB is perpendicular to BC with the foot of the perpendicular being B. Point M is a point on the extension of CB, and point N is a point on the extension of DC. Connect AM, AN, and MN such that angle MAN equals 45 degrees. Draw AH perpendicular to the extension of NM at point H, with the extension of CB being a solid line.", "ocr_zh": "在正方形ABCD中，AD垂直于CD且垂足为D，AB垂直于BC且垂足为B。点M为CB延长线上一点，点N为DC延长线上一点，连接AM，AN，MN。角MAN等于45度。过点A作AH垂直NM的延长线于点H且CB的延长线为实线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_110.jpg" }, { "index": 111, "ocr_en": "In quadrilateral ABCD, AB equals AD. Point E is a point on BC, and point F is a point on CD. Connect AE, EF, and AF with CE being a dashed line. Extend CD to point B' such that DB' equals BE. Connect AB' as an auxiliary line to form triangle ABD'. Triangles ABE and AB'D are green.", "ocr_zh": "在四边形ABCD中，AB等于AD。点E为BC上一点，点F为CD上一点。连接AE、EF、AF且CE为虚线。延长CD至点B'且DB'等于BE。连接AB‘作辅助线得三角形ABD'。三角形ABE、三角形AB'D为绿色。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_111.jpg" }, { "index": 112, "ocr_en": "In quadrilateral ABCD, AB equals CD, and AB is perpendicular to BC with the foot of the perpendicular being B. Point E is a point on BC, and point F is a point on CD. Connect AE, AF, and EF.", "ocr_zh": "在四边形ABCD中，AB等于CD，AB垂直于BC且垂足为B。点E，为BC上一点，点F为CD上一点，连接AE、AF、EF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_112.jpg" }, { "index": 113, "ocr_en": "In quadrilateral ABCD, AB equals AD. Point E is a point on BC, and point F is a point on CD. Connect AE, EF, and AF.", "ocr_zh": "在四边形ABCD中，AB等于AD。点E为BC上一点，点F为CD上一点。连接AE、EF、AF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_113.jpg" }, { "index": 114, "ocr_en": "In quadrilateral ABCD, AB equals CD, and AB is perpendicular to BC with the foot of the perpendicular being B. Point E is a point on BC, and point F is a point on CD. Connect AE, AF, and EF. Extend CB to point F' such that BF' equals DF. Connect AF' as an auxiliary line. Triangle ABF' is gray, and triangle ADF is purple.", "ocr_zh": "在四边形ABCD中，AB等于CD，AB垂直于BC且垂足为B。点E，为BC上一点，点F为CD上一点，连接AE、AF、EF。延长CB至点F'且BF'等于DF。连接AF'作辅助线。三角形ABF'为灰色，三角形ADF为紫色。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_114.jpg" }, { "index": 115, "ocr_en": "In triangle ABC, AC equals BC. Points M and N are points on AB with point M to the left of point N. Connect CM and CN. Denote AM as m, MN as x, and NB as n.", "ocr_zh": "在三角形ABC中，AC等于BC。点M、N分别为AB上两点且点M在点N左侧。连接CM，CN。将AM记为m，MN记为x，NB记为n。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_115.jpg" }, { "index": 116, "ocr_en": "In square ABCD, point Q is a point on AD, and point P is a point on AB. Connect CQ, CP, and QP.", "ocr_zh": "在正方形ABCD中，点Q为AD上一点，点P为AB上一点，连接CQ、CP、QP。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_116.jpg" }, { "index": 117, "ocr_en": "In triangle ABC, AC equals BC. Points M and N are points on AB with point M to the left of point N. Rotate triangle CBN clockwise around point C to the position of triangle CAD, where CA coincides with CB. Connect DM. Triangles CAD and CBN are gray.", "ocr_zh": "在三角形ABC中，AC等于BC。点M、N为AB上两点且点M在点N左侧。将三角形CBN绕点C顺时针旋转至三角形CAD处且CA于CB重合。连接DM。三角形CAD、三角形CBN为灰色。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_117.jpg" }, { "index": 118, "ocr_en": "In square ABCD, point E is a point on BC, and point F is a point on CD. Connect AE, AF, and ED. Point H is a point on EF, and connect AH.", "ocr_zh": "在正方形ABCD中，点E为BC上一点，点F为CD上一点，连接AE、AF、ED。点H为EF上一点，连接AH。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_118.jpg" }, { "index": 119, "ocr_en": "In triangle ABC, AB equals AC. Points D and E are points on BC with point D to the left of point E. Connect AD and AE.", "ocr_zh": "在三角形ABC中，AB等于AC。点D、E是BC边上两点且点D在点E左侧。连接AD、AE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_119.jpg" }, { "index": 120, "ocr_en": "In triangle ABC, AB equals AC. Point D is on the extension of CB, and point E is on BC. Connect AD and AE.", "ocr_zh": "在三角形ABC中，AB等于AC。点D在CB延长线上，点E在BC上，连接AD、AE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_120.jpg" }, { "index": 121, "ocr_en": "Left diagram: In square ABCD, point Q is a point on AD, and point P is a point on AB. Connect CQ, CP, and QP.", "ocr_zh": "左图：在正方形ABCD中，点Q为AD边上一点，点P为AB边上一点，连接CQ、CP、QP。右图：在正方形ABCD中，点Q为AD边上一点，点P为AB边上一点，连接CQ、CP、QP。延长AB至于点F且BF等于DQ，BF为虚线。连接CF作辅助线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_121.jpg" }, { "index": 122, "ocr_en": "Left diagram: In square ABCD, point E is a point on BC, and point F is a point on CD. Connect AE, AF, and EF. Point H is a point on EF, and connect EF.", "ocr_zh": "左图：在正方形ABCD中，点E为BC上一点，点F为CD上一点，连接AE、AF、EF。点H为EF上一点，连接EF。右图：在正方形ABCD中，点E为BC上一点，点F为CD上一点，连接AE、AF、EF。点H为EF上一点，连接EF。延长CB至点G且BG为虚线，BG等于DF，连接AG作辅助线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_122.jpg" }, { "index": 123, "ocr_en": "Upper diagram: In triangle ABC, AC equals BC. Points M and N are points on AB with point M to the left of point N. Connect CM and CN. Denote AM as m, MN as x, and NB as n.\nMiddle diagram: In triangle ABC, AC equals BC. Points M and N are points on AB with point M to the left of point N. Connect CM and CN. Denote AM as m, MN as x, and NB as n. Rotate triangle CNB clockwise around point C to the position of triangle CDA, where CA coincides with CB, and denote DA as n. Connect DM as an auxiliary line. Triangles CDA and CNB are gray.\nLower diagram: In triangle ABC, AC equals BC. Points M and N are points on AB with point M to the left of point N. Connect CM and CN. Construct triangle CPM symmetric to triangle CAM about CM, and connect PN as an auxiliary line. Triangles CAM and CPM are gray.", "ocr_zh": "上图：在三角形ABC中，AC等于BC。点M、N分别为AB上两点且点M在点N左侧。连接CM，CN。将AM记为m，MN记为x，NB记为n。中图：在三角形ABC中，AC等于BC。点M、N分别为AB上两点且点M在点N左侧。连接CM，CN。将AM记为m，MN记为x，NB记为n。将三角形CNB绕点C顺时针旋转到三角形CDA处且CA于CB重合，DA记为n,连接DM作辅助线。三角形CD、三角形CNB为灰色。下图：在三角形ABC中，AC等于BC。点M、N分别为AB上两点且点M在点N左侧。连接CM，CN。作三角形CAM关于CM对称的三角形CPM，连接PN作辅助线。三角形CAM、三角形CPM为灰色。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_123.jpg" }, { "index": 124, "ocr_en": "Lower In triangle ABC, AB equals AC. Point D is on the extension of CB, and point E is on BC. Connect AD and AE. Rotate triangle CAE clockwise around point A to the position of triangle CBE, where CA coincides with BA, and BE and AE are dashed lines. Connect DE as an auxiliary line.diagram: In triangle ABC, AC equals BC. Points M and N are points on AB with point M to the left of point N. Connect CM and CN. Construct triangle CPM symmetric to triangle CAM about CM, and connect PN as an auxiliary line. Triangles CAM and CPM are gray.", "ocr_zh": "在三角形ABC中，AB等于AC。点D在CB延长线上，点E在BC上，连接AD、AE。将三角形CAE绕点A顺时针旋转至三角形CBE处,CA与BA重合且BE、AE为虚线。连接DE作辅助线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_124.jpg" }, { "index": 125, "ocr_en": "In triangle ABC, point D is on the extension of CB, and point E is on CB. Connect AD and AE. Construct triangle AFD symmetric to triangle ABD about AD, with DF and AF being dashed lines. Connect EF as an auxiliary line.", "ocr_zh": "在三角形ABC中，点D在CB延长线上，点E为CB上一点，连接AD，AE。作三角形ABD关于AD对称的三角形AFD，DF、AF为虚线。连接EF作辅助线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_125.jpg" }, { "index": 126, "ocr_en": "In triangle ABJ, extend BJ to point C, and extend AJ to point D. Connect CD to form triangle CDJ. AB and CD are not parallel.", "ocr_zh": "在三角形ABJ中，延长BJ到点C，延长AJ到点D，连接CD得三角形CDJ。AB、CD不平行。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_126.jpg" }, { "index": 127, "ocr_en": "In a triangle with vertices A, B, and C, where AC is perpendicular to CB with the foot of the perpendicular being C, draw CD perpendicular to AB at point D, with the foot of the perpendicular being D.", "ocr_zh": "设一三角形中除A、C外的另一顶点为B。AC垂直与CB且垂足为C。过点C作CD垂直AB于点D且垂足为D。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_127.jpg" }, { "index": 128, "ocr_en": "In triangle ABJ, extend BJ to point C, and extend AJ to point D. Connect CD to form triangle CDJ. AB and CD are parallel.", "ocr_zh": "在三角形ABJ中，延长BJ到点C，延长AJ到点D，连接CD得三角形CDJ。AB、CD平行。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_128.jpg" }, { "index": 129, "ocr_en": "In triangle ABC, point D is a point on AB. Connect CD.", "ocr_zh": "在三角形ABC中，D为AB上一点。连接CD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_129.jpg" }, { "index": 130, "ocr_en": "Left diagram: In triangle ABC, point D is a point on AB, and point E is a point on AC. Connect DE such that DE is parallel to BC.\nRight diagram: In triangle ABC, point D is a point on AB, and point E is a point on AC. Connect DE such that DE is not parallel to BC.", "ocr_zh": "左图：在三角形ABC中，点D为AB上一点，点E为AC上一点，连接DE且DE平行于BC。右图：在三角形ABC中，点D为AB上一点，点E为AC上一点，连接DE且DE不平行于BC。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_130.jpg" }, { "index": 131, "ocr_en": "In triangle ADE, points B and C are points on DE with point B to the right of point C. Connect AB and AC.", "ocr_zh": "在三角形ADE中，点B、C为DE上两点且点B在点C右侧。连接AB，AC。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_131.jpg" }, { "index": 132, "ocr_en": "Left diagram: In triangle ABE, AB is perpendicular to BE with the foot of the perpendicular being B. Draw ED perpendicular to AE at point E, with the foot of the perpendicular being E, and connect AD. Draw DC perpendicular to the extension of BE at point C, with the foot of the perpendicular being C.\nRight diagram: Insufficient information to label.", "ocr_zh": "左图：在三角形ABE中，AB垂直于BE且垂足为B。过点E作ED垂直AE于点E且垂足为E，连接AD。过点D作DC垂直BE的延长线于点C，垂足为C。右图：信息不足，无法标注。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_132.jpg" }, { "index": 133, "ocr_en": "In trapezoid ABCD, AD is parallel to BC and AD is less than BC. Connect AC and BD intersecting at point O. Extend CA to point E such that BE is parallel to CD, and connect BE.", "ocr_zh": "在梯形ABCD中，AD平行于BC且AD小于BC。连接AC、BD交于点O。延长CA到点E使BE平行于CD，连接BE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_133.jpg" }, { "index": 134, "ocr_en": "In triangle ABC, point D is the midpoint of BC. Connect AD. Point E is a point on AD. Connect BE and CE.", "ocr_zh": "在三角形ABC中，D为BC的中点，连接AD。点E为AD上一点，连接BE，CE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_134.jpg" }, { "index": 135, "ocr_en": "In triangle ABC, point D is on BC, and connect AD. Construct the perpendicular bisector EF of AD, intersecting AB at point E, AC at point F, and AD at point M. Extend EF and BC to intersect at point N, and connect AN. Denote angle CAD as angle 1, angle DAB as angle 2, angle ANE as angle 3, angle BNE as angle 4, angle AME as angle 5, angle NMD as angle 6, and angle NAC as angle 7.", "ocr_zh": "在三角形ABC中，点D在BC上,连接AD。作AD的垂直平分线EF交AB于点E，交AC于点F，交AD于M。延长EEF、BC交于点N，连接AN。角CAD记为角1，角DAB记为角2，角ANE记为角3，角BNE记为角4，角AME记为角5，角NMD记为角6，角NAC记为角7。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_135.jpg" }, { "index": 136, "ocr_en": "In triangle ABC, point D is on BC, and connect AD. Construct the perpendicular bisector EF of AD, intersecting AD at point D, and extend EF to intersect the extension of BC at point F.", "ocr_zh": "在三角形ABC中，点D在BC上,连接AD。作AD的垂直平分线EF交AD于点D，延长EF交BC的延长线于点F。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_136.jpg" }, { "index": 137, "ocr_en": "In triangle ABC, AB equals BC. Draw CG parallel to AB, and connect BG intersecting AD at point E and AC at point F.", "ocr_zh": "在三角形ABC中，AB等于BC。过点C作CG平行于AB，连接BG交AD于点E，交AC于点F。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_137.jpg" }, { "index": 138, "ocr_en": "In triangle ABC, AB equals AC. Point D is a point on BC, and connect AD such that AD is the altitude of triangle ABC. Point E is a point on AC, and connect BE intersecting AD at point H. Draw EF perpendicular to BC at point F. M is the midpoint of AH, and connect BM. Extend AD to point G such that DG equals EF, and connect BG.", "ocr_zh": "在三角形ABC中，AB等于AC。D为BC边上一点，连接AD且AD为三角形ABC的高。点E为AC上一点，连接BE交AD于点H，过点E作EF垂直BC于点F。M是AH的中点，连接BM。延长AD到点G且DG等于EF。连接BG。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_138.jpg" }, { "index": 139, "ocr_en": "In triangle ABC, points D and E are points on line AC with point D to the left of point E. Point P is a point on AB. Connect DP, PE, and BE.", "ocr_zh": "在三角形ABC中，点D、E为直线AC上两点且点D在点E左侧。点P为AB上一点，连接DP、PE、BE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_139.jpg" }, { "index": 140, "ocr_en": "In triangle ABC, point D is a point on AB, and point E is a point on AC. Connect CD and BE. Point F is a point on BE. Connect DF and CF. Denote angle BCD as angle 1, angle CBE as angle 2, and angle ABE as angle 3.", "ocr_zh": "在三角形ABC中，点D为AB上一点，点E为AC上一点。连接CD，BE。点F为BE上一点，连接DF、CF。角BCD记为角1，角CBE记为角2，角ABE记为角3。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_140.jpg" }, { "index": 141, "ocr_en": "In triangle ABC, point E is a point on AB, and point D is a point on AC. Connect CE and BD.", "ocr_zh": "在三角形ABC中，点E为AB边上一点，点D为AC边上一点，连接CE、BD。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_141.jpg" }, { "index": 142, "ocr_en": "In triangle ABC, point D is a point on BC, and point E is a point on AB. Connect AD, CE, and DE.", "ocr_zh": "在三角形ABC中，点D为BC边上一点，点E为AB边上一点，连接AD、CE、DE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_142.jpg" }, { "index": 143, "ocr_en": "In triangle ABC, point E is a point on AB, point F is a point on AC, and point D is a point on BC. Connect DE and DF.", "ocr_zh": "在三角形ABC中，点E为AB上一点，点F为AC上一点，点D为BC上一点，连接DE、DF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_143.jpg" }, { "index": 144, "ocr_en": "In triangle ABC, points D and E are points on BC with point D to the left of point E. Connect AD and AE.", "ocr_zh": "在三角形ABC中，点D、E分别为BC边上的两点且点D在点E左侧。连接AD、AE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_144.jpg" }, { "index": 145, "ocr_en": "In trapezoid ABCD, AD is parallel to BC and AD is less than BC. Point P is a point on AD. Connect BP and CP.", "ocr_zh": "在梯形ABCD中，AD平行于BC且AD小于BC。点P为AD边上一点，连接BP、CP。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_145.jpg" }, { "index": 146, "ocr_en": "In triangle ABC, AB equals AC and angle BAC is an obtuse angle. Point P is a point on BC, and point Q is a point on AC. Connect AP and PQ.", "ocr_zh": "在三角形ABC中，AB等于AC且角BAC为钝角。点P为BC边上一点，点Q为AC边上一点，连接AP、PQ。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_146.jpg" }, { "index": 147, "ocr_en": "There is a trapezoid ABCD where AD is parallel to BC and AD is less than BC.", "ocr_zh": "有一梯形ABCD，AD平行于BC且AD小于BC。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_147.jpg" }, { "index": 148, "ocr_en": "In trapezoid ABCD, AD is parallel to BC and AD is less than BC. Point E is on side AB, point F is on side CD, and point M is on side BC. Connect EF, EM, and FM.", "ocr_zh": "在梯形ABCD中，AD平行于BC且AD小于BC。点E在边AB上，点F在边CD上，点M在边BC上。连接EF、EM、FM。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_148.jpg" }, { "index": 149, "ocr_en": "In triangle ABC, AB equals AC. Point D is a point on BC, and point E is a point on AC. Connect AD and DE.", "ocr_zh": "在三角形ABC中，AB等于AC。点D为BC边上一点，点E为AC边上一点，连接AD、DE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_149.jpg" }, { "index": 150, "ocr_en": "In trapezoid ABCD, AD is parallel to BC and AD is less than BC. Point E is a point on AB, and point P is a point on BC. Connect EP and DP.", "ocr_zh": "在梯形ABCD中，AD平行于BC且AD小于BC。点E为AB上一点，点P为BC上一点，连接EP、DP。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_150.jpg" }, { "index": 151, "ocr_en": "In trapezoid ABCD, AD is parallel to BC and AD is less than BC. Point E is a point on AB.", "ocr_zh": "在梯形ABCD中，AD平行于BC且AD小于BC。点E为AB上一点。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_151.jpg" }, { "index": 152, "ocr_en": "In triangle ABC, AB equals AC. Point D is a point on AB, point E is a point on BC, and point F is a point on AC. Connect DE and EF.", "ocr_zh": "在三角形ABC中，AB等于AC。点D为AB上一点，点E为BC上一点，点F为AC上一点。连接DE、EF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_152.jpg" }, { "index": 153, "ocr_en": "In rectangle ABCD, point E is a point on AB, and point P is a point on AD. Connect EP and CP.", "ocr_zh": "在矩形ABCD中，点E为AB边上一点，点P为AD边上一点，连接EP、CP。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_153.jpg" }, { "index": 154, "ocr_en": "Triangle ABC is an equilateral triangle. Point E is a point on AB, and point F is a point on BC. Connect EF. Construct an equilateral triangle EFG on the right side of EF. BG intersects AC at point M, and FG intersects AC at point N.", "ocr_zh": "三角形ABC为等边三角形，点E为AB边上一点，点F为BC边上一点，连接EF。以EF为一边向右侧作等边三角形EFG，BG交AC于点M，FG交AC于点N。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_154.jpg" }, { "index": 155, "ocr_en": "In triangle ABC, point Q is a point on BC, point P is a point on AC, and point O is a point on AB. Connect PQ and OP.", "ocr_zh": "在三角形ABC中，点Q为BC上一点，点P为AC上一点，点O为AB上一点，连接PQ、OP。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_155.jpg" }, { "index": 156, "ocr_en": "In triangle ABC, point F is a point on AC, point D is a point on BC, and point E is a point on AB. Connect DF and DE.", "ocr_zh": "在三角形ABC中，点F为AC边上一点，点D为BC边上一点，点E为AB边上一点。连接DF、DE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_156.jpg" }, { "index": 157, "ocr_en": "In triangle ABC, AC is perpendicular to BC with the foot of the perpendicular being C. Points D and F are points on CB with point D to the left of point F. Connect DE and EF, and extend EF. Extend BC.", "ocr_zh": "在三角形ABC中，AC垂直于BC且垂足为C。点D、F为CB上两点且点D在点F左侧。连接DE，EF并延长EF。延长BC。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_157.jpg" }, { "index": 158, "ocr_en": "In triangle ABC, point F is a point on AC, point D is a point on BC, and point E is a point on AB. Connect DF and DE.", "ocr_zh": "在三角形ABC中，AC垂直于BC且垂足为C。点D为BC上一点。延长BC。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_158.jpg" }, { "index": 159, "ocr_en": "In trapezoid ABCD, AB is parallel to CD and AB is less than CD. Point P is on BC. Connect AP and PQ such that PQ is not equal to DQ.", "ocr_zh": "在梯形ABCD中，AB平行于CD且AB小于CD。点P在BC上，连接AP、PQ且PQ不等于DQ。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_159.jpg" }, { "index": 160, "ocr_en": "In trapezoid ABCD, AB is parallel to CD and AB is less than CD. Point P is on BC. Connect AP and PQ such that PQ equals DQ.", "ocr_zh": "在梯形ABCD中，AB平行于CD且AB小于CD。点P在BC上，连接AP、PQ且PQ等于DQ。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_160.jpg" }, { "index": 161, "ocr_en": "In triangle ABC, point E is a point on AB, and point F is a point on AC. Connect CE and AF, intersecting at point O.", "ocr_zh": "在三角形ABC中，点E为AB上一点，点F为AC上一点，连接CE、AF交于点O。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_161.jpg" }, { "index": 162, "ocr_en": "In triangle ABC, point D is a point on AC. Connect BD.", "ocr_zh": "在三角形ABC中，点D为AC边上一点，连接BD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_162.jpg" }, { "index": 163, "ocr_en": "In triangle ABC, point M is the midpoint of BC. Draw DM perpendicular to BC at point M, intersecting the extension of CA at point D and AB at point E. Connect AM.", "ocr_zh": "在三角形ABC中，点M为BC中点。过点M作DM垂直BC于点M，与CA的延长线交于点D，于AB交于点E。连接AM。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_163.jpg" }, { "index": 164, "ocr_en": "In parallelogram ABCD, point E is a point on the extension of AB. Connect DE intersecting BC at point F.", "ocr_zh": "在平行四边形ABCD中，点E是AB延长线上一点，连接DE交BC于点F。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_164.jpg" }, { "index": 165, "ocr_en": "In triangle ABC, point D is a point on BC, and point E is a point on AC. Connect AD and AE, intersecting at point F.", "ocr_zh": "在三角形ABC中，点D为BC边上一点，点E为AC边上一点。连接AD、AE交于点F。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_165.jpg" }, { "index": 166, "ocr_en": "In triangle ABC, point D is a point on BC. Connect AD. Construct the perpendicular bisector EF of AD, intersecting the extension of BC at point F.", "ocr_zh": "在三角形ABC中，点D为BC上一点，连接AD。作AD的垂直平分线EF与BC的延长线交于点F。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_166.jpg" }, { "index": 167, "ocr_en": "In parallelogram ABCD, point E is a point on the extension of BA. Connect CE intersecting AD at point F.", "ocr_zh": "在平行四边形ABCD中，点E为BA延长线上一点，连接CE交AD于点F。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_167.jpg" }, { "index": 168, "ocr_en": "In triangle ABC, AB equals AC. Point D is a point on BC. Connect AD. Draw CF parallel to AB, and connect BF intersecting AD at point P and AC at point E.", "ocr_zh": "在三角形ABC中，AB等于AC。点D为BC上一点，连接AD。过点C作CF平行于AB，连接BF交AD于点P，交AC于点E。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_168.jpg" }, { "index": 169, "ocr_en": "In triangle ABC, AB is greater than AC. Point D is a point on AB, and point E is a point on AC such that AD equals AE. Connect DE and extend DE to intersect the extension of BC at point P.", "ocr_zh": "在三角形ABC中，AB大于AC。点D为AB边上一点，点E为AC边上一点且AD等于AE。连接DE并延长DE交BC的延长线于点P。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_169.jpg" }, { "index": 170, "ocr_en": "In the triangle ABC, the angle A is the right angle, the AD is perpendicular to BC, the perpendicular foot is D, and E is the midpoint of the AC, connecting ED and extending the extension line with AB to intersect at the point F", "ocr_zh": "三角形ABC中，角A为直角，做AD垂直于BC，垂足为D，E为AC上中点，连接ED并延长与AB的延长线交于点F", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_170.jpg" }, { "index": 171, "ocr_en": "In the parallelogram ABCD, the point E is the point on AD. Connect the BE and extend the extension cable of the BE to the CD at point F. Connect AC to BE at point O.", "ocr_zh": "在平行四边形ABCD中，点E为AD上一点。连接BE，延长BE交CD的延长线于点F。连接AC交BE于点O。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_171.jpg" }, { "index": 172, "ocr_en": "In the triangle ABC, AD is perpendicular to the point D. Point P is the midpoint of AD, which connects BP and extends BP to AC at point N. PASS POINT N AS NM AND BC PERPENDICULAR TO POINT M.", "ocr_zh": "在三角形ABC中，AD垂直BC于点D。点P为AD中点，连接BP，延长BP交AC于点N。过点N作NM垂直BC于点M。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_172.jpg" }, { "index": 173, "ocr_en": "In the triangle ABC, the BD is perpendicular to the point D. The point E is a point on the edge of AB, and the point E is EH vertical BC at point H, and EH is at point G. Connect to the CE. The extension line of the HE to CA is extended at point M.", "ocr_zh": "在三角形ABC中，BD垂直AC于点D。点E为AB边上一点，过点E作EH垂直BC于点H，EH交BD于点G。连接CE。延长HE交CA的延长线于点M。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_173.jpg" }, { "index": 174, "ocr_en": "In the triangle ABC, AD is perpendicular to the point D. Crossing point D is DE perpendicular AB to point E, DF perpendicular AC to point F. Connect EF.", "ocr_zh": "在三角形ABC中，AD垂直BC于点D。过点D作DE垂直AB于点E，DF垂直AC于点F。连接EF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_174.jpg" }, { "index": 175, "ocr_en": "In triangle ABC, point D is the midpoint of AC. Connect the BD. Passing point A as AE vertical BD to point E, connecting to CE.", "ocr_zh": "在三角形ABC中，点D为AC中点。连接BD。过点A作AE垂直BD于点E，连接CE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_175.jpg" }, { "index": 176, "ocr_en": "In the parallelogram ABCD, the point E is on AD and the point is on CD, connecting AC, EF, and AC parallel to EF. Connect BE to AC to point M, and BF to AC to point N.", "ocr_zh": "在平行四边形ABCD中，点E在AD上，点在CD上，连接AC、EF且AC平行于EF。连接BE交AC于点M，连接BF交AC于点N。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_176.jpg" }, { "index": 177, "ocr_en": "In trapezoid ABCD, AC is parallel to BD and AC is greater than BD. Connect AB such that AB equals AC. Draw CE parallel to AB intersecting the extension of DA at point E. Connect CD intersecting AB at point M, and connect BE intersecting AC at point N. Connect MN.", "ocr_zh": "在梯形ABCD中，AC平行于BD且AC大于BD。连接AB且AB等于AC。过点C作CE平行AB交DA的延长线于点E，连接CD交AB于点M，连接BE交AC于点N。连接MN。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_177.jpg" }, { "index": 178, "ocr_en": "Triangle ABC is a right triangle with AC equal to BC. Point P is on AB. Draw PE perpendicular to AC at point E, and PF perpendicular to BC at point F. Connect BE intersecting PF at point M, and connect AF intersecting PE at point N. Connect MN.", "ocr_zh": "三角形ABC为直角三角形且AC等于BC。点P在AB上，过点P作PE垂直AC于点E，作PF垂直BC于点F。连接BE交PF于点M，连接AF交PE于点N，连接MN。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_178.jpg" }, { "index": 179, "ocr_en": "There is a right triangle ABC with a square BEDF inscribed in triangle ABC, where point E is on AB, point F is on BC, and point D is on AC. Connect AF intersecting DE at point M, and connect CE intersecting DF at point N. Connect MN. AF and CE intersect at point P.", "ocr_zh": "有一直角三角形ABC，正方形BEDFD内接于三角形ABC且点E在AB上，点F在BC上，点D在AC上。连接AF交DE于点M，连接CE交DF于点N，连接MN。AF、CE交于点P。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_179.jpg" }, { "index": 180, "ocr_en": "In triangle ABC, point E is the midpoint of AC, and point F is the midpoint of AB. Connect CF and BE. Point P is on AB, and draw DP parallel to CF intersecting BC at point D. Draw DQ parallel to BE intersecting AC at point Q. Connect PQ. AE and PQ intersect at point R, CF and BE intersect at point G, and PQ and CF intersect at point S.", "ocr_zh": "在三角形ABC中，点E为AC的中点，点F为AB的中点。连接CF、BE。点P在AB上，过点P作DP平行CF交BC于点D。过点D作DQ平行BE交AC于点Q。连接PQ。AE、PQ交于点R，CF、BE交于点G，PQ、CF交于点S。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_180.jpg" }, { "index": 181, "ocr_en": "In trapezoid ABCD, AB is parallel to CD and CD is less than AB. Point M is a point on AB. Connect BD and AC. Draw MK parallel to BD intersecting AD at point K, and draw MN parallel to AC intersecting BC at point N. Connect KN intersecting AC at point P and BD at point Q. AB and CD intersect at point O, KM and AC intersect at point R, and MN and BD intersect at point S.", "ocr_zh": "在梯形ABCD中，AB平行于CD且CD小于AB。点M为AB边上一点，连接BD，AC，过点M作MK平行BD交AD于点K，作MN平行AC交BC于点N。连接KN交AC于点P，交BD于点Q。AB、CD交于点O，KM、AC交于点R，MN、BD交于点S。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_181.jpg" }, { "index": 182, "ocr_en": "In triangle ABC, AC is greater than AB. Point D is on BC such that AD is the angle bisector of angle BAC, and point E is on BC such that AE is the median of triangle ABC. Draw BF perpendicular to AD at point G, extend AG to intersect AE at point F and AC at point M. Connect EG and extend EG to intersect AB at point H.", "ocr_zh": "在三角形ABC中，AC大于AB。点D在BC上且AD是角BAC的角平分线，点E在BC上且AE是三角形ABC的中线。过点B作BF垂直AD于点G，延长AG交AE于点F，交AC于点M。连接EG，延长EG交AB于点H。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_182.jpg" }, { "index": 183, "ocr_en": "In square ABCD, point F is a point on AB, point G is a point on CD, and point E is a point on BC. Connect AE, EG, and EF. Denote angle EAB as angle 1, angle BEF as angle 2, and angle CEG as angle 3.", "ocr_zh": "在正方形ABCD中，点F为AB上一点，点G为CD上一点，点E为BC上一点，连接AE、EG、EF。角EAB记为角1，角BEF记为角2，角CEG记为角3。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_183.jpg" }, { "index": 184, "ocr_en": "In triangle ABC, point E is a point on AB, point F is a point on AC, and point D is the midpoint of BC. Connect DE and DF.", "ocr_zh": "在三角形ABC中，点E为AB上一点，点F为AC上一点，点D为BC中点。连接DE、DF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_184.jpg" }, { "index": 185, "ocr_en": "In triangle ABC, point D is a point on BC. Connect AD.", "ocr_zh": "在三角形ABC中，点D为BC边上一点，连接AD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_185.jpg" }, { "index": 186, "ocr_en": "In triangle ABC, points D and E are points on BC such that BD equals DC equals AC, and point E is the midpoint of DC. Connect AD and AE.", "ocr_zh": "在三角形ABC中，点D、E为BC边上两点且BD等于DC等于AC，点E为DC的中点。连接AD、AE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_186.jpg" }, { "index": 187, "ocr_en": "Left diagram: In triangle ABC, AB is perpendicular to AC with the foot of the perpendicular being A. Construct isosceles right triangles ABD and ACE outward with AB and AC as the legs, respectively. Connect ED. Point N is the midpoint of ED, and point M is the midpoint of BC. Connect AM and AN.\nRight diagram: In triangle ABC, construct isosceles right triangles ABD and ACE outward with AB and AC as the legs, respectively. Connect ED. Point N is the midpoint of ED, and point M is the midpoint of BC. Connect AM and AN.", "ocr_zh": "左图：在三角形ABC中，AB垂直于AC且垂足为A。分别以AB、AC为腰向外作等腰直角三角形ABD和ACE。连接ED。点N为ED中点，点M为BC中点，连接AM、AN。右图：有一三角形ABC，分别以AB、AC为腰向外作等腰直角三角形ABD和ACE。连接ED。点N为ED中点，点M为BC中点，连接AM、AN。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_187.jpg" }, { "index": 188, "ocr_en": "In triangle ABC, AD is the angle bisector of angle BAC. Extend AD to point D such that AD equals BD. Connect BD and BC.", "ocr_zh": "在三角形ABC中，AD是角BAC的角平分线。延长AD到点D，连接BD，AD等于BD。连接BC。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_188.jpg" }, { "index": 189, "ocr_en": "In quadrilateral ABCD, BC is greater than BA, and AD equals CD. Connect BD.", "ocr_zh": "在四边形ABCD中，BC大于BA且AD等于CD。连接BD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_189.jpg" }, { "index": 190, "ocr_en": "In trapezoid ABCD, AD is parallel to CB and AD is less than CB. Point E is a point on CD. Connect AE and BE.", "ocr_zh": "在梯形ABCD中，AD平行于CB且AD小于CB。点E为CD上一点，连接AE、BE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_190.jpg" }, { "index": 191, "ocr_en": "In triangle ABC, construct the angle bisector of angle CAB intersecting CB at point P, and construct the angle bisector of angle CBA intersecting AC at point Q.", "ocr_zh": "在三角形ABC中，作角CAB的角平分线角CB于点P，作角CBA的角平分线交AC于点Q。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_191.jpg" }, { "index": 192, "ocr_en": "In triangle ABC, AD is the angle bisector of angle BAC, intersecting BC at point D. Point P is a point on AD. Connect BP and CP. Denote angle BAD as angle 1 and angle DAC as angle 2.", "ocr_zh": "在三角形ABC中，AD为角BAC的角平分线，AD交BC于点D。点P为AD上一点，连接BP，CP。角BAD记为角1，角DAC记为角2。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_192.jpg" }, { "index": 193, "ocr_en": "In triangle ABC, AD bisects angle BAC and intersects BC at point D. Draw line MN perpendicular to AD at point A. Point E is on line MN and is to the right of point A. Connect AE and CE.", "ocr_zh": "在三角形ABC中，AD平分角BAC且AD交BC于点D。过点A作直线MN垂直AD于点A。点E在直线MN上且点E在点A右侧，连接AE、CE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_193.jpg" }, { "index": 194, "ocr_en": "In trapezoid ABCD, AD is parallel to BC and AD is less than BC. Point E is a point on AB. Connect DE and CE.", "ocr_zh": "在梯形ABCD中，AD平行于BC且AD小于BC。点E为AB上一点，连接DE、CE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_194.jpg" }, { "index": 195, "ocr_en": "In triangle ABC, AD bisects angle BAC and intersects BC at point D, and CE bisects angle BAC and intersects AB at point E. AD and CE intersect at point O.", "ocr_zh": "在三角形ABC中，AD平分角BAC且AD交BC于点D，CE平分角BAC且交AB于点E。AD、CE交于点O。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_195.jpg" }, { "index": 196, "ocr_en": "In triangle ABC, AD bisects angle BAC. Point D is outside triangle BAC. Draw DE perpendicular to AB at point E, and DF perpendicular to the extension of AC at point F. Point E is on side AB, and point F is on the extension of AC. Draw DG perpendicular to BC and bisecting BC at point G.", "ocr_zh": "在三角形ABC中，AD平分角BAC。点D在三角形BAC外，过点D作DE垂直AB于点E，DF垂直AC的延长线于点F。点E在边AB上，点F在AC的延长线上。过点D作DG垂直且平分BC交BC于点G。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_196.jpg" }, { "index": 197, "ocr_en": "In triangle ABC, points D and E are points on BC with point D to the left of point E, and BD equals CE. Connect AD and AE.", "ocr_zh": "在三角形ABC中，点D、E为BC边上两点且点D在点E左侧，BD等于CE。连接AD、AE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_197.jpg" }, { "index": 198, "ocr_en": "In square ABCD, point E is a point on BC, and point F is a point on CD. Connect AE, AF, and EF.", "ocr_zh": "在正方形ABCD中，点E为BC上一点，点F为CD上一点，连接AE、AF、EF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_198.jpg" }, { "index": 199, "ocr_en": "In triangle ABC, angle ACB is not a right angle. AD bisects angle BAC and intersects EC at point D, and CE bisects angle ACB and intersects AB at point E. AD and CE intersect at point F.", "ocr_zh": "在三角形ABC中，角ACB不是直角。AD平分角BAC且交EC于点D，CE平分角ACB且交AB于点E。AD、CE交于点F。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_199.jpg" }, { "index": 200, "ocr_en": "In angle MON, OP bisects angle MON.", "ocr_zh": "在角MON中，OP平分角MON。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_200.jpg" }, { "index": 201, "ocr_en": "In triangle ABC, angle ACB is a right angle. AD bisects angle BAC and intersects EC at point D, and CE bisects angle ACB and intersects AB at point E. AD and CE intersect at point F.", "ocr_zh": "在三角形ABC中，角ACB是直角。AD平分角BAC且交EC于点D，CE平分角ACB且交AB于点E。AD、CE交于点F。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_201.jpg" }, { "index": 202, "ocr_en": "Triangle ABC is an equilateral triangle. Point D is a point outside triangle ABC. Connect BD and CD such that BD equals CD and angle BDC is an obtuse angle. Point M is a point on AB, and point N is a point on AC. Connect DM, DN, and MN.", "ocr_zh": "三角形ABC为等边三角形。点D为三角形ABC外一点，连接BD、CD，BD等于CD且角BDC为钝角。点M为AB上一点，点N为AC上一点，连接DM、DN、MN。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_202.jpg" }, { "index": 203, "ocr_en": "In quadrilateral ABCD, AB is perpendicular to AD with the foot of the perpendicular being A, and BC is perpendicular to CD with the foot of the perpendicular being C, with AB equal to BC. There is an acute angle MBN with vertex B. Rotate angle MBN around point B such that BN intersects the extension of DC at point F, and BM intersects the extension of AD at point E. Connect EF.", "ocr_zh": "在四边形ABCD中，AB垂直于AD且垂足为A，BC垂直于CD且垂足为C，AB等于BC。有一以点B为顶点的锐角MBN，将角MBN绕点B旋转，BN交DC的延长线于点F，BM交AD的延长线于点E，连接EF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_203.jpg" }, { "index": 204, "ocr_en": "In quadrilateral ABCD, AB is perpendicular to AD with the foot of the perpendicular being A, and BC is perpendicular to CD with the foot of the perpendicular being C, with AB equal to BC. There is an acute angle MBN with vertex B. Rotate angle MBN around point B such that BN intersects CD at point F, and BM intersects AD at point E. Connect EF.", "ocr_zh": "在四边形ABCD中，AB垂直于AD且垂足为A，BC垂直于CD且垂足为C，AB等于BC。有一以点B为顶点的锐角MBN，将角MBN绕点B旋转，BN交CD于点F，BM交AD于点E，连接EF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_204.jpg" }, { "index": 205, "ocr_en": "There is a square ABCD. Point P is a point to the left and below square ABCD. Connect DP, AP, and MP.", "ocr_zh": "有一正方形ABCD。点P为正方形ABCD左下侧一点，连接DP、AP、MP。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_205.jpg" }, { "index": 206, "ocr_en": "Triangle ABC is an equilateral triangle. Point N is a point on the extension of CA, point M is a point on the extension of AB, and point D is a point outside triangle ABC and below BC. Connect ND, NM, DM, DC, and DB such that DC equals DB.", "ocr_zh": "三角形ABC为等边三角形。点N为CA延长线上一点，点M为AB延长线上一点，点D为三角形ABC外BC下方一点。连接ND、NM、DM、DC、DB且DC等于DB。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_206.jpg" }, { "index": 207, "ocr_en": "Left diagram: There is an equilateral triangle ABC. Point D is a point outside triangle ABC and below BC. Point M is a point on AB, and point N is a point on AC. Connect DM, DN, MN, BD, and DC such that BD equals DC, and MN is parallel to BC.\nRight diagram: There is an equilateral triangle ABC. Point D is a point outside triangle ABC and below BC. Point M is a point on AB, and point N is a point on AC. Connect DM, DN, MN, BD, and DC such that BD equals DC, and MN is not parallel to BC.", "ocr_zh": "左图：有一等边三角形ABC，点D为三角形ABC外边BC下方一点，点M为AB上一点，点N为AC上一点。连接DM、DN、MN、BD、DC且BD等于DC，MN平行于BC。右图：有一等边三角形ABC，点D为三角形ABC外边BC下方一点，点M为AB上一点，点N为AC上一点。连接DM、DN、MN、BD、DC且BD等于DC，MN不平行于BC。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_207.jpg" }, { "index": 208, "ocr_en": "In triangle ABC, point D is a point on BC. Connect AD. Construct the perpendicular bisector EF of AD, intersecting the extension of BC at point F.", "ocr_zh": "在三角形ABC中，点D为BC上一点，连接AD。作AD的垂直平分线EF与BC的延长线交于点F。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_208.jpg" }, { "index": 209, "ocr_en": "In triangle ABC, points D and E are points on BC with point D to the left of point E, and point E is the midpoint of CD. BD equals DC equals AC. Extend AE to point G, and connect AD, BG, and DG.", "ocr_zh": "在三角形ABC中，点D、E分别为BC边上两点且点D在点F左侧，点E为CD的中点，BD等于DC等于AC。延长AE至点G，连接AD、BG、DG。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_209.jpg" }, { "index": 210, "ocr_en": "In triangle ABC, point E is a point on AB, and point F is a point on AC. Connect DE, DF, and EF.", "ocr_zh": "在三角形ABC中，点E为AB上一点，点F为AC上一点，连接DE、DF、EF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_210.jpg" }, { "index": 211, "ocr_en": "In triangle ABC, D is the midpoint of BC, E is the midpoint of DC, connecting AD and AE.", "ocr_zh": "在三角形ABC中，D是BC的中点，E是DC的中点，连接AD，AE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_211.jpg" }, { "index": 212, "ocr_en": "In the quadrilateral DBCE, connect BE, CD at point A, connect DE, M on BC, and N on DE. Extend AM to G, so that MG=AM, connect BG and CG. Extend MA and DE to intersect at H.", "ocr_zh": "在四边形DBCE中，连接BE，CD交于点A，连接DE，M在BC上，N在DE上。延长AM到G，使MG=AM，连接BG和CG。延长MA与DE相交于H。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_212.jpg" }, { "index": 213, "ocr_en": "Left figure: In triangle ABC, AB is perpendicular to AC, AB=AD, AC=AE, AB is perpendicular to AD at point A, and AC is perpendicular to AE at point A. Connect DE, where M is the midpoint of BC and N is the midpoint of DE.\nRight figure: In triangle ABC, AB=AD, AC=AE, AB is perpendicular to AD at point A, and AC is perpendicular to AE at point A. Connect DE, where M is the midpoint of BC and N is the midpoint of DE.", "ocr_zh": "左图：在三角形ABC中，AB垂直于AC，AB=AD，AC=AE，AB垂直于AD于点A，AC垂直于AE于点A。连接DE，M是BC的中点，N是DE的中点。\n右图，在三角形ABC中，AB=AD，AC=AE，AB垂直于AD于点A，AC垂直于AE于点A。连接DE，M是BC的中点，N是DE的中点。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_213.jpg" }, { "index": 214, "ocr_en": "In the quadrilateral ABDC, connecting AD, BC, and F is the midpoint of AB, and connecting DF.", "ocr_zh": "四边形ABDC中，连接AD，BC，F是AB的中点，连接DF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_214.jpg" }, { "index": 215, "ocr_en": "In the quadrilateral DBCE, A is an internal point connecting AB, AC, AD, AE, BC, DE, M on BC, P on DE, N on DE, connecting AM, AN, AP, F on the AP extension line, and connecting BF.", "ocr_zh": "在四边形DBCE中，A是内部一点，连接AB，AC，AD，AE，BC，DE，M在BC上，P在DE上，N在DE上，连接AM，AN，AP，F在AP延长线上，连接BF。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_215.jpg" }, { "index": 216, "ocr_en": "Quadrilateral ABCD, E on CD, connect AE, BE.", "ocr_zh": "四边形ABCD，E在CD上，连接AE，BE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_216.jpg" }, { "index": 217, "ocr_en": "Quadrilateral ABCD, E on CD, connecting AE, BE, F on AB, as auxiliary line EF.", "ocr_zh": "四边形ABCD，E在CD上，连接AE，BE，F在AB上，作辅助线EF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_217.jpg" }, { "index": 218, "ocr_en": "In triangle ABC, P is on BC and Q is on AC, connecting BQ and AP.", "ocr_zh": "三角形ABC中，P在BC上，Q在AC上，连接BQ，AP。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_218.jpg" }, { "index": 219, "ocr_en": "In triangle ABC, D is on BC, connected to AD, P is on AD, connected to BP, CP, angle 1 is angle BAD, angle 2 is angle CAD.", "ocr_zh": "三角形ABC中，D在BC上，连接AD，P在AD上，连接BP，CP，角1是角BAD，角2是角CAD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_219.jpg" }, { "index": 220, "ocr_en": "In triangle ABC, D is on BC, connected to AD, P is on AD, connected to BP, CP, and F on the extension line of AC, serving as auxiliary line PF, angle 1 is angle BAD, and angle 2 is angle CAD.", "ocr_zh": "三角形ABC中，D在BC上，连接AD，P在AD上，连接BP，CP，F在AC延长线上，作辅助线PF，角1是角BAD，角2是角CAD。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_220.jpg" }, { "index": 221, "ocr_en": "Quadrilateral ABCD, E on AB, connecting DE, CE.", "ocr_zh": "四边形ABCD，E在AB上，连接DE，CE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_221.jpg" }, { "index": 222, "ocr_en": "Quadrilateral ABCD, E on AB, connecting DE, CE, as auxiliary line AC, F on AC, as auxiliary line EF.", "ocr_zh": "四边形ABCD，E在AB上，连接DE，CE，作辅助线AC，F在AC上，作辅助线EF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_222.jpg" }, { "index": 223, "ocr_en": "Quadrilateral ABCE, connecting AC, BE, D on BC, connecting AD, M on EA extension line, N on AE extension line.", "ocr_zh": "四边形ABCE，连接AC，BE，D在BC上，连接AD，M在EA延长线上，N在AE延长线上。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_223.jpg" }, { "index": 224, "ocr_en": "In triangle ABC, D, M, and E are on BC, connecting AD, AM, AE, and P on AB as auxiliary lines PD and extending, intersecting with the AM extension line at N and connecting BN.", "ocr_zh": "三角形ABC中，D，M，E在BC上，连接AD，AM，AE，P在AB上，作辅助线PD并延长，与AM延长线交于N，连接BN。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_224.jpg" }, { "index": 225, "ocr_en": "Triangle ABC, D is outside triangle ABC, E is on AB, G is on BC, F is on the AC extension line, connecting AD, DE, DF, DG.", "ocr_zh": "三角形ABC，D在三角形ABC外部，E在AB上，G在BC上，F在AC延长线上，连接AD,DE，DF，DG。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_225.jpg" }, { "index": 226, "ocr_en": "In triangle ABC, E is on AB, D is on BC, connecting AD and CE at point F.", "ocr_zh": "在三角形ABC中，E在AB上，D在BC上，连接AD，CE交于点F。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_226.jpg" }, { "index": 227, "ocr_en": "In triangle ABC, AC is perpendicular to BC, the vertical foot is C, E is on AB, D is on BC, connecting AD, and CE intersects at point F.", "ocr_zh": "在三角形ABC中，AC垂直于BC，垂足是C，E在AB上，D在BC上，连接AD，CE交于点F。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_227.jpg" }, { "index": 228, "ocr_en": "In the acute angle MON, OP is its angle bisector.", "ocr_zh": "锐角MON中，OP是其角平分线。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_228.jpg" }, { "index": 229, "ocr_en": "In triangle ABC, E is on AB, D is on BC, connecting AD and CE at point F. Angle EAD is equal to angle CAD, angle ECD is equal to angle ECA, G is on AC, serving as auxiliary line FG, angle AFE is equal to angle AFG, angle 1 is angle BAD, angle 2 is angle CAD, angle 3 is angle ECA, and angle 4 is angle BCE.", "ocr_zh": "在三角形ABC中，E在AB上，D在BC上，连接AD，CE交于点F，角EAD等于角CAD，角ECD等于角ECA，G在AC上，作辅助线FG，角AFE等于角AFG，角1是角BAD，角2是角CAD，角3是角ECA，角4是角BCE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_229.jpg" }, { "index": 230, "ocr_en": "In squares ABCD, E is on BC and F is on CD, connecting AE, AF, and EF.", "ocr_zh": "正方形ABCD中，E在BC上，F在CD上，连接AE，AF，EF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_230.jpg" }, { "index": 231, "ocr_en": "In triangle ABC, E is on AB, D is on BC, connecting AD, CE intersects at point F, G is on BE, H is on CD, forming auxiliary lines FG and FH. Angle EAD equals angle CAD, angle ECD equals angle ECA, angle 1 is angle BAD, angle 2 is angle CAD, angle 3 is angle ECA, and angle 4 is angle BCE.", "ocr_zh": "在三角形ABC中，E在AB上，D在BC上，连接AD，CE交于点F，G在BE上，H在CD上，作辅助线FG和FH，角EAD等于角CAD，角ECD等于角ECA，角1是角BAD，角2是角CAD，角3是角ECA，角4是角BCE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_231.jpg" }, { "index": 232, "ocr_en": "In the quadrilateral ABDC, M is on AB, N is on AC, connecting BC, DM, DN, MN, AD, F are the intersection points of the extension lines of AC and BD, and E is on CF, serving as the auxiliary line DE.", "ocr_zh": "四边形ABDC中，M在AB上，N在AC上，连接BC，DM，DN，MN，AD，F是AC和BD延长线的交点，E在CF上，作辅助线DE。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_232.jpg" }, { "index": 233, "ocr_en": "In the isosceles right triangle ABC, CA equals CB, AC is perpendicular to BC to C, D is on AB, E is on BC, F is on AC, connecting CD, DE, DF, extending DE to point M, and extending DF to point N.", "ocr_zh": "等腰直角三角形ABC中，CA等于CB，AC垂直于BC于C，D在AB上，E在BC上，F在AC上，连接CD，DE，DF，延长DE至点M，延长DF至点N。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_233.jpg" }, { "index": 234, "ocr_en": "Quadrilateral ABCD, AB perpendicular to AD, vertical foot A, CB perpendicular to CD, vertical foot C, F on the DC extension line, N on the BF extension line, M on the AD extension line, E on the DM, connecting BE and EF.", "ocr_zh": "四边形ABCD，AB垂直于AD，垂足是A，CB垂直于CD，垂足是C，F在DC延长线上，N在BF延长线上，M在AD延长线上，E在DM上，连接BE，EF。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_234.jpg" }, { "index": 235, "ocr_en": "Quadrilateral ABCD, AB perpendicular to AD, vertical foot A, CB perpendicular to CD, vertical foot C, F on CD, E on AD, connecting BE, BF, EF, N on BF extension line, M on BE extension line.", "ocr_zh": "四边形ABCD，AB垂直于AD，垂足是A，CB垂直于CD，垂足是C，F在CD上，E在AD上，连接BE，BF，EF，N在BF延长线上，M在BE延长线上。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_235.jpg" }, { "index": 236, "ocr_en": "There is a point P outside the squares ABCD, connecting AP, BP, and DP, and the three line segments AP, BP, and DP have no intersection. E is on PB and serves as an auxiliary line AE.", "ocr_zh": "正方形ABCD外有一点P，连接AP，BP，DP，且AP，BP，DP三条线段无交点，E在PB上，作辅助线AE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_236.jpg" }, { "index": 237, "ocr_en": "There is a point P outside the squares ABCD, connecting AP, BP, and DP, and the three line segments AP, BP, and DP have no intersection.", "ocr_zh": "正方形ABCD外有一点P，连接AP，BP，DP，且AP，BP，DP三条线段无交点。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_237.jpg" }, { "index": 238, "ocr_en": "Quadrilateral ABCD, AB is perpendicular to AD, the vertical foot is A, CB is perpendicular to CD, the vertical foot is C, F is on CD, E is on AD, connecting BE, BF, EF, N on the BF extension line, M on the BE extension line, K on the DC extension line, as auxiliary line BK.", "ocr_zh": "四边形ABCD，AB垂直于AD，垂足是A，CB垂直于CD，垂足是C，F在CD上，E在AD上，连接BE，BF，EF，N在BF延长线上，M在BE延长线上，K在DC延长线上，作辅助线BK。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_238.jpg" }, { "index": 239, "ocr_en": "There is a point P outside the squares ABCD, connecting AP, BP, and DP, and the three line segments AP, BP, and DP have no intersection. E is on PB and serves as the auxiliary line AE. P 'is outside the quadrilateral PBCD, and P' is close to PD and serves as the auxiliary lines P'P, P'A, P'B, and P'D.", "ocr_zh": "正方形ABCD外有一点P，连接AP，BP，DP，且AP，BP，DP三条线段无交点，E在PB上，作辅助线AE，P'在四边形PBCD外部，P'靠近PD，作辅助线P'P，P'A，P'B，P'D。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_239.jpg" }, { "index": 240, "ocr_en": "There is a point P outside the squares ABCD, connecting AP, BP, and DP, and the three line segments AP, BP, and DP have no intersection. E is on PB and serves as the auxiliary line AE. P 'is outside the quadrilateral PBCD, and P' is close to AP and serves as the auxiliary line PD, P'A, P'B, and P'P.", "ocr_zh": "正方形ABCD外有一点P，连接AP，BP，DP，且AP，BP，DP三条线段无交点，E在PB上，作辅助线AE，P'在四边形PBCD外部，P'靠近AP，作辅助线PD，P'A，P'B，P'P。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_240.jpg" }, { "index": 241, "ocr_en": "Square ABCD, P is its external point, connecting AP, BP, extending BP to point P ', as auxiliary lines P'A, PD.", "ocr_zh": "正方形ABCD，P是其外部一点，连接AP，BP，延长BP至点P'，作辅助线P'A，PD。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_241.jpg" }, { "index": 242, "ocr_en": "In quadrilateral MDCN, connect ND, A on CN, connect AM, B on AM, and connect BC, BD.", "ocr_zh": "在四边形MDCN中，连接ND，A在CN上，连接AM，B在AM上，连接BC，BD。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_242.jpg" }, { "index": 243, "ocr_en": "Left figure: In the quadrilateral ABDC, connect BC, AB=AC=BC, DB=DC, M on AB, N on AC, connect BC, DM, DN, MN, MN parallel to BC.\nRight figure: In the quadrilateral ABDC, connect BC, AB=AC=BC, DB=DC, M on AB, N on AC, connect BC, DM, DN, MN.", "ocr_zh": "左图：四边形ABDC中，连接BC，AB=AC=BC，DB=DC，M在AB上，N在AC上，连接BC，DM，DN，MN，MN平行于BC。\n右图：四边形ABDC中，连接BC，AB=AC=BC，DB=DC，M在AB上，N在AC上，连接BC，DM，DN，MN。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_243.jpg" }, { "index": 244, "ocr_en": "In the quadrilateral ABDC, connect BC, AB=AC=BC, DB=DC, N on the CA extension line, M on the AB extension line, connect ND, MD, H on AC, as auxiliary lines DH.", "ocr_zh": "四边形ABDC中，连接BC，AB=AC=BC，DB=DC，N在CA延长线上，M在AB延长线上，连接ND，MD，H在AC上，作辅助线DH。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_244.jpg" }, { "index": 245, "ocr_en": "In the quadrilateral ABDC, connect BC, AB=AC=BC, DB=DC, M on AB, N on AC, connect BC, DM, DN, MN, MN parallel to BC.", "ocr_zh": "四边形ABDC中，连接BC，AB=AC=BC，DB=DC，M在AB上，N在AC上，连接BC，DM，DN，MN，MN平行于BC。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_245.jpg" }, { "index": 246, "ocr_en": "In the quadrilateral ABDC, connect BC, AB=AC=BC, DB=DC, N on the CA extension line, M on the AB extension line, F on the AC extension line, connect ND, MD as auxiliary lines DF.", "ocr_zh": "四边形ABDC中，连接BC，AB=AC=BC，DB=DC，N在CA延长线上，M在AB延长线上，F在AC延长线上，连接ND，MD，作辅助线DF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_246.jpg" }, { "index": 247, "ocr_en": "Make a circle O with AB as the diameter, and intersect the chord PQ with AB.", "ocr_zh": "以AB为直径作圆O，弦PQ与AB相交。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_247.jpg" }, { "index": 248, "ocr_en": "There is a circle O in the figure, and AB is the chord of circle O, serving as auxiliary lines OA and OB.", "ocr_zh": "图中有一圆O，AB是圆O的弦，作辅助线OA，OB。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_248.jpg" }, { "index": 249, "ocr_en": "In the above figure, there is a circle O with a point A outside it. Starting from A, draw a straight line through O and intersect with circle O at points C and D. Starting from A, draw a straight line and intersect with circle O at points B and E, where points B and C are close to A and connect OE.\nBelow: There is a circle O with a point A outside it. The extension of AO intersects with circle O and B, and a straight line is drawn from A to intersect with circle O. The intersection points are D and C, where point D is close to A and connects BC.", "ocr_zh": "上图：有一圆O，圆外有一点A，从A出发作一条直线过O，与圆O交于C，D，从A出发做一条直线与圆O相交，交点是B,E，其中点B，C靠近A，连接OE。\n下图：有一圆O，圆外有一点A，AO的延长线交圆O于B，从A出发做一条直线与圆O相交，交点是D，C，其中点D靠近A，连接BC。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_249.jpg" }, { "index": 250, "ocr_en": "Make a circle O with AB as the diameter, and the chord DC intersects AB at point E.", "ocr_zh": "以AB为直径作圆O，弦DC与AB交于点E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_250.jpg" }, { "index": 251, "ocr_en": "Take AB as the diameter of circle O, C as the outer point of circle O, AC intersects with circle O at E, BC intersects with circle O at D, and connects BE.", "ocr_zh": "以AB为直径作圆O，C是圆O外一点，AC与圆O交于E，BC与圆O交于D，连接BE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_251.jpg" }, { "index": 252, "ocr_en": "There is a circle O on the diagram, where AB is the chord connecting OA, OB, OE perpendicular to AB, and the vertical foot is E.", "ocr_zh": "图上有一圆O，AB是圆O的弦，连接OA，OB，OE垂直于AB，垂足是E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_252.jpg" }, { "index": 253, "ocr_en": "Use AB as the diameter to form circle O, and C as auxiliary lines CA and CB on circle O.", "ocr_zh": "以AB为直径作圆O，C在圆O上，作辅助线CA，CB。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_253.jpg" }, { "index": 254, "ocr_en": "There is a circle O on the diagram, and AB is tangent to circle O at point C, serving as the auxiliary line OC.", "ocr_zh": "图上有一圆O，AB与圆O相切于C，作辅助线OC。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_254.jpg" }, { "index": 255, "ocr_en": "There is a circle O on the diagram, AB is the chord of circle O, CD is the chord of circle O, AB is perpendicular to CD, the vertical foot is E, connecting OE.", "ocr_zh": "图上有一圆O，AB是圆O的弦，CD是圆O的弦，AB垂直于CD，垂足是E，连接OE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_255.jpg" }, { "index": 256, "ocr_en": "Make a circle O with AB as the diameter, CD is the chord of circle O, and CD intersects with AO at E.", "ocr_zh": "以AB为直径作圆O，CD是圆O的弦，CD与AO交于E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_256.jpg" }, { "index": 257, "ocr_en": "There is a point O inside triangle ABC. Draw the circumcircle of triangle ABC with O as the center and the tangent DE of circle O passing through A.", "ocr_zh": "三角形ABC内有一点O，作以O为圆心作三角形ABC的外接圆，过A作圆O的切线DE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_257.jpg" }, { "index": 258, "ocr_en": "Take AB as the diameter of circle O, C is a point outside the circle, AC is tangent to circle O at point A, connected to OC, D is a point outside circle O, CD is tangent to circle O, connected to BD.", "ocr_zh": "以AB为直径作圆O，C是圆外一点，AC与圆O相切于点A，连接OC，D是圆O外一点，CD与圆O相切，连接BD。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_258.jpg" }, { "index": 259, "ocr_en": "There is a circle O on the diagram, where OA and OB are the radii of circle O, and OA is perpendicular to OB. The vertical foot is O, P is on OA, Q is the intersection point of BP extension line and circle O, and the tangent to circle O passing through Q intersects with OA extension line at R.", "ocr_zh": "图上有一圆O，OA、OB是圆O的半径，且OA垂直于OB，垂足是O，P在OA上，Q是BP延长线与圆O的交点，过Q作圆O的切线与OA的延长线交于R。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_259.jpg" }, { "index": 260, "ocr_en": "Triangle ABC, with AB as the diameter, forms circle O. Circle O intersects with BC at D, intersects with AC at E, forms DF perpendicular to AC, and its vertical foot is F.", "ocr_zh": "三角形ABC，以AB为直径作圆O，圆O与BC交于D，与AC交于E，作DF垂直于AC，垂足是F。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_260.jpg" }, { "index": 261, "ocr_en": "There is a circle on the diagram with O as the center and OB as the radius. A is a point outside the circle, connecting AB and extending it to intersect the circle at C. B is between A and C, passing through A to make a tangent to the circle O. The tangent point is D. M is a point on BC, connecting DM and extending it to intersect the circle O at E.", "ocr_zh": "图上有一个以O为圆心，OB为半径的圆，A是该圆外一点，连接AB并延长交该圆于C，B在A和C之间，过A作圆O的切线，切点是D，M是BC上一点，连接DM并延长交圆O于E。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_261.jpg" }, { "index": 262, "ocr_en": "In the coordinate system xOy, the diamond OABC is in the first quadrant, C is on the positive x-axis, the straight line l passing through the first quadrant is parallel to the y-axis, l intersects with OA at M, and l intersects with OC at N.", "ocr_zh": "坐标系xOy下，菱形OABC在第一象限，C在x轴正半轴，过第一象限的直线l与y轴平行，l与OA交于M，l与OC交于N。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_262.jpg" }, { "index": 263, "ocr_en": "In the right angled triangle ABC, AC is perpendicular to BC, and the perpendicular foot is C. Circle O intersects with triangle ABC and intersects points D, E, and F with AB, BC, and AC, respectively. Auxiliary lines OA, OD, OE, OF, P are on AF, H is on AB, PH is perpendicular to AB, and the perpendicular foot is H.", "ocr_zh": "直角三角形ABC中，AC垂直于BC，垂足是C，圆O与三角形ABC内切，分别与AB，BC，AC交于点D，E，F，作辅助线OA，OD，OE，OF，P在AF上，H在AB上，PH垂直于AB，垂足是H。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_263.jpg" }, { "index": 264, "ocr_en": "In the coordinate system xOy, the center A of circle A is in the second quadrant. Circle A intersects with the negative x-axis at points D, C, and D to the left of C. Circle A is tangent to the positive y-axis at point B. A parabolic curve with an upward opening passes through points B, C, D, and E, which are the lowest points of the parabolic curve and connect EC.", "ocr_zh": "坐标系xOy下，圆A的圆心A在第二象限，圆A与x轴负半轴交于点D，C，D在C左，圆A与y轴正半轴相切于点B，作开口向上的抛物线过B，C，D，E是抛物线的最低点，连接EC。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_264.jpg" }, { "index": 265, "ocr_en": "Triangle ABC, E is on the extension line of BC, D is outside triangle ABC, D is close to AC, connected to DB, DC.", "ocr_zh": "三角形ABC，E在BC延长线上，D在三角形ABC外，D靠近AC，连接DB，DC。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_265.jpg" }, { "index": 266, "ocr_en": "There is a point P inside triangle ABC, connecting PB and PC.", "ocr_zh": "三角形ABC内有一点P，连接PB，PC。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_266.jpg" }, { "index": 267, "ocr_en": "Quadrilateral ABPC, connecting BC, F on AB extension line, E on AC extension line.", "ocr_zh": "四边形ABPC，连接BC，F在AB延长线上，E在AC延长线上。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_267.jpg" }, { "index": 268, "ocr_en": "Triangle ABC, E is on the extension line of BC, P is outside triangle ABC, P is close to AC, connecting PB, PC.", "ocr_zh": "三角形ABC，E在BC延长线上，P在三角形ABC外，P靠近AC，连接PB，PC。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_268.jpg" }, { "index": 269, "ocr_en": "Triangles ABC and D are on the extension line of BC, while A1, A2, and A3 are all outside triangle ABC and close to AC, connecting A1B, A1C, A2B, A2C, A3B, and A3C.", "ocr_zh": "三角形ABC，D在BC延长线上，A1，A2，A3都在三角形ABC外且都靠近AC，连接A1B，A1C，A2B，A2C，A3B，A3C。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_269.jpg" }, { "index": 270, "ocr_en": "There is a point O inside triangle ABC, connecting OB and OC.", "ocr_zh": "三角形ABC内有一点O，连接OB，OC。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_270.jpg" }, { "index": 271, "ocr_en": "In triangle ABC, BA is equal to BC, M is on AC, AM is equal to CM, P is on BM, serve as auxiliary lines PC, D is on the extension of BM, connect CD, and use P as the center and PA as the radius to form an auxiliary circle intersecting CD at Q, connecting PQ and PA.", "ocr_zh": "三角形ABC中，BA等于BC，M在AC上，AM等于CM，P在BM上，作辅助线PC，D在BM延长线上，连接CD，以P为圆心、PA为半径作辅助圆与CD交于Q，连接PQ，PA。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_271.jpg" }, { "index": 272, "ocr_en": "In triangle ABC, BA is equal to BC, M is on AC, P and M coincide, AM is equal to CM, Q is outside triangle ABC, Q is close to AC, and connects MQ.", "ocr_zh": "三角形ABC中，BA等于BC，M在AC上，P和M重合，AM等于CM，Q在三角形ABC外，Q靠近AC，连接MQ。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_272.jpg" }, { "index": 273, "ocr_en": "In triangle ABC, BA is equal to BC, M is on AC, AM is equal to CM, P is on BM, Q is outside triangle ABC, Q is close to AC, and connects PQ.", "ocr_zh": "三角形ABC中，BA等于BC，M在AC上，AM等于CM，P在BM上，Q在三角形ABC外，Q靠近AC，连接PQ。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_273.jpg" }, { "index": 274, "ocr_en": "In the quadrilateral ABCD, P is on BC, connected to AC, BD, intersecting with O, M is on AO, N is on DO, connected to MN, MP, NP.", "ocr_zh": "四边形ABCD中，P在BC上，连接AC，BD，交于O，M在AO上，N在DO上，连接MN，MP，NP。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_274.jpg" }, { "index": 275, "ocr_en": "In the quadrilateral ABCD, AB is parallel to CD, P is on BC, connecting AC, BD, intersecting O, M is on AO, N is on DO, connecting MN, MP, NP.", "ocr_zh": "四边形ABCD中，AB平行于CD，P在BC上，连接AC，BD，交于O，M在AO上，N在DO上，连接MN，MP，NP。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_275.jpg" }, { "index": 276, "ocr_en": "In the coordinate system AOB, M is in the first quadrant, connected to OM, P is on OM, passing through P makes PH perpendicular to OA, vertical foot is H, passing through P makes PN perpendicular to OB, vertical foot is N, C is on OH, R is on the PC extension line, D is on OB and to the right of N, S is on the PD extension line, CP is perpendicular to DP, vertical foot is P, connected to DC, DC intersects with OP at G, angle 1 is angle HPC, angle 2 is angle NPD, and angle 3 is angle PDC.", "ocr_zh": "坐标系AOB下，M在第一象限，连接OM，P在OM上，过P作PH垂直于OA，垂足是H，过P作PN垂直于OB，垂足是N，C在OH上，R在PC延长线上，D在OB上且在N右边，S在PD延长线上，CP垂直于DP，垂足是P，连接DC，DC与OP交于G，角1是角HPC，角2是角NPD，角3是角PDC。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_276.jpg" }, { "index": 277, "ocr_en": "In the coordinate system AOB, M is in the first quadrant, connecting OM, P is on OM, crossing P makes PH perpendicular to OA, vertical foot is H, crossing P makes PN perpendicular to OB, vertical foot is N, C is on OH, R is on the PC extension line, D is on OB and to the right of N, S is on the PD extension line, CP is perpendicular to DP, vertical foot is P, connecting DC, DC and OP intersecting at G.", "ocr_zh": "坐标系AOB下，M在第一象限，连接OM，P在OM上，过P作PH垂直于OA，垂足是H，过P作PN垂直于OB，垂足是N，C在OH上，R在PC延长线上，D在OB上且在N右边，S在PD延长线上，CP垂直于DP，垂足是P，连接DC，DC与OP交于G。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_277.jpg" }, { "index": 278, "ocr_en": "In the quadrilateral ABCD, P is on BC, connecting AC and BD, intersecting O, M is on AO, N is on DO, connecting MN, MP, NP, and serving as auxiliary lines BM and NC.", "ocr_zh": "四边形ABCD中，P在BC上，连接AC，BD，交于O，M在AO上，N在DO上，连接MN，MP，NP，作辅助线BM，NC。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_278.jpg" }, { "index": 279, "ocr_en": "In rectangles ABCD, E and B coincide, C and F coincide, P is on AD, connecting BP, CP, BP is perpendicular to CP, and the vertical foot is P.", "ocr_zh": "长方形ABCD中，E和B重合，C和F重合，P在AD上，连接BP，CP，BP垂直于CP，垂足是P。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_279.jpg" }, { "index": 280, "ocr_en": "In rectangles ABCD, P is on AD, E is on AB, F is on BC, connecting EP, FP, EF, EP is perpendicular to FP, and the vertical foot is P.", "ocr_zh": "长方形ABCD中，P在AD上，E在AB上，F在BC上，连接EP，FP，EF，EP垂直于FP，垂足是P。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_280.jpg" }, { "index": 281, "ocr_en": "Square ABCD, M on BC, E is a point outside of square ABCD, E is close to CD, connects AM, EM, EC, and serves as auxiliary lines AC, AE.", "ocr_zh": "正方形ABCD，M在BC上，E是正方形ABCD外一点，E靠近CD，连接AM，EM，EC，作辅助线AC，AE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_281.jpg" }, { "index": 282, "ocr_en": "Square ABCD, M on BC, E is a point outside of square ABCD, E is close to CD, connecting AM, EM, EC.", "ocr_zh": "正方形ABCD，M在BC上，E是正方形ABCD外一点，E靠近CD，连接AM，EM，EC。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_282.jpg" }, { "index": 283, "ocr_en": "Left image: Square ABCD, M on BC, E is a point outside of square ABCD, E is close to CD, connecting AM, EM, EC.\nRight image: Square ABCD, M on BC, E is the outer point of square ABCD, E is close to CD, connecting AM, EM, EC as auxiliary lines AC, AE.", "ocr_zh": "左图：正方形ABCD，M在BC上，E是正方形ABCD外一点，E靠近CD，连接AM，EM，EC。\n右图：正方形ABCD，M在BC上，E是正方形ABCD外一点，E靠近CD，连接AM，EM，EC，作辅助线AC，AE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_283.jpg" }, { "index": 284, "ocr_en": "In rectangles ABCD, P is on AD, E is on AB, F is on BC, connecting EP, FP, EF, EP is perpendicular to FP, P is perpendicular to the vertical foot, FG is perpendicular to AD as an auxiliary line, and G is perpendicular to the vertical foot.", "ocr_zh": "长方形ABCD中，P在AD上，E在AB上，F在BC上，连接EP，FP，EF，EP垂直于FP，垂足是P，作辅助线FG垂直于AD，垂足是G。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_284.jpg" }, { "index": 285, "ocr_en": "In squares ABCD, E is on BC, M is on the AE extension line, F is on CD, N is on the AF extension line, connecting AE, AF, EF, BD, BD intersects AE at P, BD intersects AF at Q, and EQ is connected.", "ocr_zh": "正方形ABCD中，E在BC上，M在AE延长线上，F在CD上，N在AF延长线上，连接AE，AF，EF，BD，BD与AE交于P，BD与AF交于Q，连接EQ。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_285.jpg" }, { "index": 286, "ocr_en": "In the quadrilateral ABCD, connect AC.", "ocr_zh": "四边形ABCD中，连接AC。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_286.jpg" }, { "index": 287, "ocr_en": "In the quadrilateral PBCQ, A is an internal point that connects AB, AC, AP, AQ. AC is perpendicular to BC, and the vertical foot is C. A circle with the midpoint O1 of AB as the center, AB as the diameter, intersects BP with D, AC as the midpoint O2 as the center, AC as the diameter, intersects CQ with E, and BC as the midpoint F.", "ocr_zh": "四边形PBCQ中，A是内部一点，连接AB，AC，AP，AQ，AC垂直于BC，垂足是C，以AB中点O1为圆心，AB为直径作圆交BP于D，以AC中点O2为圆心，AC为直径作圆交CQ于E，BC中点是F。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_287.jpg" }, { "index": 288, "ocr_en": "In the quadrilateral PBCQ, A is an internal point, connecting AB, AC, AP, AQ, and F is the midpoint of BC, connecting FA and extending it so that FA is perpendicular to PQ. The vertical foot can be denoted as G. Taking the midpoint O1 of AB as the center and AB as the diameter, a circle intersection BP is made with D, and taking the midpoint O2 of AC as the center and AC as the diameter, a circle intersection CQ is made with E.", "ocr_zh": "四边形PBCQ中，A是内部一点，连接AB，AC，AP，AQ，F是BC中点，连接FA并延长，使FA垂直于PQ，垂足可记为G，以AB中点O1为圆心，AB为直径作圆交BP于D，以AC中点O2为圆心，AC为直径作圆交CQ于E。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_288.jpg" }, { "index": 289, "ocr_en": "In the quadrilateral PBCQ, F is the midpoint of BC, A is an internal point connecting AB, AC, AP, AQ. AC is perpendicular to BC, and the vertical foot is C. Taking the midpoint O1 of AB as the center, AB is the diameter of the circle intersecting BP and D, and taking the midpoint O2 of AC as the center, AC is the diameter of the circle intersecting CQ and E. Extending CA to G outside the quadrilateral PBCQ serves as auxiliary lines AE and BG.", "ocr_zh": "四边形PBCQ中，F是BC中点，A是内部一点，连接AB，AC，AP，AQ，AC垂直于BC，垂足是C，以AB中点O1为圆心，AB为直径作圆交BP于D，以AC中点O2为圆心，AC为直径作圆交CQ于E，延长CA至四边形PBCQ外的G，作辅助线AE，BG。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_289.jpg" }, { "index": 290, "ocr_en": "In the quadrilateral PBCQ, A is an internal point that connects AB, AC, AP, AQ, and F on BC, connects FA and extends it. When FA is perpendicular to PQ, the vertical foot is M. Taking the midpoint O1 of AB as the center, AB as the diameter makes a circular intersection BP to D, AC as the midpoint O2, AC as the diameter makes a circular intersection CQ to E, auxiliary line CS is perpendicular to AF, the vertical foot is S, auxiliary line BR is perpendicular to AF extension line, the vertical foot is R, auxiliary lines DR, DA, DM, angle 1 is angle MQA, angle 2 is angle MAQ, angle 3 is angle FAC, angle 4 is angle ACS, angle 5 is angle BD. R, angle 6 is angle PAM, angle 7 is angle PDM, angle 8 is angle BAR, and angle 9 is angle MDA.", "ocr_zh": "四边形PBCQ中，A是内部一点，连接AB，AC，AP，AQ，F在BC上，连接FA并延长，时FA垂直于PQ，垂足为M，以AB中点O1为圆心，AB为直径作圆交BP于D，以AC中点O2为圆心，AC为直径作圆交CQ于E，作辅助线CS垂直于AF，垂足是S，作辅助线BR垂直于AF延长线，垂足是R，作辅助线DR，DA，DM，角1是角MQA，角2是角MAQ，角3是角FAC，角4是角ACS，角5是角BDR，角6是角PAM，角7是角PDM，角8是角BAR，角9是角MDA。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_290.jpg" }, { "index": 291, "ocr_en": "In the acute triangle ABC, F is the midpoint of BC, with the midpoint O1 of AB as the center and AB as the diameter, forming a semicircle outside the triangle ABC. The midpoint of this arc is D, with the midpoint O2 of AC as the center and AC as the diameter, forming a semicircle outside the triangle ABC. The midpoint of this arc is E, connecting O1D, O1F, DF, O2E, O2F, EF.", "ocr_zh": "锐角三角形ABC中，F是BC中点，以AB中点O1为圆心，AB为直径向三角形ABC外作半圆，该圆弧中点为D，以AC中点O2为圆心，AC为直径向三角形ABC外作半圆，该圆弧中点为E，连接O1D，O1F，DF，O2E，O2F，EF。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_291.jpg" }, { "index": 292, "ocr_en": "There is a circle O on the diagram, where AB is the chord of circle O. There are P and C outside the circle, with P and C on one side of AB and O on the other side of AB. PABC forms a quadrilateral, where AP is perpendicular to CP and P is the perpendicular foot, connecting PB and AC.", "ocr_zh": "图上有一圆O，AB是圆O的弦，圆外有P和C，P与C在AB一侧，O在AB另一侧，PABC构成四边形，其中AP垂直于CP，垂足是P，连接PB，AC。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_292.jpg" }, { "index": 293, "ocr_en": "In the pentagonal ABCDE, connect AC and AD as auxiliary lines BD and CE.", "ocr_zh": "五边形ABCDE中，连接AC，AD，作辅助线BD，CE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_293.jpg" }, { "index": 294, "ocr_en": "In the pentagon ABCDE, connect AC and AD.", "ocr_zh": "五边形ABCDE中，连接AC，AD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_294.jpg" }, { "index": 295, "ocr_en": "Left image: In squares ABCD, F is on AD, E is on DC, connected to CF, BE intersects at point H, and connected to AH.\nRight figure: In squares ABCD, F is on AD, E is on DC, connected to CF, BE intersects at point H, connected to AH, as auxiliary line BF", "ocr_zh": "左图：正方形ABCD中，F在AD上，E在DC上，连接CF，BE交于点H，连接AH。\n右图：正方形ABCD中，F在AD上，E在DC上，连接CF，BE交于点H，连接AH，作辅助线BF。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_295.jpg" }, { "index": 296, "ocr_en": "In squares ABCD, F is on AD, E is on DC, connected to CF, BE intersects at point H, and connected to AH.", "ocr_zh": "正方形ABCD中，F在AD上，E在DC上，连接CF，BE交于点H，连接AH。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_296.jpg" }, { "index": 297, "ocr_en": "In squares ABCD, F is on AD, E is on DC, connecting CF, BE intersects at point H, connecting AH, as auxiliary line BF.", "ocr_zh": "正方形ABCD中，F在AD上，E在DC上，连接CF，BE交于点H，连接AH，作辅助线BF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_297.jpg" }, { "index": 298, "ocr_en": "In the coordinate system xOy, C coordinate (3,0), B coordinate (0,5), A coordinate (-3,0), connect AB, BC.", "ocr_zh": "在坐标系xOy中，C坐标(3,0)，B坐标(0,5)，A坐标(-3,0)，连接AB，BC。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_298.jpg" }, { "index": 299, "ocr_en": "In the coordinate system xOy, C coordinate (3,0), B coordinate (0,5), A coordinate (-3,0), E coordinate (0, -5), connect AB, BC, P in the fourth quadrant, EP parallel to AC, connect AP, CE.", "ocr_zh": "在坐标系xOy中，C坐标(3,0)，B坐标(0,5)，A坐标(-3,0)，E坐标(0,-5)，连接AB，BC，P在第四象限，EP平行于AC，连接AP，CE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_299.jpg" }, { "index": 300, "ocr_en": "Left image: In squares ABCD, O is its center, P is a point inside it, and P is close to A, connecting OP, PA, and PB.\nRight figure: In squares ABCD, O is its center, P is a point inside it, P is close to A, connecting OP, PA, PB as auxiliary lines OA, OB.", "ocr_zh": "左图：正方形ABCD中，O是其中心，P是其内部一点，P靠近A，连接OP，PA，PB。\n右图：正方形ABCD中，O是其中心，P是其内部一点，P靠近A，连接OP，PA，PB，作辅助线OA，OB。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_300.jpg" }, { "index": 301, "ocr_en": "Left image: In squares ABCD, connect AC, P is a point on AC, E is a point on CD, connect PD, PE.\nMiddle figure: In squares ABCD, connect AC, P is a point on AC, E is a point on CD, connect PD, PE, as auxiliary line PB.\nRight figure: In squares ABCD, connect AC, P is a point on AC, E is a point on CD, connect PD, PE, as auxiliary line PB.", "ocr_zh": "左图：正方形ABCD中，连接AC，P是AC上一点，E是CD上一点，连接PD，PE。\n中图：正方形ABCD中，连接AC，P是AC上一点，E是CD上一点，连接PD，PE，作辅助线PB。\n右图：正方形ABCD中，连接AC，P是AC上一点，E是CD上一点，连接PD，PE，作辅助线PB。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_301.jpg" }, { "index": 302, "ocr_en": "Left figure: In triangle ABC, AC is perpendicular to BC, with the vertical foot C, P on AB, D on AC, and E on BC. Connect DP, DE, and PE to make PDCE a rectangle.\nMiddle figure: In triangle ABC, AC is perpendicular to BC, the vertical foot is C, P is on AB, D is on AC, E is on BC, connect DP, DE, PE to make PDCE a rectangle, and use it as an auxiliary line PC.\nRight figure: In triangle ABC, AC is perpendicular to BC, the vertical foot is C, P is on AB, P is close to point B, D is on AC, E is on BC, connect DP, DE, PE to make PDCE rectangular, and serve as auxiliary line PC.", "ocr_zh": "左图：三角形ABC中，AC垂直于BC，垂足是C，P在AB上，D在AC上，E在BC上，连接DP，DE，PE，使PDCE是矩形。\n中图：三角形ABC中，AC垂直于BC，垂足是C，P在AB上，D在AC上，E在BC上，连接DP，DE，PE，使PDCE是矩形，作辅助线PC。\n右图：三角形ABC中，AC垂直于BC，垂足是C，P在AB上，P靠近B带点，D在AC上，E在BC上，连接DP，DE，PE，使PDCE是矩形，作辅助线PC。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_302.jpg" }, { "index": 303, "ocr_en": "Left image: In the squares ABCD, connect AC, where P is a point on AC, connect DP, and E within the triangle ADC, connect AE, BE, and PE.\nRight figure: In squares ABCD, connect AC, P is a point on AC, connect DP, E inside triangle ADC, connect AE, BE, PE, as auxiliary line PB.", "ocr_zh": "左图：正方形ABCD中，连接AC，P是AC上一点，连接DP，E在三角形ADC内，连接AE，BE，PE。\n右图：正方形ABCD中，连接AC，P是AC上一点，连接DP，E在三角形ADC内，连接AE，BE，PE，作辅助线PB。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_303.jpg" }, { "index": 304, "ocr_en": "Left image: In diamond ABCD, A 'is an internal point, M is the midpoint of AD, N is on AB, connecting MN, A'M, A'N, A'C.\nRight figure: In the diamond ABCD, M is the midpoint of AD, with M as the center and AD as the diameter, a semicircle is formed inside the diamond. A 'is on this semicircle, N is on AB, connecting MN, A'M, A'N, A'C.", "ocr_zh": "左图：菱形ABCD中，A'是其内部一点，M是AD中点，N在AB上，连接MN，A'M，A'N，A'C。\n右图：菱形ABCD中，M是AD中点，以M为圆心，AD为直径在菱形内部作半圆，A'在该半圆上，N在AB上，连接MN，A'M，A'N，A'C。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_304.jpg" }, { "index": 305, "ocr_en": "Left image: In diamond ABCD, P is an internal point.\nMiddle figure: In diamond ABCD, DE is perpendicular to AB, the vertical foot is E, P is on DE, PF is perpendicular to AD, the vertical foot is F, connected to PA as auxiliary line DB.\nRight figure: In diamond ABCD, DE is perpendicular to AB, the vertical foot is E, P is on DE, PF is perpendicular to AD, the vertical foot is F, connecting PA and PB as auxiliary lines DB.", "ocr_zh": "左图：在菱形ABCD中，P是内部一点。\n中图：在菱形ABCD中，DE垂直于AB，垂足是E，P在DE上，PF垂直于AD，垂足是F，连接PA，作辅助线DB。\n右图：在菱形ABCD中，DE垂直于AB，垂足是E，P在DE上，PF垂直于AD，垂足是F，连接PA，PB，作辅助线DB。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_305.jpg" }, { "index": 306, "ocr_en": "Left image: In the coordinate system MON, A is on OM, B is on ON, C and D are in the first quadrant, connecting AB, BC, CD, DA so that the quadrilaterals ABCD are rectangles.\nRight figure: In the coordinate system MON, A is on OM, B is on ON, C and D are in the first quadrant, connecting AB, BC, CD, DA so that quadrilaterals ABCD are rectangles and E is the midpoint of AB, connecting OD, OE, DE.", "ocr_zh": "左图：在坐标系MON中，A在OM上，B在ON上，C和D在第一象限，连接AB，BC，CD，DA，使四边形ABCD是矩形。\n右图：在坐标系MON中，A在OM上，B在ON上，C和D在第一象限，连接AB，BC，CD，DA，使四边形ABCD是矩形，E是AB中点，连接OD，OE，DE。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_306.jpg" }, { "index": 307, "ocr_en": "In the coordinate system MON, A is on OM, B is on ON, C and D are in the first quadrant, connecting AB, BC, CD, DA so that the quadrilaterals ABCD are rectangles.", "ocr_zh": "在坐标系MON中，A在OM上，B在ON上，C和D在第一象限，连接AB，BC，CD，DA，使四边形ABCD是矩形。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_307.jpg" }, { "index": 308, "ocr_en": "In the coordinate system MON, A is on OM, B is on ON, C and D are in the first quadrant, connecting AB, BC, CD, DA so that quadrilaterals ABCD are rectangles and E is the midpoint of AB, connecting OD, OE, DE.", "ocr_zh": "在坐标系MON中，A在OM上，B在ON上，C和D在第一象限，连接AB，BC，CD，DA，使四边形ABCD是矩形，E是AB中点，连接OD，OE，DE。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_308.jpg" }, { "index": 309, "ocr_en": "Left image: There is a point B outside the square ACDE, which is close to the AC side and connects BA, BC, and BE.\nRight figure: There is a point B outside the square ACDE, which is close to the AC side and connects BA, BC, BE, and B 'outside the triangle BAC. It serves as an auxiliary line B'A, B'B, and B'C.", "ocr_zh": "左图：正方形ACDE外有一点B，B靠近AC边，连接BA，BC，BE。\n右图：正方形ACDE外有一点B，B靠近AC边，连接BA，BC，BE，B'在三角形BAC外，作辅助线B'A，B'B，B'C。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_309.jpg" }, { "index": 310, "ocr_en": "Left image: There is a point D 'inside triangle ABC, D is on AC, connecting D'A, D'B, F is on D'B, and connecting CF.\nRight figure: There is a point D 'inside triangle ABC, D is on AC, connecting D'A, D'B, F on D'B, connecting CF, M on AB, as auxiliary lines MC, MF.", "ocr_zh": "左图：三角形ABC内有一点D'，D在AC上，连接D'A，D'B，F在D'B上，连接CF。\n右图：三角形ABC内有一点D'，D在AC上，连接D'A，D'B，F在D'B上，连接CF，M在AB上，作辅助线MC，MF。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_310.jpg" }, { "index": 311, "ocr_en": "Left figure: In the coordinate system xOy, A is on the x-axis positive half axis, B is on the y-axis positive half axis, C is on OB, D is on OA, connecting AB, with CD as the diameter, making a semicircle on the right side of CD, assuming the center of the circle is P. The semicircle P intersects with AB at points E and F, where E is closer to B than F.\nRight figure: In the coordinate system xOy, A is on the positive half axis of the x-axis, B is on the positive half axis of the y-axis, C is on OB, D is on OA, connecting AB, with CD as the diameter, making a semicircle on the right side of CD, assuming the center of the circle is P. The semicircle P intersects with AB at points E, F, where E is closer to B than F, making auxiliary line OP, making auxiliary line PQ perpendicular to EF, and the vertical foot is Q.", "ocr_zh": "左图：坐标系xOy中，A在x轴正半轴，B在y轴正半轴，C在OB上，D在OA上，连接AB，以CD为直径，在CD右边作半圆，设圆心为P，半圆P与AB交于点E，F，其中E比F更加靠近B。\n右图：坐标系xOy中，A在x轴正半轴，B在y轴正半轴，C在OB上，D在OA上，连接AB，以CD为直径，在CD右边作半圆，设圆心为P，半圆P与AB交于点E，F，其中E比F更加靠近B，作辅助线OP，作辅助线PQ垂直于EF，垂足是Q。", "class": "Analytic Geometry", "difficulty": 3, "image_path": "fig_311.jpg" }, { "index": 312, "ocr_en": "Left image: Within squares ABCD, G is on AD, E is on CD, F is on AB, connected to DF, GE intersects with P, and connected to AP.\nRight figure: Within squares ABCD, G is on AD, E is on CD, F is on AB, connecting DF, GE intersects with P, making GE perpendicular to DF and perpendicular to P. Connect AP, with the midpoint of DE as the center, to form a semicircle inside ABCD, with P on the semicircle.", "ocr_zh": "左图：正方形ABCD内，G在AD上，E在CD上，F在AB上，连接DF，GE交于P，连接AP。\n右图：正方形ABCD内，G在AD上，E在CD上，F在AB上，连接DF，GE交于P，使得GE垂直于DF，垂足是P，连接AP，以DE中点为圆心，在ABCD内部作半圆，P在半圆上。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_312.jpg" }, { "index": 313, "ocr_en": "In the coordinate system xOy, a parabola with an opening facing downwards passes through point O and point B on the positive half axis of the x-axis. The vertex of the parabola is A, and passing through A forms a straight line parallel to the y-axis. P in the first quadrant is directly below A, connected to OP as auxiliary line OA, and D is a point on OA as auxiliary line BD.", "ocr_zh": "在坐标系xOy中，开口向下的抛物线过O点和x轴正半轴的B点，抛物线顶点是A，过A作平行于y轴的直线，第一象限内的P在A正下方，连接OP，作辅助线OA，D是OA上一点，作辅助线BD。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_313.jpg" }, { "index": 314, "ocr_en": "In rectangles ABCD, P is on BC, E is on AD, F is on CD, connecting AP, EF, G is on EF, and PG is connected.", "ocr_zh": "矩形ABCD中，P在BC上，E在AD上，F在CD上，连接AP，EF，G在EF上，连接PG。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_314.jpg" }, { "index": 315, "ocr_en": "In the coordinate system xOy, a parabola with an opening facing downwards passes through point O and point B on the positive half axis of the x-axis. The vertex of the parabola is A, and a straight line parallel to the y-axis is drawn through point A. P in the first quadrant is directly below point A, connecting point OP.", "ocr_zh": "在坐标系xOy中，开口向下的抛物线过O点和x轴正半轴的B点，抛物线顶点是A，过A作平行于y轴的直线，第一象限内的P在A正下方，连接OP。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_315.jpg" }, { "index": 316, "ocr_en": "In rectangles ABCD, P is on BC, E is on AD, F is on CD, connecting AP, EF, G is on EF, connecting PG, with D as the center and DG as the radius. Use a dashed line to form a quarter circle inside the rectangle, with A 'on the extension of AB, AB equal to A'B, as the auxiliary line A'P.", "ocr_zh": "矩形ABCD中，P在BC上，E在AD上，F在CD上，连接AP，EF，G在EF上，连接PG，以D为圆心，DG为半径在矩形内部用虚线作四分之一圆，A'在AB延长线上，AB等于A'B，作辅助线A'P。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_316.jpg" }, { "index": 317, "ocr_en": "There is a circle O on the diagram, where AB is the chord of circle O, OM is perpendicular to AB, the vertical foot is M, CD is the other chord of circle O, CD intersects with AB, and the intersection point is close to B. N is the midpoint of CD, connecting MN.", "ocr_zh": "图上有一圆O，AB是圆O的弦，OM垂直于AB，垂足是M，CD是圆O的另一个弦，CD与AB相交，交点靠近B，N是CD中点，连接MN。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_317.jpg" }, { "index": 318, "ocr_en": "On the diagram, there is a circle O with A and B, connected by AB and C. C is a point inside circle O, CA is perpendicular to AB, and the vertical foot is A. It connects BC and OC.", "ocr_zh": "图上有一圆O，圆O上有A和B，连接AB，C是圆O内一点，CA垂直于AB，垂足是A，连接BC，OC。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_318.jpg" }, { "index": 319, "ocr_en": "In the coordinate system xOy, A is on the y-axis positive half axis, C is on the x-axis positive half axis, B is in the first quadrant, connect AB, BC, make the quadrilateral ABCO rectangular, D is on AB, connect CD, and make a square CDEF in the first quadrant.", "ocr_zh": "在坐标系xOy中，A在y轴正半轴，C在x轴正半轴，B在第一象限，连接AB，BC，使四边形ABCO是矩形，D在AB上，连接CD，在第一象限内作正方形CDEF。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_319.jpg" }, { "index": 320, "ocr_en": "There is a circle with O as the center and P1P2 as the diameter on the diagram. There is a point P on the circle, and a point A on the extension line of P2P1, connecting AP.", "ocr_zh": "图上有一以O为圆心，P1P2为直径的圆，圆上有一点P，P2P1延长线上有一点A，连接AP。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_320.jpg" }, { "index": 321, "ocr_en": "In the coordinate system xOy, B is on the positive y-axis, C is on the positive x-axis, A is in the first quadrant, connecting AB, AC, BC, M on BC, P on OC, and PM.", "ocr_zh": "在坐标系xOy中，B在y轴正半轴，C在x轴正半轴，A在第一象限，连接AB，AC，BC，M在BC上，P在OC上，连接PM。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_321.jpg" }, { "index": 322, "ocr_en": "Point P is a moving point on line l, A is a point outside the line, connecting AP, passing through A makes AP 'perpendicular to line l, and the vertical foot is P'", "ocr_zh": "点P是直线l上一动点，A是直线外一点，连接AP，过A作AP’垂直于直线l，垂足是P'", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_322.jpg" }, { "index": 323, "ocr_en": "In squares ABCD, E is on BC, BF is perpendicular to AE, and the vertical foot is G. BF intersects with CD at F, connecting CG.", "ocr_zh": "正方形ABCD中，E在BC上，作BF垂直于AE，垂足是G，BF与CD交于F，连接CG。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_323.jpg" }, { "index": 324, "ocr_en": "In the coordinate system xOy, A is on the positive half axis of the x-axis, B is on OA, with A as the center and AB as the radius, making a circle intersecting the positive half axis of the x-axis at C, C is on the right side of A, D is on the circle with A as the center and AB as the radius, and D is in the fourth quadrant, connecting AD, and making a tangent to the circle with A as the center and AB as the radius through D. This tangent intersects the negative half axis of the y-axis at P.", "ocr_zh": "在坐标系xOy中，A在x轴正半轴，B在OA上，以A为圆心，AB为半径作圆与x轴正半轴交于C，C在A正右边，D在以A为圆心，AB为半径的圆上且D在第四象限，连接AD，过D作以A为圆心，AB为半径的圆的切线，该切线与y轴负半轴交于P。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_324.jpg" }, { "index": 325, "ocr_en": "In the coordinate system xOy, A is on the positive x-axis, B is on the positive y-axis, and C is in the first quadrant, connecting OC, AB, CB, and CA.", "ocr_zh": "坐标系xOy中，A在x轴正半轴，B在y轴正半轴，C在第一象限，连接OC，AB，CB，CA。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_325.jpg" }, { "index": 326, "ocr_en": "In triangle ABC, AC is perpendicular to AB, the vertical foot is C, and D is on AB. Draw triangle ADE outward from triangle ABC, making AE=AE=AD, making DF perpendicular to DE, perpendicular to F, connecting EF, G is the midpoint of EF, connecting CG.", "ocr_zh": "三角形ABC中，AC垂直于AB，垂足是C，D在AB上，向三角形ABC外作三角形ADE，使AE=AE=AD，作DF垂直于DE，垂直是F，连接EF，G是EF中点，连接CG。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_326.jpg" }, { "index": 327, "ocr_en": "In the coordinate system xOy, A is on the positive x-axis, B and C are in the first quadrant, connecting AC, CB, BO, OC, and AB.", "ocr_zh": "坐标系xOy中，A在x轴正半轴，B，C在第一象限，连接AC，CB，BO，OC，AB。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_327.jpg" }, { "index": 328, "ocr_en": "In the right angled triangle ABC, AC is perpendicular to BC, with a vertical foot of C. D is a moving point on BC, connecting AD, and a circle with CD as the center and CD as the diameter intersects AD at point E, connecting BE.", "ocr_zh": "在直角三角形ABC中，AC垂直于BC，垂足是C，D是BC上一动点，连接AD，以CD中点为圆心，CD为直径的圆与AD交于点E，连接BE。（缺失的字母跑到22、单线段最值问题整合_page_002_figure_6.png中去了）", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_328.jpg" }, { "index": 329, "ocr_en": "In the acute triangle ABC, D is on BC, connected to AD, E is on AD, connected to CE, and a triangle CDF is formed with CD as the side facing the outside of triangle ABC.", "ocr_zh": "锐角三角形ABC中，D在BC上，连接AD，E在AD上，连接CE，以CD为边向三角形ABC外部作三角形CDF。（多了上图22、单线段最值问题整合_page_002_figure_5.png的字母噪音）", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_329.jpg" }, { "index": 330, "ocr_en": "In the right angled triangle ABC, AC is perpendicular to BC, the vertical foot is C, D is on AB, DE is perpendicular to AC, the vertical foot is E, DF is perpendicular to BC, the vertical foot is F, and EF is connected.", "ocr_zh": "在直角三角形ABC中，AC垂直于BC，垂足是C，D在AB上，作DE垂直于AC，垂足是E，作DF垂直于BC，垂足是F，连接EF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_330.jpg" }, { "index": 331, "ocr_en": "In triangle ABC, D is on AC, E is on AB, A 'is on BC, connecting DE, DA', EA '.", "ocr_zh": "在三角形ABC中，D在AC上，E在AB上，A'在BC上，连接DE，DA'，EA'。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_331.jpg" }, { "index": 332, "ocr_en": "In the right angled triangle ABC, AC is perpendicular to BC, with the legs C and D on AC, E on BC, and C 'inside the triangle ABC, connecting C'A, C'D, C'E, and DE.", "ocr_zh": "在直角三角形ABC中，AC垂直于BC，垂足是C，D在AC上，E在BC上，C'在三角形ABC内部，连接C'A，C'D，C'E，DE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_332.jpg" }, { "index": 333, "ocr_en": "In obtuse triangle ABC, angle A is obtuse, D is on BC, E is on AB, F is on AC, connecting DE, DF, EF, and AD.", "ocr_zh": "钝角三角形ABC中，角A是钝角，D在BC上，E在AB上，F在AC上，连接DE，DF，EF，AD。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_333.jpg" }, { "index": 334, "ocr_en": "In the right angled triangle ABC, AC is perpendicular to BC, with the legs C and D on AC, E on BC, and C 'inside the triangle ABC, connecting C'B, C'D, C'E, and DE.", "ocr_zh": "在直角三角形ABC中，AC垂直于BC，垂足是C，D在AC上，E在BC上，C'在三角形ABC内部，连接C'B，C'D，C'E，DE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_334.jpg" }, { "index": 335, "ocr_en": "In the coordinate system xOy, A is on the negative half axis of the x-axis, B is on the positive half axis of the x-axis, OA is greater than OB, C is on the negative half axis of the y-axis, and an upward opening parabola passes through points ABC, connecting AC, BC, E on OB, D on BC, F on AC, EF, AD, ED, FD, P on AD, and PE, PF.", "ocr_zh": "在坐标系xOy中，A在x轴负半轴，B在x轴正半轴，OA大于OB，C在y轴负半轴，开口向上的抛物线过ABC三点，连接AC，BC，E在OB上，D在BC上，F在AC上，连接EF，AD，ED，FD，P在AD上，连接PE，PF。", "class": "Analytic Geometry", "difficulty": 4, "image_path": "fig_335.jpg" }, { "index": 336, "ocr_en": "Quadrilateral ABCD, with A as the center and AB as the radius, forms a circle connecting AD and BC.", "ocr_zh": "四边形ABCD，以A为圆心，AB为半径作圆，连接AD，BC。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_336.jpg" }, { "index": 337, "ocr_en": "In the coordinate system xOy, A is on the y-axis positive half axis, C is on the x-axis positive half axis, B is in the first quadrant, connect AB, BC, make the quadrilateral ABCO rectangular, D is on AB, connect CD, and make a square CDEF in the first quadrant.", "ocr_zh": "在坐标系xOy中，A在y轴正半轴，C在x轴正半轴，B在第一象限，连接AB，BC，使四边形ABCO是矩形，D在AB上，连接CD，在第一象限内作正方形CDEF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_337.jpg" }, { "index": 338, "ocr_en": "Quadrilateral ABCD, with A as the center and AB as the radius, forms a circle connecting AD and BC.", "ocr_zh": "四边形ABCD，以A为圆心，AB为半径作圆，连接AD，BC。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_338.jpg" }, { "index": 339, "ocr_en": "Quadrilateral ABCD, connecting AD and BC.", "ocr_zh": "四边形ABCD，连接AD，BC。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_339.jpg" }, { "index": 340, "ocr_en": "Triangle ABC is an equilateral triangle, D is the midpoint of BC, connecting AD, E is on AD, F is outside triangle ABC, F is close to the BC side, connecting CE, DF, CF.", "ocr_zh": "三角形ABC是等边三角形，D是BC中点，连接AD，E在AD上，F在三角形ABC外，F靠近BC边，连接CE，DF，CF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_340.jpg" }, { "index": 341, "ocr_en": "There is a triangle AOP on the diagram, where O 'is on AO and M is on AP. O' M is parallel to OP, with O as the center and OP as the radius for the circle. O 'is the center and O' is the radius for the circle.", "ocr_zh": "图上有三角形AOP，O'在AO上，M在AP上，O'M平行于OP，以O为圆心，OP为半径作圆，以O'为圆心，O'M为半径作圆。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_341.jpg" }, { "index": 342, "ocr_en": "On the diagram, there are triangles ABC, B 'on AB, C' on AC, connecting B'C 'parallel to BC, P on BC, connecting AP, and AP intersecting with B'C' at M.", "ocr_zh": "图上有三角形ABC，B'在AB上，C'在AC上，连接B'C'，B'C'平行于BC，P在BC上，连接AP，AP与B'C'相交于M。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_342.jpg" }, { "index": 343, "ocr_en": "Left image: There is a moving point M on the line segment AP.\nRight figure: There is a triangle ABC on the figure, with B 'on AB and C' on AC, connecting B'C 'parallel to BC, P on BC, connecting AP, and AP intersecting with B'C' at M.", "ocr_zh": "左图：线段AP上有一动点M。\n右图：图上有三角形ABC，B'在AB上，C'在AC上，连接B'C'，B'C'平行于BC，P在BC上，连接AP，AP与B'C'相交于M。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_343.jpg" }, { "index": 344, "ocr_en": "Top left: There is a triangular APQ.\nBottom left: There is a point B 'outside the quadrilateral ABPQ, where C is on the BP extension line, connecting BB', AB ', B'Q, and C' on the B'Q extension line.\nBottom right: There is a circle with O as its center and OP as its radius, and a circle with O 'as its center and O'Q as its radius. The circle O with OP as its radius is separated from the circle O' with O'Q as its radius. A is an external point that connects AP, AQ, AO, AO ', OO', PQ, OP, and O'Q. ", "ocr_zh": "左上：有三角形APQ。\n左下：四边形ABPQ外有一点B'，C在BP延长线上，连接BB'，AB'，B'Q，C'在B'Q延长线上。\n右下：有一个以O为圆心，OP为半径的圆，有一个以O'为圆心，O'Q为半径的圆，以OP为半径的圆O与以O'Q为半径的圆O'相离，A是外部一点，连接AP，AQ，AO，AO'，OO'，PQ，OP，O'Q。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_344.jpg" }, { "index": 345, "ocr_en": "Left image: There is triangle ABB 'and triangle APQ, connecting BP and C on the extension line of BP, connecting B'Q and C' on the extension line of B'Q, with angle BAQ equal to angle PAQ.\nRight figure: There is a circle with O as the center and OP as the radius, and a circle with O 'as the center and O'Q as the radius. The circle O with OP as the radius is separated from the circle O' with O'Q as the radius. A is an external point that connects AP, AQ, AO, AO ', OO', PQ, OP, and O'Q.", "ocr_zh": "左图：有三角形ABB'，还有三角形APQ，连接BP，C在BP延长线上，连接B'Q，C'在B'Q延长线上，角BAQ等于角PAQ。\n右图：有一个以O为圆心，OP为半径的圆，有一个以O'为圆心，O'Q为半径的圆，以OP为半径的圆O与以O'Q为半径的圆O'相离，A是外部一点，连接AP，AQ，AO，AO'，OO'，PQ，OP，O'Q。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_345.jpg" }, { "index": 346, "ocr_en": "Left image: There is a point B 'outside the quadrilateral ABPQ, where C is on the BP extension line, connecting BB', AB ', B'Q, and C' on the B'Q extension line.\nRight figure: There is a circle with O as the center and OP as the radius, and a circle with O 'as the center and O'Q as the radius. The circle O with OP as the radius is separated from the circle O' with O'Q as the radius. A is an external point that connects AP, AQ, AO, AO ', OO', PQ, OP, and O'Q.", "ocr_zh": "左图：四边形ABPQ外有一点B'，C在BP延长线上，连接BB'，AB'，B'Q，C'在B'Q延长线上。\n右图：有一个以O为圆心，OP为半径的圆，有一个以O'为圆心，O'Q为半径的圆，以OP为半径的圆O与以O'Q为半径的圆O'相离，A是外部一点，连接AP，AQ，AO，AO'，OO'，PQ，OP，O'Q。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_346.jpg" }, { "index": 347, "ocr_en": "There is an obtuse triangle ABC, angle ABC is obtuse, there is a point O on AB, OA is less than OB, P is on a circle with O as the center and OA as the radius, and P is inside triangle ABC, connecting BP and CP.", "ocr_zh": "有钝角三角形ABC，角ABC是钝角，AB上有一点O，OA小于OB，P在以O为圆心，OA为半径的圆上，且P在三角形ABC内部，连接BP，CP。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_347.jpg" }, { "index": 348, "ocr_en": "In the coordinate system xOy, A is on the positive x-axis, B and C are in the first quadrant, B moves on a straight line passing through the first, second, and fourth quadrants, and triangle ABC is an equilateral triangle.", "ocr_zh": "在坐标系xOy中，A在x轴正半轴，B和C在第一象限，B在一条过第一象限、第二象限和第四象限的直线上运动，三角形ABC是等边三角形。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_348.jpg" }, { "index": 349, "ocr_en": "In the coordinate system xOy, C is on the positive x-axis, B is on the positive y-axis, A is in the first quadrant, connecting AB, AC, BC, M on BC, P on OC, and PM.", "ocr_zh": "在坐标系xOy中，C在x轴正半轴，B在y轴正半轴，A在第一象限，连接AB，AC，BC，M在BC上，P在OC上，连接PM。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_349.jpg" }, { "index": 350, "ocr_en": "In the quadrilateral ABCD, connecting AC, BD, and E is a point on BD and connecting CE.", "ocr_zh": "四边形ABCD中，连接AC，BD，E是BD上一点，连接CE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_350.jpg" }, { "index": 351, "ocr_en": "There is a circle on the diagram with O as the center and AB as the diameter. There is a point P on the circle, and a point D inside the circle. D and P are on the same side as AB, and C is on OB, connecting PC, CD, and DP.", "ocr_zh": "图上有一个以O为圆心，AB为直径的圆，该圆上有一点P，该圆内有一点D，D和P在AB同侧，C在OB上，连接PC，CD，DP。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_351.jpg" }, { "index": 352, "ocr_en": "In diamond ABCD, connect AC, BD, P on CD, and connect AP.", "ocr_zh": "菱形ABCD中，连接AC，BD，P在CD上，连接AP。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_352.jpg" }, { "index": 353, "ocr_en": "In the coordinate system xOy, there is a rectangle ABCD in the first and second quadrants, B on the positive x-axis, A in the second quadrant and on the line y=- x, C in the first quadrant, and D in the second quadrant.", "ocr_zh": "在坐标系xOy中，有一个矩形ABCD在第一象限和第二象限内，B在x轴正半轴，A在第二象限且在y=-x的直线上，C在第一象限，D在第二象限。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_353.jpg" }, { "index": 354, "ocr_en": "In the acute triangle ABC, E is on AC, D is on BC, connecting AD, BE intersects at point F, and connects CF.", "ocr_zh": "锐角三角形ABC中，E在AC上，D在BC上，连接AD，BE交于点F，连接CF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_354.jpg" }, { "index": 355, "ocr_en": "There is a circle on the diagram with O as the center and OB as the radius, and there is a triangle ABC inside it. A and B are on the circle, and C is inside the circle.", "ocr_zh": "图上有一个以O为圆心，OB为半径的圆，其内部有一个三角形ABC，A和B在该圆上，C在该圆内。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_355.jpg" }, { "index": 356, "ocr_en": "There is an obtuse triangle ACD, angle CAD is obtuse, with a point O inside. OA equals OC, and a circle with O as the center and OA as the radius intersects CD with B.", "ocr_zh": "有一个钝角三角形ACD，角CAD是钝角，内部有一点O，OA等于OC，以O为圆心，OA为半径的圆交CD于B。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_356.jpg" }, { "index": 357, "ocr_en": "In the coordinate system xOy, A is on the positive x-axis, C is on the negative y-axis, P is in the fourth quadrant, CP is less than OC, and a circle is formed with C as the center and CP as the radius, connecting AP. M is the midpoint of AP and connects OM.", "ocr_zh": "在坐标系xOy中，A在x轴正半轴，C在y轴负半轴，P在第四象限，CP小于OC，以C为圆心，CP为半径作圆，连接AP，M是AP中点，连接OM。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_357.jpg" }, { "index": 358, "ocr_en": "In triangle ABC, D is on AB, E is on AC, A 'is on BC, connecting DE, A'D, A'E.", "ocr_zh": "三角形ABC中，D在AB上，E在AC上，A'在BC上，连接DE，A'D，A'E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_358.jpg" }, { "index": 359, "ocr_en": "There is a circle on the diagram with O as the center and AB as the diameter. C and D are points on the circle, and D is between B and C. It connects CD, AD, and E, which are points on AD, and connects CE and BE.", "ocr_zh": "图上有一以O为圆心，AB为直径的圆，C和D是该圆上的点，D在B和C之间，连接CD，AD，E是AD上一点，连接CE，BE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_359.jpg" }, { "index": 360, "ocr_en": "In diamond ABCD, Q is on AD, P is on BC, connecting PQ and PD.", "ocr_zh": "菱形ABCD中，Q在AD上，P在BC上，连接PQ，PD。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_360.jpg" }, { "index": 361, "ocr_en": "There is a point P near AB outside the squares ABCD, connecting AP and BP.", "ocr_zh": "正方形ABCD外有一点靠近AB的点P，连接AP，BP。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_361.jpg" }, { "index": 362, "ocr_en": "In the acute triangle ABC, D is on BC, connected to AD, M is a point on AD, N is inside the triangle ADC, connected to DN.", "ocr_zh": "锐角三角形ABC中，D在BC上，连接AD，M是AD上一点，N在三角形ADC内，连接DN。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_362.jpg" }, { "index": 363, "ocr_en": "In acute triangle ABC, P is on AC, D is outside triangle ABC and close to AC, connecting PD and BD.", "ocr_zh": "锐角三角形ABC中，P在AC上，D在三角形ABC外且靠近AC，连接PD，BD。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_363.jpg" }, { "index": 364, "ocr_en": "In triangle ABC, D is on BC, E is outside triangle ABC and close to AC, connecting AE, AD, F is a point on BE and F is inside triangle ABC, connecting CF.", "ocr_zh": "三角形ABC中，D在BC上，E在三角形ABC外且靠近AC，连接AE，AD，F是BE上一点且F在三角形ABC内，连接CF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_364.jpg" }, { "index": 365, "ocr_en": "In the coordinate system xOy, the vertices A and B of triangle ABP are in the first quadrant, vertex P is in the fourth quadrant, AM is perpendicular to the x-axis, and the vertical foot is M. Connect OP and extend the intersecting AM extension line at point N.", "ocr_zh": "在坐标系xOy中，三角形ABP的顶点A和B第一象限，顶点P在第四象限，AM垂直于x轴，垂足是M，连接OP并延长交AM延长线于点N。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_365.jpg" }, { "index": 366, "ocr_en": "In diamond ABCD, G is on AD, E is on DC, and F is on AB, connecting GE, GF, and EF.", "ocr_zh": "菱形ABCD中，G在AD上，E在DC上，F在AB上，连接GE，GF，EF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_366.jpg" }, { "index": 367, "ocr_en": "In the diamond ABCD, G is on AD, F is on AB, connected to GF, and serves as an auxiliary line AE perpendicular to GF. The vertical foot is H, and AE intersects with CD at E. It connects EG, EF, and serves as an auxiliary line EB.", "ocr_zh": "菱形ABCD中，G在AD上，F在AB上，连接GF，作辅助线AE垂直于GF，垂足是H，AE与CD交于E，连接EG，EF，作辅助线EB。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_367.jpg" }, { "index": 368, "ocr_en": "In diamond ABCD, G is on AD, F is on AB, GF is connected, auxiliary line AE is perpendicular to GF, vertical foot is H, AE intersects with CD at E, EG, EF are connected, auxiliary line EB is perpendicular to AB, vertical foot is B, auxiliary line BE is perpendicular to CD, vertical foot is E, DE equals EC equals 1, AB equals 2, angle BCD is 60 °, BE length is root 3, AH equals HE, length is half root 7, FH length is quarter root 21.", "ocr_zh": "菱形ABCD中，G在AD上，F在AB上，连接GF，作辅助线AE垂直于GF，垂足是H，AE与CD交于E，连接EG，EF，作辅助线EB,BE垂直于AB，垂足是B，作辅助线BE垂直于CD，垂足是E，DE等于EC等于1，AB等于2，角BCD是60°，BE长度是根号3，AH等于HE，长度是二分之根号七，FH长度是四分之根号二十一。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_368.jpg" }, { "index": 369, "ocr_en": "In diamond ABCD, G is on AD, F is on AB, connecting GF, auxiliary line AE is perpendicular to GF, vertical foot is H, AE intersects with CD at E, connecting EG, EF, auxiliary line EB, BE is perpendicular to AB, vertical foot is B, auxiliary line BE is perpendicular to CD, vertical foot is E, auxiliary line BD, DE equals EC equals 1, angle BCD is 60 °, BE length is root 3, angle 1 is angle AFG, angle 2 is angle GFE, angle 3 is angle BAE.", "ocr_zh": "菱形ABCD中，G在AD上，F在AB上，连接GF，作辅助线AE垂直于GF，垂足是H，AE与CD交于E，连接EG，EF，作辅助线EB,BE垂直于AB，垂足是B，作辅助线BE垂直于CD，垂足是E，作辅助线BD，DE等于EC等于1，角BCD是60°，BE长度是根号3，角1是角AFG，角2是角GFE，角三是角BAE。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_369.jpg" }, { "index": 370, "ocr_en": "In diamond ABCD, E is on AB, F is on BC, G is on AD, connecting AC, EF, EG, GF, the intersection of GF and AC is H, and the angle AEG is equal to 30 °.", "ocr_zh": "菱形ABCD中，E在AB上，F在BC上，G在AD上，连接AC，EF，EG，GF，GF和AC交点是H，角AEG等于30°。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_370.jpg" }, { "index": 371, "ocr_en": "In the diamond ABCD, E is on AB, F is on BC, G is on AD, connecting AC, EF, EG, GF, and the intersection point of GF and AC is H, serving as the auxiliary line BD. BD is perpendicular to AC, with a vertical foot of O. The angle AEG is equal to 30 °, and the angle ABD is equal to 30 °.", "ocr_zh": "菱形ABCD中，E在AB上，F在BC上，G在AD上，连接AC，EF，EG，GF，GF和AC交点是H，作辅助线BD，BD垂直于AC，垂足是O，角AEG等于30°，角ABD等于30°。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_371.jpg" }, { "index": 372, "ocr_en": "In the diamond ABCD, E is on AB, F is on BC, G is on AD, connecting AC, EF, EG, GF, GF, and AC at the intersection point of H. The angle AEG is equal to the angle AGE, which is 30 °, the angle ABC is equal to the angle EGF, which is 60 °, and M is on BC. The auxiliary line AM is perpendicular to BC, and the vertical foot is M. The length of AB is six times the root number two.", "ocr_zh": "菱形ABCD中，E在AB上，F在BC上，G在AD上，连接AC，EF，EG，GF，GF和AC交点是H，角AEG等于角AGE等于30°，角ABC等于角EGF等于60°，M在BC上，作辅助线AM垂直于BC，垂足是M，AB长度等于六倍根号二。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_372.jpg" }, { "index": 373, "ocr_en": "In the diamond ABCD, E is on AB, F is on BC, G is on AD, connecting AC, EF, EG, GF, and the intersection point of GF and AC is H. The angle AEG is equal to 30 °, the angle ABC is equal to 60 °, and the length of AB is equal to BC. The length is six times the root two.", "ocr_zh": "菱形ABCD中，E在AB上，F在BC上，G在AD上，连接AC，EF，EG，GF，GF和AC交点是H，角AEG等于30°，角ABC等于60°，AB长度等于BC长度等于六倍根号二。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_373.jpg" }, { "index": 374, "ocr_en": "In the diamond ABCD, E is on AB, F is on BC, G is on AD, connecting AC, EF, EG, GF, GF, and AC at the intersection point H, and N is on BF, serving as the auxiliary line EN. The angle BFE is equal to the angle GFE, which is 45 °. If the intersection point of AC and GE is P, then AP is equal to x. The length of AE is twice that of AP, P is the midpoint of EG. The lengths of EP, GP, and BN are three times the root of AP, BE is twice that of EP, and NF is three times that of BN.", "ocr_zh": "菱形ABCD中，E在AB上，F在BC上，G在AD上，连接AC，EF，EG，GF，GF和AC交点是H，N在BF上，作辅助线EN，角BFE等于角GFE等于45°，设AC和GE的交点是P，则AP等于x，AE的长度是AP的两倍，P是EG的中点，EP和GP和BN的长度是AP的根号三倍，BE的长度是EP的两倍，NF的长度是BN的根号三倍。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_374.jpg" }, { "index": 375, "ocr_en": "In diamond ABCD, M is on AD, N is on AB, A 'is on DC, connecting BD, A'M, A'N, MN.", "ocr_zh": "菱形ABCD中，M在AD上，N在AB上，A'在DC上，连接BD，A'M，A'N，MN。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_375.jpg" }, { "index": 376, "ocr_en": "In diamond ABCD, M is on AD, N is on AB, A 'is on DC, connecting BD, A'M, A'N, MN, MN with a length of root six, angle DAB is equal to 60 °, angle DMA' is equal to 30 °, angle ANM is equal to 45 °, H is on AN, auxiliary line MH is perpendicular to AB, vertical foot is H. Assuming the length of MD and A'D is x, and the intersection point of BD and A'M is P, then the length of DP is half of DM, the length of A'M is three times the root of DM, the length of MH and HN is three times DP, the length of AH is three times the root of DP, and the length of AM is H. It's twice as much as AH.", "ocr_zh": "菱形ABCD中，M在AD上，N在AB上，A'在DC上，连接BD，A'M，A'N，MN，MN长度是根号六，角DAB等于60°，角DMA'等于30°，角ANM等于45°，H在AN上，作辅助线MH垂直于AB，垂足是H，设MD和A'D的长度是x，且BD和A'M的交点是P，那么DP的长度是DM的一半，A'M的长度是DM的根号三倍，MH和HN的长度是DP的三倍，AH的长度是DP的根号三倍，AM的长度是AH的两倍。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_376.jpg" }, { "index": 377, "ocr_en": "In rectangles ABCD, connect AC, where E is a point on BC, F is a point on AC, connect AE, EF, where EF is perpendicular to AC, the vertical foot is F, AB length is 3, AD length is 4, AF length is 3, CF length is 2, let the lengths of EF and BE be x, and the length of EC be 4-x.", "ocr_zh": "矩形ABCD中，连接AC，E是BC上一点，F是AC上一点，连接AE，EF，其中EF垂直于AC，垂足是F，AB长度是3，AD长度是4，AF长度是3，CF长度是2，设EF和BE的长度为x，EC长度为4-x。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_377.jpg" }, { "index": 378, "ocr_en": "In rectangles ABCD, F is on AD, E is on BC, connecting AE, EF, FC, AB and BE. The length of EF is 3, and the length of EC is 1.", "ocr_zh": "矩形ABCD中，F在AD上，E在BC上，连接AE，EF，FC，AB和BE和EF长度均为3，EC长度为1。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_378.jpg" }, { "index": 379, "ocr_en": "In rectangles ABCD, F is an internal point, E is on BC, connecting AF, AE, FE, and FC.", "ocr_zh": "矩形ABCD中，F是内部一点，E在BC上，连接AF，AE，FE，FC。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_379.jpg" }, { "index": 380, "ocr_en": "In rectangles ABCD, C 'is an internal point, E is on BC, connecting C'A, C'B, C'D, C'E, DE, where AB equals 2 and AD equals 4.", "ocr_zh": "矩形ABCD中，C'是内部一点，E在BC上，连接C'A，C'B，C'D，C'E，DE，其中AB等于2，AD等于4。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_380.jpg" }, { "index": 381, "ocr_en": "In rectangles ABCD, C 'is on AD, E is on BC, connecting BC', C'E, ED, where AB equals AC ', C'E equals C'D equals CD equals 2, and AD equals 4.", "ocr_zh": "矩形ABCD中，C'在AD上，E在BC上，连接BC'，C'E，ED，其中AB等于AC'等于C'E等于C'D等于CD等于2，AD等于4。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_381.jpg" }, { "index": 382, "ocr_en": "In the rectangle ABCD, C 'is an internal point, E is on BC, connecting C'A, C'B, C'D, C'E, DE, EC' perpendicular to C'D, the vertical foot is C ', M is on AD, N is on BE, C'M is perpendicular to AD, the vertical foot is M, C'N is perpendicular to BC, and the vertical foot is N, where AB is equal to CD, C'D is equal to 2, AD is equal to 4, C'M is equal to C'N is equal to 1, MD is equal to the third of the root, and C'E is equal to the lower third of the root.", "ocr_zh": "矩形ABCD中，C'是内部一点，E在BC上，连接C'A，C'B，C'D，C'E，DE，EC'垂直于C'D，垂足是C'，M在AD上，N在BE上，C'M垂直于AD，垂足是M，C'N垂直于BC，垂足是N，其中AB等于CD等于C'D等于2，AD等于4，C'M等于C'N等于1，MD等于根号三，C'E等于根号下三分之四。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_382.jpg" }, { "index": 383, "ocr_en": "In rectangles ABCD, F is an internal point, E is on BC, connecting AF, CF, FE, AE.", "ocr_zh": "矩形ABCD中，F是内部一点，E在BC上，连接AF，CF，FE，AE。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_383.jpg" }, { "index": 384, "ocr_en": "In rectangles ABCD, F is an internal point, E is on BC, connecting AF, CF, FE, AE, passing through F makes FG perpendicular to BC, perpendicular foot is G, passing through F makes FH perpendicular to H, perpendicular foot is H, BE=EF=EC=3，AB=AF=4，AE=5，AH=4x，FH=4-3x，FG=3x。", "ocr_zh": "矩形ABCD中，F是内部一点，E在BC上，连接AF，CF，FE，AE，过F作FG垂直于BC，垂足是G，过F作FH垂直于H，垂足是H，其中，BE=EF=EC=3，AB=AF=4，AE=5，AH=4x，FH=4-3x，FG=3x。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_384.jpg" }, { "index": 385, "ocr_en": "In rectangles ABCD, F is an internal point, E is on BC, connecting AF, CF, FE, AE, BF, where BF and AE intersect at G, BG=AG=12/5, AB=4, BE=3.", "ocr_zh": "矩形ABCD中，F是内部一点，E在BC上，连接AF，CF，FE，AE，BF，其中BF和AE交于G，BG=AG=12/5，AB=4，BE=3。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_385.jpg" }, { "index": 386, "ocr_en": "There are points G, E on AD, F on AB, and H on CD within rectangles ABCD, connecting EH, HG, GF, FE, GC, and GB to make EFGH a rectangle.", "ocr_zh": "在矩形ABCD内有一点G，E在AD上，F在AB上，H在CD上，连接EH，HG，GF，FE，GC，GB，使EFGH是矩形。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_386.jpg" }, { "index": 387, "ocr_en": "In rectangles ABCD, F is on DC and E is on AB, connecting AF and FE.", "ocr_zh": "矩形ABCD中，F在DC上，E在AB上，连接AF，FE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_387.jpg" }, { "index": 388, "ocr_en": "Left image: There is a point A 'inside rectangle ABCD, with E and G on AD, and E between A and G, connecting A'A, A'B, A'E, A'G.\nRight picture: There is a rectangle on the picture.", "ocr_zh": "左图：矩形ABCD内有一点A'，E和G在AD上，且E在A和G之间，连接A'A，A'B，A'E，A'G。\n右图：图上有一矩形。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_388.jpg" }, { "index": 389, "ocr_en": "In triangle ABC, D is on AB, E is on BC, B1 is outside triangle ABC and close to BC, connecting B1E, B1C, B1D, DE, DC, where CD is perpendicular to DE and the vertical foot is D.", "ocr_zh": "三角形ABC中，D在AB上，E在BC上，B1在三角形ABC外部且靠近BC，连接B1E，B1C，B1D，DE，DC，其中，CD垂直于DE，垂足是D。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_389.jpg" }, { "index": 390, "ocr_en": "In triangle ABC, D is on AB, E is on BC, B1 is outside triangle ABC and close to BC, connecting B1E, B1, B1D, DE, DC, where CD is perpendicular to DE, perpendicular foot is D, angle 1 is angle BDE, angle 2 is angle B1DE, angle 3 is angle B1DC, and angle 4 is angle ADC.", "ocr_zh": "三角形ABC中，D在AB上，E在BC上，B1在三角形ABC外部且靠近BC，连接B1E，B1，B1D，DE，DC，其中，CD垂直于DE，垂足是D，角1是角BDE，角2是角B1DE，角三是角B1DC，角4是角ADC。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_390.jpg" }, { "index": 391, "ocr_en": "In triangle ABC, BA is perpendicular to CA, the vertical foot is A, D is on AB, E is on BC, B1 is outside triangle ABC and close to BC, connecting B1E, B1, B1D, DE, DC, and M are the intersection points of B1D and BC, extending B1E to intersect BD at N, where CD is perpendicular to DE, the vertical foot is D, and angle 5 is angle EB1D.", "ocr_zh": "三角形ABC中，BA垂直于CA，垂足是A，D在AB上，E在BC上，B1在三角形ABC外部且靠近BC，连接B1E，B1，B1D，DE，DC，M是B1D和BC的交点，延长B1E交BD于N其中，CD垂直于DE，垂足是D，角5是角EB1D。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_391.jpg" }, { "index": 392, "ocr_en": "In triangle ABC, BA is perpendicular to CA, the vertical foot is A, D is on AB, E is on BC, B1 is outside triangle ABC and close to BC, connecting B1E, B1, B1D, DE, DC, M are the intersection points of B1D and BC, extending B1E to intersect BD at N, where CD is perpendicular to DE, the vertical foot is D, angle 5 is angle EB1D, DM=AD=DN=x, BN=8-2x.", "ocr_zh": "三角形ABC中，BA垂直于CA，垂足是A，D在AB上，E在BC上，B1在三角形ABC外部且靠近BC，连接B1E，B1，B1D，DE，DC，M是B1D和BC的交点，延长B1E交BD于N其中，CD垂直于DE，垂足是D，角5是角EB1D，DM=AD=DN=x，BN=8-2x。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_392.jpg" }, { "index": 393, "ocr_en": "There is a circle on the diagram with O as the center and OA as the radius. AB is the chord of the circle, and its position satisfies the condition that after folding the circle along AB, the inferior arc AB can make O exactly on the folded arc. C is a point on the circle O, connecting AC and BC so that O is inside the acute angle BAC. Connect AC and the folded arc at D.", "ocr_zh": "图上有一个圆，该圆以O为圆心，OA为半径，AB是该圆的弦，AB的位置满足沿AB翻折该圆的劣弧AB后，能够恰好使O在翻折后的圆弧上，C是该圆O上一点，连接AC，BC，使得O在锐角BAC内部，连接AC与翻折后的圆弧交于D。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_393.jpg" }, { "index": 394, "ocr_en": "There is a circle on the diagram with O as the center and OA as the radius. AB is the chord of the circle, and its position satisfies the condition that after folding the inferior arc AB of the circle along AB, O can be placed within the folded arch. C is a point on the circle O, connecting AC and BC so that O is inside the acute angle BAC, connecting AC and the folded arc at D, and connecting AD so that O is exactly on AD.", "ocr_zh": "图上有一个圆，该圆以O为圆心，OA为半径，AB是该圆的弦，AB的位置满足沿AB翻折该圆的劣弧AB后，能够使O在翻折后的弓形内，C是该圆O上一点，连接AC，BC，使得O在锐角BAC内部，连接AC与翻折后的圆弧交于D，连接AD使O恰好在AD上。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_394.jpg" }, { "index": 395, "ocr_en": "There is a circle on the diagram with O as the center and OA as the radius. AB is the chord of the circle, and its position satisfies the condition that after folding the circle along AB, the inferior arc AB can make O outside the arch after folding. C is a point on the arc AB after folding, connecting OC.", "ocr_zh": "图上有一个圆，该圆以O为圆心，OA为半径，AB是该圆的弦，AB的位置满足沿AB翻折该圆的劣弧AB后，能够使O在翻折后的弓形外，C是翻折后圆弧AB上一点，连接OC。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_395.jpg" }, { "index": 396, "ocr_en": "There is a semicircle on the diagram, with O as the center, AB as the diameter, and C as a point on the semicircle. Connect BC and fold the arc BC with O as the center and OB as the radius along BC, so that O is inside the folded arch, and the folded arch intersects with AO at D.", "ocr_zh": "图上有一半圆，该半圆以O为圆心，AB为直径，C是该半圆上一点，连接BC，将以O为圆心，OB为半径的弧BC沿BC翻折，使O在翻折后的弓形内部，翻折后的弓形与AO交于D。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_396.jpg" }, { "index": 397, "ocr_en": "There is a circle on the diagram with O as the center, OA as the radius, AB as the chord of the circle, C on the circle, O inside the acute angle CAB, connected to AC, folded along AC with O as the center, OA as the radius of the inferior arc AC, so that O is inside the folded arch, and the folded arch intersects with AB at D.", "ocr_zh": "图上有一圆，该圆以O为圆心，OA为半径，AB是该圆的弦，C在该圆上，O在锐角CAB内部，连接AC，沿AC翻折以O为圆心，OA为半径的劣弧AC，使得O在翻折后的弓形内部，翻折后的弓形与AB交于D。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_397.jpg" }, { "index": 398, "ocr_en": "There is a semicircle on the diagram, with O as the center, AB as the diameter, D as a point on the semicircle, C directly below the semicircle, connecting CA and CB, folding the arc AD along AD so that the folded arc is exactly tangent to BC, and the folded arc intersects with OB at F.", "ocr_zh": "图上有一半圆，该半圆以O为圆心，AB为直径，D是该半圆上一点，C在半圆正下方，连接CA，CB，沿AD翻折圆弧AD，使得翻折后的圆弧恰与BC相切，翻折后的圆弧与OB交于F。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_398.jpg" }, { "index": 399, "ocr_en": "There is a circle on the diagram with O as the center, OA as the radius, AB as the chord of the circle, C as a point on the circle O, connecting AC and BC so that O is inside the acute angle BAC. After folding the inferior arc BC of the circle along BC, O can be inside the folded bow shape, and the folded bow shape intersects with AB at D.", "ocr_zh": "图上有一个圆，该圆以O为圆心，OA为半径，AB是该圆的弦，C是该圆O上一点，连接AC，BC，使得O在锐角BAC内部，沿BC翻折该圆的劣弧BC后，能够使O在翻折后的弓形内部，翻折后的弓形和AB交于D。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_399.jpg" }, { "index": 400, "ocr_en": "There is a circle on the diagram with O as the center, OA as the radius, AB as the chord of the circle, C as a point on the circle O, connecting AC and BC so that O is inside the acute angle BAC. After folding the inferior arc BC of the circle along BC, O can be inside the folded bow shape, and the folded bow shape intersects with AB at D.", "ocr_zh": "图上有一个圆，该圆以O为圆心，OA为半径，AB是该圆的弦，C是该圆O上一点，连接AC，BC，使得O在锐角BAC内部，沿BC翻折该圆的劣弧BC后，能够使O在翻折后的弓形内部，翻折后的弓形和AB交于D。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_400.jpg" }, { "index": 401, "ocr_en": "There is a circle on the diagram with O as the center, AB as the diameter, and C as a point on the circle O. It is connected to AC, and when the inferior arc AC of the circle is folded along AC, O can be placed inside the folded arch, and the folded arch intersects with OB at D.", "ocr_zh": "图上有一个圆，该圆以O为圆心，AB为直径，C是该圆O上一点，连接AC，沿AC翻折该圆的劣弧AC后，能够使O在翻折后的弓形内部，翻折后的弓形和OB交于D。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_401.jpg" }, { "index": 402, "ocr_en": "There is a circle on the diagram with O as the center, AB as the diameter, and C as a point on the circle O. It connects BC, and by folding the inferior arc AC of the circle along BC, O can be placed inside the folded arch, and the folded arch intersects with OA at D. By folding the inferior arc BD along BD, the folded inferior arc intersects with BC at E.", "ocr_zh": "图上有一个圆，该圆以O为圆心，AB为直径，C是该圆O上一点，连接BC，沿BC翻折该圆的劣弧AC后，能够使O在翻折后的弓形内部，翻折后的弓形和OA交于D，沿BD翻折劣弧BD，翻折后的该劣弧交BC于E。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_402.jpg" }, { "index": 403, "ocr_en": "On the diagram, there is a circle with O as the center and OA as the radius. AB is the chord of the circle. Fold the inferior arc AB along AB so that O is outside the folded arch, and C is a point on the inferior arc after folding. Connect AC and BC, and extend AC to intersect the circle O with D.", "ocr_zh": "图上有一以O为圆心，OA为半径的圆，AB是该圆的弦，沿AB将劣弧AB翻折，使O在翻折的弓形之外，C是翻折后劣弧上的一点，连接AC，BC，延长AC交该圆O于D。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_403.jpg" }, { "index": 404, "ocr_en": "There is a semicircle with diameter AB on the diagram, and C is a point on the semicircle. Fold the inferior arc BC along BC, and the inferior arc intersects with AB at D.", "ocr_zh": "图上有一以AB为直径的半圆，C是该半圆上一点，沿BC翻折劣弧BC，该劣弧与AB交于D。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_404.jpg" }, { "index": 405, "ocr_en": "On the diagram, there is a semicircle with O as the center and AB as the diameter. C is a point on the semicircle, and the inferior arc BC is folded along BC, so that O is inside the arch after folding, and the inferior arc intersects with AB at D.", "ocr_zh": "图上有一以O为圆心，AB为直径的半圆，C是该半圆上一点，沿BC翻折劣弧BC，使O在翻折后的弓形内部，该劣弧与AB交于D。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_405.jpg" }, { "index": 406, "ocr_en": "There is a semicircle on the diagram with O as the center and AB as the diameter. E and F are points on the semicircle. Fold the inferior arc EF along EF so that the folded inferior arc is exactly tangent to AB, with the tangent point on OB, denoted as D.", "ocr_zh": "图上有一以O为圆心，AB为直径的半圆，E和F是该半圆上的点，沿EF翻折劣弧EF，使翻折后的劣弧恰好与AB相切，切点在OB上，记为D。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_406.jpg" }, { "index": 407, "ocr_en": "There is a circle on the diagram with O as the center and OA as the radius. A, B, and D are on this circle, connecting AB, AD, BD, and O inside triangle ABD. Fold the inferior arc AB along AB so that O is exactly on the folded inferior arc. The folded inferior arc intersects with AD at C and connects BC.", "ocr_zh": "图上有一以O为圆心，OA为半径的圆，A，B，D在该圆上，连接AB，AD，BD，O在三角形ABD内部，沿AB翻折劣弧AB，使O恰好在翻折后的劣弧上，翻折后的劣弧与AD交于C，连接BC。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_407.jpg" }, { "index": 408, "ocr_en": "There is a quarter circle on the diagram, with O as the center and OA as the radius. B is on this quarter circle, with BO perpendicular to AO, O as the vertical foot, P on OB, and Q on the inferior arc AB. Connect PQ and fold the inferior arc BQ along PQ so that the folded inferior arc is exactly tangent to OA. The tangent point is C, and the point where B is folded is denoted as B '.", "ocr_zh": "图上有一四分之一圆，该圆以O为圆心，OA为半径，B在该四分之一圆上，BO垂直于AO，垂足是O，P在OB上，Q在劣弧AB上，连接PQ并将劣弧BQ沿PQ翻折使翻折后的的劣弧恰好与OA相切，切点是C，B经过翻折后的点记为B'。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_408.jpg" }, { "index": 409, "ocr_en": "There is a quarter circle on the diagram, with O as the center and OA as the radius. B is on the quarter circle, BO is perpendicular to AO, O is the vertical foot, P is on OB, Q is on the inferior arc AB, and PQPQ is parallel to OA.", "ocr_zh": "图上有一四分之一圆，该圆以O为圆心，OA为半径，B在该四分之一圆上，BO垂直于AO，垂足是O，P在OB上，Q在劣弧AB上，连接PQPQ平行于OA。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_409.jpg" }, { "index": 410, "ocr_en": "There is a circle on the diagram with O as the center and AB as the diameter. C is a point on the circle, and C is close to A. Fold the inferior arc BC along BC, so that O is inside the folded arch. The folded inferior arc intersects with OA at D, connects CD and extends to intersect the circle at E, connects BE, and serves as auxiliary lines AE and OE.", "ocr_zh": "图上有一以O为圆心，AB为直径的圆，C是该圆上一点，C靠近A，沿BC翻折劣弧BC，使O在翻折后的弓形内部，翻折后的劣弧与OA交于D，连接CD并延长交该圆于E，连接BE，作辅助线AE，OE。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_410.jpg" }, { "index": 411, "ocr_en": "There is a circle on the diagram with O as the center and AB as the diameter. C is a point on the circle, and C is close to A. Fold the inferior arc BC along BC, so that O is inside the folded arch. The folded inferior arc intersects with OA at D. Fold the folded inferior arc BD along BD again and intersect with BC at F.", "ocr_zh": "图上有一以O为圆心，AB为直径的圆，C是该圆上一点，C靠近A，沿BC翻折劣弧BC，使O在翻折后的弓形内部，翻折后的劣弧与OA交于D，将上述翻折后的劣弧BD沿BD再次翻折与BC交于F。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_411.jpg" }, { "index": 412, "ocr_en": "There is a circle on the diagram with O as the center and AB as the diameter. C is a point on the circle, and C is close to A. Fold the inferior arc BC along BC so that O is inside the folded arch. The folded inferior arc intersects with OA at D, connects CD and extends to intersect the circle at E, connects BE, and O 'is the symmetry point of O about BC. O' is the center of the circle, and O 'C is the radius used as an auxiliary circle.", "ocr_zh": "图上有一以O为圆心，AB为直径的圆，C是该圆上一点，C靠近A，沿BC翻折劣弧BC，使O在翻折后的弓形内部，翻折后的劣弧与OA交于D，连接CD并延长交该圆于E，连接BE，O'是O关于BC的对称点，以O'为圆心，O'C为半径作辅助圆。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_412.jpg" }, { "index": 413, "ocr_en": "There is a circle on the diagram with O as the center and AB as the diameter. C is a point on the circle, and C is close to A. Fold the inferior arc BC along BC, so that O is inside the folded arch. The folded inferior arc intersects with OA at D, connects CD and extends to intersect the circle at E, and connects BE.", "ocr_zh": "图上有一以O为圆心，AB为直径的圆，C是该圆上一点，C靠近A，沿BC翻折劣弧BC，使O在翻折后的弓形内部，翻折后的劣弧与OA交于D，连接CD并延长交该圆于E，连接BE。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_413.jpg" }, { "index": 414, "ocr_en": "There is a sector with O as the center, OA and OB as radii, angle AOB less than 180 °, Q on inferior arc AB, P on OB, connecting QP.", "ocr_zh": "图上有一以O为圆心，OA和OB为半径的扇形，角AOB小于180°，Q在劣弧AB上，P在OB上，连接QP。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_414.jpg" }, { "index": 415, "ocr_en": "There is a sector with O as the center, OA and OB as radii, angle AOB less than 180 °, Q on the inferior arc AB, P on OB, connecting QP, QP perpendicular to OB, and the vertical foot is P.", "ocr_zh": "图上有一以O为圆心，OA和OB为半径的扇形，角AOB小于180°，Q在劣弧AB上，P在OB上，连接QP，QP垂直于OB，垂足是P。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_415.jpg" }, { "index": 416, "ocr_en": "There is a circle on the diagram with O as the center and AB as the diameter. P is a point on the extension line of BA, and the tangent lines PM and PC of the circle O are drawn through P. C is the tangent point and is close to A, connecting BC. A point D on the inferior arc BC of the circle satisfies the symmetry of D with respect to BC. Fold the inferior arc BC along BC to form an auxiliary arc BOC, connecting AD, OC, and CD.", "ocr_zh": "图上有一以O为圆心，AB为直径的圆，P是BA延长线上一点，过P作该圆O的切线PM和PC，其中，C是切点且靠近A，连接BC，该圆劣弧BC上一点D满足D是O关于BC的对称点，沿BC翻折劣弧BC作辅助弧线BOC，连接AD，OC，CD。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_416.jpg" }, { "index": 417, "ocr_en": "On the diagram, there is a semicircle with O as the center and AB as the diameter. Both E and F are on this semicircle, with E close to A and E close to B. Fold the inferior arc EF of the circle along EF so that the folded inferior arc is exactly tangent to AB, with the tangent point on OB, denoted as D.", "ocr_zh": "图上有一以O为圆心，AB为直径的半圆，E和F均在该半圆上，E靠近A，E靠近B，沿EF翻折该圆的劣弧EF使翻折后的劣弧恰好与AB相切，切点在OB上，记为D。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_417.jpg" }, { "index": 418, "ocr_en": "On the diagram, there is a circle with O as the center and AB as the diameter. C and E are on this circle, with C near A and E near B. AE, BC, BE, and D are connected by a point on the extension line of BE, and CD is connected. The inferior arc BC of the circle is folded along BC, and the folded inferior arc intersects OB with F.", "ocr_zh": "图上有一以O为圆心，AB为直径的圆，C和E在该圆上，C靠近A，E靠近B，连接AE，BC，BE，D是BE延长线上一点，连接CD，沿BC翻折该圆的劣弧BC，翻折后的劣弧交OB于F。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_418.jpg" }, { "index": 419, "ocr_en": "On the diagram, there is a circle with O as the center and AB as the diameter. C and E are on this circle, with C near A and E near B. They connect AE, BC, BE, and D, which are a point on the extension line of BE, and connect CD.", "ocr_zh": "图上有一以O为圆心，AB为直径的圆，C和E在该圆上，C靠近A，E靠近B，连接AE，BC，BE，D是BE延长线上一点，连接CD。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_419.jpg" }, { "index": 420, "ocr_en": "There is a semicircle on the diagram with O as the center and AB as the diameter. F is on this semicircle, and F is close to B. It connects AF, BF, and D on AF, and forms a square BCDE with BD as the diagonal. C is between B and O.", "ocr_zh": "图上有一以O为圆心，AB为直径的半圆，F在该半圆上，F靠近B，连接AF，BF，D在AF上，以BD为对角线作正方形BCDE，C在B和O之间。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_420.jpg" }, { "index": 421, "ocr_en": "On the diagram, there is a semicircle with O as the center and AB as the diameter. F is on this semicircle, and F is close to B. It connects AF, BF, and D on AF, and forms a square BCDE with BD as the diagonal. C is between B and O. Fold the inferior arc AF of the circle along AF, and the folded inferior arc intersects with CD at G. O is inside the arch after folding.", "ocr_zh": "图上有一以O为圆心，AB为直径的半圆，F在该半圆上，F靠近B，连接AF，BF，D在AF上，以BD为对角线作正方形BCDE，C在B和O之间，沿AF翻折该圆的劣弧AF，翻折后的劣弧与CD交于G，O在翻折后的弓形内部。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_421.jpg" }, { "index": 422, "ocr_en": "There is a circle in the figure with O as the center and OE as the radius. AB is the chord of the circle, and O and E are on both sides of AB.", "ocr_zh": "图中有一以O为圆心，OE为半径的圆，AB是该圆的弦，O和E在AB两侧。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_422.jpg" }, { "index": 423, "ocr_en": "In the figure, there is a circle with O as the center and OE as the radius. AB is the chord of the circle, and O and E are on both sides of AB. Fold the inferior arc AB of the circle along AB, and after folding, E coincides with O.", "ocr_zh": "图中有一以O为圆心，OE为半径的圆，AB是该圆的弦，O和E在AB两侧，沿AB翻折该圆的劣弧AB，翻折后E恰好与O重合。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_423.jpg" }, { "index": 424, "ocr_en": "In the figure, there is a circle with O as the center and OE as the radius. AB is the chord of the circle, and O and E are on both sides of AB. Fold the inferior arc AB of the circle along AB, and E happens to coincide with O. Make a tangent to the inferior arc OA after folding, intersecting with the circle at C and D, where C is close to A.", "ocr_zh": "图中有一以O为圆心，OE为半径的圆，AB是该圆的弦，O和E在AB两侧，沿AB翻折该圆的劣弧AB，翻折后E恰好与O重合，在翻折后的劣弧OA上去一点作该劣弧的切线，与该圆交于C和D，其中，C靠近A。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_424.jpg" }, { "index": 425, "ocr_en": "Left image: In acute triangle ABC, P is an internal point that connects PA, PB, PC, and Q outside triangle ABC and near AB, serving as auxiliary lines QA, QB, QP.\nMiddle image: Square ABCD with one point P inside and one point Q outside, connecting PA, PB, PC, QB, QC as auxiliary line PQ.\nRight figure: In diamond ABCD, connect BD, E on AD, F on CD, connect BE, EF, BF.", "ocr_zh": "左图：锐角三角形ABC中，P是内部一点，连接PA，PB，PC，Q在三角形ABC外且靠近AB，作辅助线QA，QB，QP。\n中图：正方形ABCD，其内有一点P，其外有一点Q，连接PA，PB，PC，QB，QC，作辅助线PQ。\n右图：菱形ABCD中，连接BD，E在AD上，F在CD上，连接BE，EF，BF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_425.jpg" }, { "index": 426, "ocr_en": "In trapezoidal ABDO, AB is parallel to OD, E is on BD, OB, AE, OE, C are connected on the AE extension line, OC, CD are connected, AE and OB intersect at P, a point Q is taken on EP to connect OQ, OC and ED intersect at M, a point N is taken on MD to connect ON.", "ocr_zh": "梯形ABDO中，AB平行于OD，E在BD上，连接OB，AE，OE，C在AE延长线上，连接OC，CD，设AE和OB交于P，在EP上取一点Q连接OQ，设OC与ED交于M点，在MD上取一点N，连接ON。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_426.jpg" }, { "index": 427, "ocr_en": "In the trapezoidal ABDO, AB is parallel to OD, E is on BD, OB, AE, OE, C are connected on the AE extension line, and OC, CD are connected.", "ocr_zh": "梯形ABDO中，AB平行于OD，E在BD上，连接OB，AE，OE，C在AE延长线上，连接OC，CD。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_427.jpg" }, { "index": 428, "ocr_en": "There is a point D outside the acute triangle ABC, which is close to BC. Connect DA, DB, DC, and take a point E on DC as the auxiliary line AE. Take a point F on the extension line of DB and connect it to AF.", "ocr_zh": "锐角三角形ABC外有一点D，D靠近BC，连接DA，DB，DC，DC上取一点E，作辅助线AE，DB延长线上取一点F，连接AF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_428.jpg" }, { "index": 429, "ocr_en": "There is a point D outside the acute triangle ABC, which is close to BC. Take a point E on the extension line connecting DA, DB, DC, and DC as the auxiliary line AE.", "ocr_zh": "锐角三角形ABC外有一点D，D靠近BC。连接DA，DB，DC，DC延长线上取一点E，作辅助线AE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_429.jpg" }, { "index": 430, "ocr_en": "There is a point D outside the acute triangle ABC, which is close to BC. Connect DA, DB, DC, and take a point E inside the triangle ABC on AD as the auxiliary line BE.", "ocr_zh": "锐角三角形ABC外有一点D，D靠近BC。连接DA，DB，DC，在AD上取一个在三角形ABC内部的点E，作辅助线BE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_430.jpg" }, { "index": 431, "ocr_en": "There are two points G and F inside the squares ABCD, and one point E outside. E is close to AB, and D is collinear with G and F, connecting DG, GF, CG, BF, EF, EB, EG.", "ocr_zh": "正方形ABCD内部有两点G和F，外部有一点E，E靠近AB，D和G和F共线，连接DG，GF，CG，BF，EF，EB，EG。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_431.jpg" }, { "index": 432, "ocr_en": "There are two points G and F inside the square ABCD, and one point E outside. E is close to AB, and D is collinear with G and F. It connects DG, GF, CG, BF, EF, EB, EG as auxiliary lines CE, and extends EG to H. H is outside the square ABCD and serves as auxiliary lines DH and CH.", "ocr_zh": "正方形ABCD内部有两点G和F，外部有一点E，E靠近AB，D和G和F共线，连接DG，GF，CG，BF，EF，EB，EG，作辅助线CE，延长EG至H，H在正方形ABCD外部，作辅助线DH，CH。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_432.jpg" }, { "index": 433, "ocr_en": "Left image: In the obtuse triangle BDE, the angle BDE is obtuse, A is outside the triangle BDE, A is close to BD, C is a point on BD, P is a point on BE, connecting AB, AC, AP, CE, DP.\nRight figure: In the obtuse triangle BDE, the angle BDE is obtuse, A is outside the triangle BDE, A is close to BD, C is a point on BD, P is a point on BE, connecting AB, AC, AP, CE, DP, extending AP to point G as auxiliary lines DG, EG.", "ocr_zh": "左图：钝角三角形BDE中，角BDE是钝角，A在三角形BDE外部，A靠近BD，C是BD上一点，P是BE上一点，连接AB，AC，AP，CE，DP。\n右图：钝角三角形BDE中，角BDE是钝角，A在三角形BDE外部，A靠近BD，C是BD上一点，P是BE上一点，连接AB，AC，AP，CE，DP，延长AP于点G，作辅助线DG，EG。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_433.jpg" }, { "index": 434, "ocr_en": "Left figure: In trapezoid ABCD, AD is parallel to BC, F is on DC, E is on BC, connected to EF, M is on EF, connected to BM, DM.\nRight figure: In trapezoid ABCD, AD is parallel to BC, F is on DC, E is on BC, connected to EF, M is on EF, connected to BM and DM, serving as auxiliary lines BD, extending DM to a point G outside the trapezoid, serving as auxiliary lines GB and GE.", "ocr_zh": "左图：梯形ABCD中，AD平行于BC，F在DC上，E在BC上，连接EF，M在EF上，连接BM，DM。\n右图：梯形ABCD中，AD平行于BC，F在DC上，E在BC上，连接EF，M在EF上，连接BM，DM，作辅助线BD，延长DM至梯形外一点G，作辅助线GB，GE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_434.jpg" }, { "index": 435, "ocr_en": "In the obtuse triangle BDE, the angle BDE is obtuse, A is outside the triangle BDE, A is close to BD, C is a point on BD, P is a point on BE, connecting AB, AC, AP, CE, DP.", "ocr_zh": "钝角三角形BDE中，角BDE是钝角，A在三角形BDE外部，A靠近BD，C是BD上一点，P是BE上一点，连接AB，AC，AP，CE，DP。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_435.jpg" }, { "index": 436, "ocr_en": "In quadrilateral ABCD, point E is on BC, point F is on CD, and point M is on EF. Connect BM and DM. Draw auxiliary line BD. There is an auxiliary point outside quadrilateral ABCD; connect this point with B, M, and E. The lengths of AB and BE are equal, the lengths of AD and DF are equal, and the lengths of EM and MF are equal.", "ocr_zh": "在四边形ABCD中，点E在BC上，点F在CD上，点M在EF上，连接BM、DM，辅助线BD，四边形ABCD外有一辅助点并连接该点与点B、点M、点E，AB=BE，AD=DF，EM=MF", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_436.jpg" }, { "index": 437, "ocr_en": "In triangle BDE and triangle ABC, point P is the midpoint of BE. Point C is on BD, and point P is on BE. There are line segments AP, DP, and CE. Draw auxiliary line AD. Extend AP by the same length to get a point, and connect this point with D and E to construct auxiliary lines. The lengths of AB and AC are equal, and the lengths of CD and DE are equal.", "ocr_zh": "有三角形BDE和三角形ABC，点P为BE中点，点C在BD上，点P在BE上，连接AP、DP、CE，辅助线AD，延长AP相同长度得到一点并连接点D、点E构建辅助线，AB=AC，CD=DE", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_437.jpg" }, { "index": 438, "ocr_en": "In quadrilateral ABCD, point E is on BC, point F is on CD. Connect EF. Point M is on EF. Connect DM and BM. Point M is the midpoint of EF. AB = BE, AD = DF.", "ocr_zh": "有四边形ABCD，点E在BC上，点F在CD上，连接EF，点M在EF上，连接DM、BM，点M为EF中点，AB=BE，AD=DF", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_438.jpg" }, { "index": 439, "ocr_en": "There are two equilateral triangles ABC and ADE, with point D located on BC. There exist line segments CE and DE. A point on AB is connected to point D to form an auxiliary line.", "ocr_zh": "有等边三角形ABC、等边三角形ADE，点D在BC上，连接CE、DE，AB上有一点与点D相连成辅助线", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_439.jpg" }, { "index": 440, "ocr_en": "In triangle ABC, rotate it counterclockwise around point A to get triangle AB'C'. B'C' intersects BC at E and AC at F. Connect AE.", "ocr_zh": "有三角形ABC，绕A点逆时针旋转得三角形AB'C',B'C'与BC交于E，与AC交于F，连接AE", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_440.jpg" }, { "index": 441, "ocr_en": "In triangle ABC, there are equilateral triangles ABD and ACE. Connect CD and BE, which intersect at point O. Connect OA.", "ocr_zh": "有三角形ABC、正三角形ABD和正三角形ACE，连接CD、BE相较于点O，连接OA", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_441.jpg" }, { "index": 442, "ocr_en": "In the left diagram, there is triangle ABC with point D on BC. Outside triangle ABC, there is a point E. Connect DE, AE, CE, and AD. There is another auxiliary point outside triangle ABC, and it is connected to points A and D. ∠BAC and ∠ADE are equal in measure.\n\nIn the middle diagram, there is triangle ABC with point D on BC. Outside triangle ABC, there is a point E. Connect ED, EC, AD, and AE. AB = AC, DA = DE, and ∠BAC is equal to ∠ADE. There is an auxiliary point on AC connected to point D.\n\nIn the right diagram, there is an obtuse triangle ABC and an obtuse triangle ADE. The remaining elements are the same as in the left diagram.\n", "ocr_zh": "左图有三角形ABC，点D在BC上，三角形ABC外有点E，连接DE、AE、CE、AD，三角形ABC外另有一辅助点并连接该点与点A、点D，角BAC和角ADE大小相等；中图有三角形ABC，点D在BC上，三角形ABC外有一点E，连接ED、EC、AD、AE，AB=AC，DA=DE，角BAC与角ADE大小相等，AC上有一辅助点并连接该点与点D；右图有钝角三角形ABC，钝角三角形ADE，其余要素与左图相同", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_442.jpg" }, { "index": 443, "ocr_en": "In the left diagram, there are isosceles triangles ABC and DEF. Point E is on AB. There is an auxiliary point on AF, and it is connected to point E. Connect ED, EF, FD, and AD. AB = AC, DE = DF, and ∠BAC = ∠EDF.\n\nIn the right diagram, AB = AD. There is an auxiliary point on AF, and it is connected to point D.\n", "ocr_zh": "左图有等腰三角形ABC、等腰三角形DEF，点E在AB上，点AF上有一辅助点并连接该点与点E，连接ED、EF、FD、AD，AB=AC,DE=DF，角BAC与角EDF相等；右图另有AB=AD，AF上有一辅助点并连接该点与点D", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_443.jpg" }, { "index": 444, "ocr_en": "There is a right triangle ABC, with point D on AB and point E on AC. D is the midpoint of AB, E is the midpoint of AC, and angle A is a right angle.", "ocr_zh": "有直角三角形ABC，点D在AB上，点E在AC上，D为AB中点，E是AC中点，角A为直角", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_444.jpg" }, { "index": 445, "ocr_en": "Given isosceles right triangle ABC and right triangle CDE. D is on AB. M is the midpoint of EB. Angle A is 90 degrees. Angle CDE is 90 degrees. Angle ADE is 15 degrees. Connect ED, EB, and auxiliary line AM. There is an auxiliary point N. Connect ND, NA, NB.", "ocr_zh": "有等腰直角三角形ABC和直角三角形CDE，D在AB上，M为EB中点，角A为90度，角CDE为90度，角ADE为15度，连接ED、EB、辅助线AM，有辅助点N，连接ND、NA、NB", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_445.jpg" }, { "index": 446, "ocr_en": "In right triangle ABC, there is a point O inside. Connect AO, BO, CO. There is an auxiliary point outside triangle ABC. Connect this point to points A, O, B. Angle AOC is equal to angle BOC. AO is perpendicular to BO.", "ocr_zh": "有直角三角形ABC，其内有一点O，连接AO、BO、CO，三角形ABC外有一辅助点，连接该点与点A、点O、点B，角AOC和角BOC大小相等，AO与BO垂直于点O", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_446.jpg" }, { "index": 447, "ocr_en": "In right triangle ABC, there is a point D outside triangle ABC. Connect AD, BD, CD. There is an auxiliary point, connect this point to points A, B, D. AB = AC.", "ocr_zh": "有直角三角形ABC，三角形ABC外有一点D，连接AD、BD、CD，另有一辅助点并连接该点与点A、点B、点D，AB=AC", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_447.jpg" }, { "index": 448, "ocr_en": "In right triangle ABC, point D is on AC. O is the midpoint of BC. Connect BD. Point E is on BD. Connect AE, EO, auxiliary line AO. There is another auxiliary point on BD, connect this point to point O. Angle BAC is 90 degrees.", "ocr_zh": "有直角三角形ABC，点D在AC上，O为BC的中点，连接BD，点E在BD上，连接AE、EO、辅助线AO，BD上有另一辅助点并连接该点与点O，角BAC为90度", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_448.jpg" }, { "index": 449, "ocr_en": "In right triangle ABC, D is a point outside triangle ABC. Connect AD, BD, CD. There is another auxiliary point connected to points A and D. AB = AC. Angle BAC is 90 degrees.", "ocr_zh": "有直角三角形ABC，D为三角形ABC外一点，连接AD、BD、CD，另有一辅助点并连接该点与点A和点D，AB=AC，角BAC为90度", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_449.jpg" }, { "index": 450, "ocr_en": "Given an equilateral triangle ABC. Point D is on BC. There are two points E and F outside triangle ABC. There is an auxiliary point inside triangle ABF. Connect this auxiliary point to points F and B. Point E is on the line connecting the auxiliary point and B. Connect AF, DE, EF, BF, AD.", "ocr_zh": "有等边三角形ABC，点D在BC上，三角形ABC外有两个点E与点F，三角形ABF内有一辅助点并连接该点与点F、点B，点E在辅助点与B的连线上，连接AF、DE、EF、BF、AD", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_450.jpg" }, { "index": 451, "ocr_en": "Given isosceles right triangle ABC and isosceles right triangle CPD. Point P is on AB. D is a point inside triangle ABC. Connect CP, CD, AD, DP. There is an auxiliary point outside triangle ABC. Connect this point to points C, D, A.", "ocr_zh": "有等腰直角三角形ABC、等腰直角三角形CPD，点P在AB上，D为三角形ABC内一点，连接CP、CD、AD、DP，三角形ABC外有一辅助点，连接该点与点C、点D、点A", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_451.jpg" }, { "index": 452, "ocr_en": "Given triangle ABC, side AB is rotated clockwise around point A to obtain AD. Side AC is rotated counter-clockwise around point A to obtain AE. Connect BD, CE, BE, CD, with CD and BE intersecting at point O inside triangle ABC. Connect AO.", "ocr_zh": "有三角形ABC，边AB绕点A顺时针旋转得到AD， 边AC绕点A逆时针旋转得到AE，连接BD、CE、BE、CD与BE相交于点O于三角形ABC内，连接AO", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_452.jpg" }, { "index": 453, "ocr_en": "Given equilateral triangle ABC and equilateral triangle DEC. CA is rotated 30 degrees clockwise around point C to point in the direction of CE. F is the midpoint of AB. Connect BD intersecting AC at point G. Connect BE, FG.", "ocr_zh": "有等边三角形ABC、等边三角形DEC，CA绕点C顺时针旋转30度指向CE的方向，F为AB中点，连接BD交AC于点G，连接BE、FG", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_453.jpg" }, { "index": 454, "ocr_en": "Given equilateral triangle ABC and equilateral triangle DEC. Point E is on AC. Connect AD, BE.", "ocr_zh": "有等边三角形ABC和等边三角形DEC共点C，点E在AC上，连接AD、BE", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_454.jpg" }, { "index": 455, "ocr_en": "Given square ABCD and square BEFG. Square BEFG rotates clockwise around point B, forming a circular locus with its side length as the radius. Connect AG, CE, DG.", "ocr_zh": "有正方形ABCD、正方形BEFG，正方形BEFG绕点B顺时针旋转以其边长为半径形成轨迹圆，连接AG、CE、DG", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_455.jpg" }, { "index": 456, "ocr_en": "Given square ABCD and square BEFG. Connect AG, CE.", "ocr_zh": "有正方形ABCD、正方形BEFG共点B，连接AG、CE", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_456.jpg" }, { "index": 457, "ocr_en": "In triangle ABC, points D and E are on side BC. Connect AD, AE. AB = AC.", "ocr_zh": "在三角形ABC中，点D、E在边BC上，连接AD、AE，AB=AC", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_457.jpg" }, { "index": 458, "ocr_en": "In right triangle ABC, D is on AB. ED intersects AC at point F. Connect EA, EC.", "ocr_zh": "在直角三角形ABC中，D在AB上，ED交AC于点F，连接EA、EC，角ABC为90度", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_458.jpg" }, { "index": 459, "ocr_en": "Given triangle CNB, triangle CNB is rotated clockwise around point C to obtain triangle CDA. Points M and N are on AB. Connect CM, CN. Auxiliary line DM.", "ocr_zh": "有三角形CNB，三角形CNB顺时针绕点C旋转至A点位于BN延长线上得到三角形CDA，点M、点N在AB上，连接CM、CN，辅助线DM", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_459.jpg" }, { "index": 460, "ocr_en": "In quadrilateral ABCD, point E is on BC, and point F is on CD. Connect EF. Point H is on EF. Connect AE, AH, AF.", "ocr_zh": "有长方形ABCD，点E在BC上，点F在CD上，连接EF，点H在EF上，连接AE、AH、AF", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_460.jpg" }, { "index": 461, "ocr_en": "In quadrilateral ABCD, point E is on BC, and point F is on CD. Connect EF. Point H is on EF. Connect AE, AH, AF. Extend CB to point G to construct auxiliary line CG. Connect auxiliary line AG.", "ocr_zh": "有长方形ABCD，点E在BC上，点F在CD上，连接EF，点H在EF上，连接AE、AH、AF，延长CB到点G构建辅助线CG，连接辅助线AG", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_461.jpg" }, { "index": 462, "ocr_en": "In the left diagram, there is triangle CNB. Triangle CNB is rotated clockwise about point C so that point A lies on the extension of BN, forming triangle CDA. Points M and N are on AB. Connect CM and CN, and draw auxiliary line DM.\n\nIn the middle diagram, there is rectangle ABCD. Point E is on BC, and point F is on CD. Connect EF. Point H is on EF. Connect AE, AH, and AF.\n\nIn the right diagram, there is rectangle ABCD. Point E is on BC, and point F is on CD. Connect EF. Point H is on EF. Connect AE, AH, and AF. Extend CB to point G and draw auxiliary line CG. Connect auxiliary line AG.\n", "ocr_zh": "左图有三角形CNB，三角形CNB顺时针绕点C旋转至A点位于BN延长线上得到三角形CDA，点M、点N在AB上，连接CM、CN，辅助线DM；中图有长方形ABCD，点E在BC上，点F在CD上，连接EF，点H在EF上，连接AE、AH、AF；右图有长方形ABCD，点E在BC上，点F在CD上，连接EF，点H在EF上，连接AE、AH、AF，延长CB到点G构建辅助线CG，连接辅助线AG", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_462.jpg" }, { "index": 463, "ocr_en": "In square ABCD, connect BD. Point P is on CD. Point E is on BD. Connect EP, BP. O is the midpoint of BP. Extend CO to intersect BD at point F.", "ocr_zh": "在正方形ABCD中，连接BD，点P在CD中，点E在BD上，连接EP、BP，O为BP中点，连接CO延长交BD于点F", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_463.jpg" }, { "index": 464, "ocr_en": "In square ABCD, connect BD. Point P is on CD. Point E is on BD. Connect EP, BP. O is the midpoint of BP. Extend CO to intersect BD at point F. Connect EO.", "ocr_zh": "在正方形ABCD中，连接BD，点P在CD中，点E在BD上，连接EP、BP，O为BP中点，连接CO延长交BD于点F，连接EO", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_464.jpg" }, { "index": 465, "ocr_en": "In triangle ABC, point E is on BC. Connect AE. Ray BM is inside angle ABC and intersects AE at point F. Connect CF.", "ocr_zh": "在三角形ABC中，点E在BC上，连接AE，射线BM在角ABC内部交AE于点F，连接CF", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_465.jpg" }, { "index": 466, "ocr_en": "In triangle ABC, point E is on BC. Connect AE. Ray BM is inside angle ABC, intersecting AE at point F and AC at point G. Ray BN is inside angle ABC, intersecting AE at point D and AC at point H. Connect CF and CD. Extend AE to an auxiliary point outside triangle ABC and connect this point to B and C.", "ocr_zh": "在三角形ABC中，点E在BC上，连接AE，射线BM在角ABC内部交AE于点F、交AC于点G，射线BN在角ABC内部交AE于点D、交AC于点H，连接CF、CD，延长AE至三角形ABC外一辅助点并连接该点与点B、点C", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_466.jpg" }, { "index": 467, "ocr_en": "A cylinder without bases has AB as a height on its lateral surface.", "ocr_zh": "有无底圆柱体，AB为侧面上的一条高", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_467.jpg" }, { "index": 468, "ocr_en": "The lateral surface development of a cone in a plane appears as a sector OAA'. Q is the vertex of the cone, and QA and QA' are the generatrices of the cone. The curve connecting point A and point A' is the arc of the sector, and the dashed line connecting point A and point A' is a chord of this sector.", "ocr_zh": "有圆锥的侧面展开图在平面上呈现为一个扇形OAA',Q为圆锥顶点，有圆锥母线QA和QA'，连接A点与A'点的曲线是扇形的弧，连接A点和A'点的虚线是该扇形的一条弦", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_468.jpg" }, { "index": 469, "ocr_en": "In the unfolded diagram of a rectangular prism's side and top faces, point A is the bottom-left corner of the front face, and point B is the top-right corner of the top face. Connect point A and point B in the unfolded diagram.", "ocr_zh": "有长方形其为一长方体侧面和顶面的展开图，点A为正面左下角点，点B为顶面的右上角点，在展开图中连接点A、点B", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_469.jpg" }, { "index": 470, "ocr_en": "In a rectangular prism, point A is the bottom-left corner of the front face, and point B is the top-right corner of the top face.", "ocr_zh": "有长方体，点A为长方体正面左下角点，点B为长方体顶面的右上角", "class": "Solid Geometry", "difficulty": 1, "image_path": "fig_470.jpg" }, { "index": 471, "ocr_en": "In the unfolded diagram of a rectangular prism's front and right faces, point B is the top-right corner of the right face, and point A is the bottom-left corner of the front face. Connect point A and point B in the unfolded diagram.", "ocr_zh": "有长方形其为一长方体正面和右侧面的展开图，点B为右侧面右上角点，点A为正面的左下角点，在展开图中连接点A、点B", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_471.jpg" }, { "index": 472, "ocr_en": "As shown in the figure, there is a cylinder, where AB and CD are the heights.", "ocr_zh": "如图有一圆柱体，AB与CD分别为高", "class": "Solid Geometry", "difficulty": 1, "image_path": "fig_472.jpg" }, { "index": 473, "ocr_en": "In a cylinder, A, B, C, D are points on one of its cross-sections. The trace connecting A and B is the diagonal of its lateral surface development.", "ocr_zh": "有圆柱体，A、B、C、D是其一横截面，连接A、B轨迹为其侧面展开图的对角线", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_473.jpg" }, { "index": 474, "ocr_en": "There is a rectangle ABA'B'. Point C is on BB', point D is on AA'. Connect AC.", "ocr_zh": "有长方形ABA'B', 点C在BB'上，点D在AA'上，连接AC", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_474.jpg" }, { "index": 475, "ocr_en": "There is a rectangle. Point B is on one side of the rectangle, and point A is inside the rectangle. Point A is reflected across the side perpendicular to the side containing point B to obtain point A'. Connect A'B, which intersects the side of reflection at a point. Connect an auxiliary line from this point to A. Also, connect auxiliary line AA'.", "ocr_zh": "有一长方形，点B在长方形的一条边上，点A在长方形内，点A沿与点B所在边垂直的边做对称得到点A'，连接A'B与对称的边交于一点并连接辅助线该点与点A，连接辅助线AA'", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_475.jpg" }, { "index": 476, "ocr_en": "In a cylinder, points A and B are two distinct, opposite points on its lateral surface.", "ocr_zh": "有圆柱体，点A、点B为圆柱体侧面不重合的相对的两个点", "class": "Solid Geometry", "difficulty": 1, "image_path": "fig_476.jpg" }, { "index": 477, "ocr_en": "Given a cone OAB with O as its vertex. In its lateral surface development, connect chord AB. The path obtained after folding intersects OB at point C.", "ocr_zh": "有圆锥体OAB，0为顶点，在其侧面展开图中连接弦AB折叠后得到轨迹并与OB相交与点C", "class": "Solid Geometry", "difficulty": 1, "image_path": "fig_477.jpg" }, { "index": 478, "ocr_en": "Top Left Figure: In the lateral surface development of a cone OA(A), connect auxiliary generatrix OB. Point C is on OB. Connect AC. There is a point on OA; connect this point to C and B.\n\nTop Right Figure: In the lateral surface development of a cone OA(A), connect auxiliary generatrix OB. Point C is on OB. Connect AC. There is a point on OA; connect this point to C, extending to intersect arc A(A) at a point.\n\nBottom Left Figure: In the lateral surface development of a cone OA(A), connect auxiliary generatrix OB. Point C is on OB. Connect AC. There is a point on OA; connect this point to B. There is a point on O(A); connect this point to C.\n\nBottom Right Figure: In the lateral surface development of a cone OA(A), connect auxiliary generatrix OB. Point C is on OB. Connect C(A). There is a point on OA; connect this point to B. There is a point on O(A); connect this point to C.", "ocr_zh": "左上图为圆锥体侧面展开图OA(A),连接辅助母线OB，点C在OB上，连接AC，OA上有一点并连接该点与点C、点B；右上图为圆锥体侧面展开图OA(A)，连接辅助母线OB，点C在OB上，连接AC，OA上有一点并连接该点与点C延长交弧A(A)于一点；左下图为圆锥体侧面展开图OA(A)，连接辅助母线OB，点C在OB上，连接AC，OA上有一点并连接该点于点B，O(A)上有一点并连接该点与点C；右下图为圆锥体侧面展开图OA(A)，连接辅助母线OB，点C在OB上，连接C(A),OA上有一点并连接该点与点B，O(A)上有一点并连接该点与点C", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_478.jpg" }, { "index": 479, "ocr_en": "Given a cube. Point A is the bottom-left corner of its front face, and point B is the top-right corner of its back face. Unfold the front face, right face, and back face. Connect AB in the unfolded diagram; it intersects the right edge of the front face at point C.", "ocr_zh": "有一正方体，点A为正面左下角点，点B为背面右上角点，展开正面、右侧面、后面并连接AB与正面右边的棱交于点C", "class": "Solid Geometry", "difficulty": 1, "image_path": "fig_479.jpg" }, { "index": 480, "ocr_en": "There is a net of a cube. Point B is the top right corner of the front face, and point A is the bottom left corner of the back face. Connect AB, intersecting the right edge of the back face at point C.", "ocr_zh": "有正方体展开图，点B为正面右上角点，点A为后面左下角点，连接AB交后面右边棱于点C", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_480.jpg" }, { "index": 481, "ocr_en": "There is a cube. Point P is the bottom left corner of the front face, point A is the top right corner of the front face, and point Q is on an edge of the top face.", "ocr_zh": "有正方体，点P为正面左下角点，点A为正面右上角点，点Q在上底面棱上", "class": "Solid Geometry", "difficulty": 1, "image_path": "fig_481.jpg" }, { "index": 482, "ocr_en": "As shown in the figure, there are three steps. Each step is a rectangular prism with a length of 5, a width of 3, and a height of 1. Points A, B, and C are endpoints.", "ocr_zh": "如图为三级台阶，每级台阶为一长为5、宽为3、高为1的长方体，点A、B、C为端点", "class": "Solid Geometry", "difficulty": 1, "image_path": "fig_482.jpg" }, { "index": 483, "ocr_en": "In square ABCD, point E is on BC, and point G is on CD. Connect BD and EG. Connect AE, intersecting BD at point Q. Connect AG, intersecting BD at point P. EP is perpendicular to AG.", "ocr_zh": "有正方形ABCD，点E在BC上，点G在CD上，连接BD、EG，连接AE交BD于点Q，连接AG交BD于点P，连接EP与AG相互垂直于点P", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_483.jpg" }, { "index": 484, "ocr_en": "There is a square ABCD. Take a point G on side CD and connect AG, which intersects BD at point P. Draw PE perpendicular to AG at point P, and let PE intersect BC at point E. Connect BD and EG. Connect AE, which intersects BD at point Q. Connect AG, which intersects BD at point P. Connect EP, which is perpendicular to AG at point P. Connect PF, which is perpendicular to BC at point F.\n", "ocr_zh": "有正方形ABCD，CD边上取一点G，连接AG交BD于P点，过点P作PE垂直AG于P，PE与BC交于点E，连接BD、EG，连接AE交BD于点Q，连接AG交BD于点P，连接EP与AG相互垂直于点P，连接PF与BC相互垂直于点F", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_484.jpg" }, { "index": 485, "ocr_en": "In square ABCD, point F is on AD, point E is on AB, and point M is on CD. Connect EF and FM. EF is perpendicular to FM.", "ocr_zh": "有正方形ABCD，点F在AD上，点E在AB上，点M在CD上，连接EF、FM，EF垂直于FM于点F", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_485.jpg" }, { "index": 486, "ocr_en": "In square ABCD, point F is on AD, and point E is on CD. Connect FB, EB, EF, and AC. AC intersects BF at point G. Point H is on BE. Connect EG. GH is perpendicular to BE.", "ocr_zh": "有正方形ABCD，点F在AD上，点E在CD上，连接FB、EB、EF、AC，AC与BF相交于点G，点H在BE上，连接EG，GH垂直于BE于点H", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_486.jpg" }, { "index": 487, "ocr_en": "In square ABCD, M is a moving point on side AD (not coinciding with A or D). Construct an isosceles trapezoid BMNC, where BM is parallel to CN, and MN intersects CD at point P.", "ocr_zh": "有正方形ABCD中，M在边AD上（不与A、D重合），作等腰梯形BMNC，有BM与CN相互平行，MN交CD于点P", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_487.jpg" }, { "index": 488, "ocr_en": "In square ABCD, point H is on AD, point G is on AB, point F is on CD, and point E is on BC. Connect EH and GF, which intersect at point P, parallel to the sides. Connect AF and AE.", "ocr_zh": "有正方形ABCD，点H在AD上、点G在AB上、点F在CD上、点E在BC上，连接EH、GF分别与边平行相较于点P，连接AF、AE", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_488.jpg" }, { "index": 489, "ocr_en": "In square ABCD, E and F are two interior points. Connect AF, AE, BE, CE, CF, and FD. The measure of angle EAF is equal to the measure of angle ECF.", "ocr_zh": "有正方形ABCD，E、F为其内部两点，连接AF、AE、BE、CE、CF、FD，角EAF与角ECF大小相等", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_489.jpg" }, { "index": 490, "ocr_en": "The left figure shows square ABCD, with points E, F, G, and H on sides AB, CD, BC, and AD respectively. Connect HG and EF, which intersect at point P. The right figure shows square ABCD, with points E, F, G, and H on sides AB, CD, BC, and AD respectively. Connect HG, EF, auxiliary line HN, and auxiliary line MF.", "ocr_zh": "左图有正方形ABCD，E、F、G、H分别为AB、CD、BC、AD边上的点，连接HG、EF相交于点P；右图有正方形ABCD，，E、F、G、H分别为AB、CD、BC、AD边上的点，连接HG、EF、辅助线HN、辅助线MF", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_490.jpg" }, { "index": 491, "ocr_en": "In square ABCD, E, F, G, H are points on sides AB, CD, BC, AD respectively. H' is on AD, G' is on BC. Connect EF, HG, and auxiliary line H'G'.", "ocr_zh": "有正方形ABCD，E、F、G、H分别为AB、CD、BC、AD边上的点，H'在AD上、G'在BC上，连接EF、HG、辅助线H'G'", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_491.jpg" }, { "index": 492, "ocr_en": "In square ABCD, point N is on AD, and point M is on CD. Connect AM and BN, which intersect at point P, and BN is perpendicular to AM.", "ocr_zh": "有正方形ABCD，点N在AD上、点M在CD上，连接AM、BN相交于点P且BN垂直于AM于点P", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_492.jpg" }, { "index": 493, "ocr_en": "In square ABCD, E is the midpoint of CD. F is on AD, and G is on BC. Connect FG. Fold square ABCD along FG so that point A coincides with point E. Let the new position of point B be B'. Connect auxiliary lines FE, EB', and GB'.", "ocr_zh": "有正方形ABCD，E为CD中点，F在AD上、G在BC上，连接FG，将正方形ABCD的点A绕FG折叠至点E，此时点B所在位置记为B'，连接辅助线FE、辅助线EB'、辅助线GB'", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_493.jpg" }, { "index": 494, "ocr_en": "In rectangle ABCD, E and H are on AD, and G and F are on BC. Connect EF and HG, and EF is perpendicular to HG. The measure of angle EFG is $\\alpha$.", "ocr_zh": "有矩形ABCD，E、H在AD上，G、F在BC上，连接EF、HG且EF与HG相互垂直，角EFG大小为α", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_494.jpg" }, { "index": 495, "ocr_en": "In rectangle ABCD, E, F, G, H are points on sides AD, BC, AB, CD respectively. Connect HG and EF, and EF is perpendicular to HG. Connect auxiliary line MF and auxiliary line GN.", "ocr_zh": "有矩形ABCD中，E、F、G、H分别为AD、BC、AB、CD边上的点，连接HG、EF且EF与HG相互垂直，连接辅助线MF、辅助线GN", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_495.jpg" }, { "index": 496, "ocr_en": "In rectangle ABCD, point E is on AD, point H is on CD, point F is on BC, and point G is on AB. Connect EF and HG, which intersect at one point.", "ocr_zh": "有矩形ABCD，点E在AD上、点H在CD上、点F在BC上、点G在AB上，连接EF、HG相交于一点", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_496.jpg" }, { "index": 497, "ocr_en": "In rectangle ABCD, point E is on AD. Connect CE and BD, and they are perpendicular to each other.", "ocr_zh": "有矩形ABCD，点E在AD上，连接CE、BD且两者相互垂直", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_497.jpg" }, { "index": 498, "ocr_en": "In a Cartesian coordinate system, a line intersects the x-axis and y-axis at points B and A respectively, forming triangle AOB. Fold triangle AOB along AB so that point O falls on point D. Connect auxiliary lines AD and BD.", "ocr_zh": "在平面直角坐标系中有直线与x轴、y轴分别交于B、A两点构成三角形AOB，将三角形AOB沿着AB翻折使得点O落在点D上，连接辅助线AD、辅助线BD", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_498.jpg" }, { "index": 499, "ocr_en": "In a Cartesian coordinate system, a line intersects the x-axis and y-axis at points B and A respectively, forming triangle AOB. Fold triangle AOB along AB so that point O falls on point D. Connect auxiliary lines AD and BD. An auxiliary line segment EF passes through point D, is parallel to OB, and its endpoint E is on the y-axis, with EF equal to OB. Connect auxiliary line OD, which intersects line AB at point G, and AB is perpendicular to OD.", "ocr_zh": "在平面直角坐标系中有直线与x轴、y轴分别交于B、A两点构成三角形AOB，将三角形AOB沿着AB翻折使得点O落在点D上，连接辅助线AD、辅助线BD，过点D有辅助线段EF与OB平行且端点E在y轴，EF与OB相等，连接辅助线OD与直线AB相交于点G且AB与OD相互垂直于点G", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_499.jpg" }, { "index": 500, "ocr_en": "In rectangle ABCD, point M is on AD, and point N is on BC. Connect MN.", "ocr_zh": "有矩形ABCD，点M在AD上、点N在BC上，连接MN", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_500.jpg" }, { "index": 501, "ocr_en": "In rectangle ACBH, point G is on AH, and point D is on AC. Connect BD and CE, which intersect at point F, and BD is perpendicular to CE. Extend CE to form an auxiliary line intersecting AH at point G.", "ocr_zh": "有矩形ACBH，点G在AH上、点D在AC上，连接BD、CE相交于点F且BD与CE相互垂直于点F，延长CE构建辅助线交AH于点G", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_501.jpg" }, { "index": 502, "ocr_en": "In right triangle ACB, angle ACB is a right angle. Point D is on AC, and point E is on AB. Connect BD and CE, which intersect at point F.", "ocr_zh": "有直角三角形ACB，角ACB为直角，点D在AC上、点E在AB上，连接BD、CE相交于点F", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_502.jpg" }, { "index": 503, "ocr_en": "In right triangle ABC, angle ABC is a right angle. Point D is on BC, and point F is on AC. Connect BF and AD, which intersect at point E, and BF is perpendicular to AD.", "ocr_zh": "有直角三角形ABC，角ABC为直角，点D在BC上、点F在AC上，连接BF、AD相交于点E且BF与AD相互垂直于点E", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_503.jpg" }, { "index": 504, "ocr_en": "In rectangle ACBH, point G is on AH, and point D is on AC. Connect BD and CE, which intersect at point F, and BD is perpendicular to CE. Extend CE to form an auxiliary line intersecting AH at point G.", "ocr_zh": "有矩形ACBH，点G在AH上、点D在AC上，连接BD、CE相交于点F且BD与CE相互垂直于点F，延长CE构建辅助线交AH于点G", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_504.jpg" }, { "index": 505, "ocr_en": "In rectangle BEDF, points A and M are on ED, and points N and C are on BF. Connect AB, MN, and CD. Connect auxiliary line BD. MN is perpendicular to BD. Angle EAB is $45^\\circ$, and angle DCF is $45^\\circ$. The lengths of AE, BE, CF, and DF are all 2.", "ocr_zh": "有矩形BEDF，点A、M在ED上，点N、C在BF上，连接AB、MN、CD、辅助线BD，MN与BD相互垂直，角EAB为45度，角DCF为45度，AE、BE、CF、DF长都为2", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_505.jpg" }, { "index": 506, "ocr_en": "In quadrilateral ABCD, point E is on AB, and point F is on AD. Connect ED and CF, which intersect at point G.", "ocr_zh": "有四边形ABCD，点E在AB上，点F在AD上，俩姐ED、CF相交于点G", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_506.jpg" }, { "index": 507, "ocr_en": "In parallelogram ABCD, point M is on AD, and point N is on BC. Connect MN.", "ocr_zh": "有平行四边形ABCD，点M在AD上，点N在BC上，连接MN", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_507.jpg" }, { "index": 508, "ocr_en": "In rectangle ADNM, point F is on AD, point C is on MN, and points E and B are on AM. Connect CF and ED, which intersect at point G. Connect BC and CD. Angles AMC and CND are both right angles. The length of BC is 6, the length of CD is 8, and the length of BE is 6. The length of BM is $x$, the length of AD is 8, and the length of DN is $6+x$.", "ocr_zh": "有矩形ADNM，点F在AD上，点C在MN上，点E、B在AM上，连接CF、ED相交于点G，连接BC、CD，角AMC、角CND都为直角，BC长为6、CD长为8、BE长为6，BM长为x、AD长为8、DN长为6+x", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_508.jpg" }, { "index": 509, "ocr_en": "In rectangle ABCD, point G is on AD, point H is on BC, point F is on CD, and point E is on AB. Connect EF and GH, and EF is perpendicular to GH.", "ocr_zh": "有矩形ABCD，点G在AD上、点H在BC上、点F在CD上、点E在AB上，连接EF、GH，且EF与GH相互垂直", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_509.jpg" }, { "index": 510, "ocr_en": "In rectangle ABCD, points F and N are on CD, points M and H are on BC, point E is on AB, and point G is on AD. Connect EF, NB, GH, and AM. GH is perpendicular to EF, and AM is perpendicular to BN.", "ocr_zh": "有矩形ABCD，点F、N在CD上，点M、H在BC上，点E在AB上，点G在AD上，连接EF、NB、GH、AM，且GH与EF相互垂直、AM与BN相互垂直", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_510.jpg" }, { "index": 511, "ocr_en": "There is an equilateral triangle, and within it, there is another equilateral triangle with a smaller side length. The sides of the smaller equilateral triangle are extended to intersect at the vertices of the larger equilateral triangle.", "ocr_zh": "有一正三角形，其内有另一边长较小的正三角形，延长小正三角形各边交于正三角形的顶点", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_511.jpg" }, { "index": 512, "ocr_en": "There is a square, and within it, there is another square with a smaller side length. The sides of the smaller square are extended to intersect at the vertices of the larger square.", "ocr_zh": "有一正方形，其内有另一边长较小的正方形，延长小正方形各边交于正方形的顶点", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_512.jpg" }, { "index": 513, "ocr_en": "There is a regular hexagon, and concentrically within it, there is a smaller regular hexagon whose vertices lie on the sides of the larger regular hexagon. Connect point O (the common center) to one vertex of the smaller regular hexagon, forming a line segment with length $r$.", "ocr_zh": "有正六边形，同心点O其内有一小正六边形且小正六边形的各个顶点正好在正六边形的边上，连接点O与小正六边形的一个顶点构建线段长度为r", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_513.jpg" }, { "index": 514, "ocr_en": "There is a square, and within it, there is a smaller square. The smaller square is rotated so that its vertices lie precisely on the sides of the larger square.", "ocr_zh": "有一个正方形，其内有一小正方形，旋转小正方形使得其顶点均恰好在大正方形的边上", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_514.jpg" }, { "index": 515, "ocr_en": "There is a square, and within it, there is a smaller square rotated so that all its vertices lie precisely on the sides of the larger square.", "ocr_zh": "有一个正方形，其内有一小正方形，旋转小正方形使得其顶点均恰好在大正方形的边上", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_515.jpg" }, { "index": 516, "ocr_en": "There is a square, and within it, there is a smaller square. The sides of the smaller square are extended precisely to the vertices of the larger square.", "ocr_zh": "有一个正方形，其内有一小正方形，延长小正方形各边恰好至大正方形的各个顶点", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_516.jpg" }, { "index": 517, "ocr_en": "Consider a triangle. Extend each of its sides in both directions to points A, B, C, D, E, and F. Connect AB, CD, and EF.", "ocr_zh": "有一三角形，假设其顶点为XYZ，沿YX、ZX方向延长到A、B点，连接AB；沿XY、ZY方向延长到D、C点，连接DC；沿YZ、XZ方向延长到F、E点，连接EF", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_517.jpg" }, { "index": 518, "ocr_en": "Connect vertices A, B, C, D, E to form a pentagram by drawing segments AB, BC, CD, DE, and EA.", "ocr_zh": "有顶点A、B、C、D、E，连接AB、BC、CD、DE、EA构成五角星", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_518.jpg" }, { "index": 519, "ocr_en": "In triangle BOC, extend CO to point A, extend BO to point D, and connect AD.", "ocr_zh": "有三角形BOC，延长CO至点A，延长BO至点D，连接AD", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_519.jpg" }, { "index": 520, "ocr_en": "In quadrilateral ABCD, connect AC and BD intersecting at point O.", "ocr_zh": "有四边形ABCD，连接AC、BD相交于点O", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_520.jpg" }, { "index": 521, "ocr_en": "In triangle BAC, extend CA to point D, extend BA to point E, and connect DE. Angle DEA is marked as Angle 1, and angle ACB is marked as Angle 2.", "ocr_zh": "有三角形BAC，延长CA至点D，延长BA至点E，连接DE，角DEA标记为角1，角ACB标记为角2", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_521.jpg" }, { "index": 522, "ocr_en": "In triangle BAC, extend CA to point D, extend BA to point E, and connect DE. Angle EDA is marked as Angle 1, and angle ABC is marked as Angle 2.", "ocr_zh": "有三角形BAC，延长CA至点D，延长BA至点E，连接DE，角EDA标记为角1，角ABC标记为角2", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_522.jpg" }, { "index": 523, "ocr_en": "In equilateral triangle ABC, D is a point on AC. Connect BD and extend it to point E. Extend BC to point F. Connect CE.", "ocr_zh": "有等边三角形ABC，D为AC上一点，连接BD延长至点E，延长BC延长至点F，连接CE", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_523.jpg" }, { "index": 524, "ocr_en": "In parallelogram ABCD, AC and BD intersect at point O. E is the midpoint of OD. Connect AE and extend it to intersect DC at point F.", "ocr_zh": "在平行四边形ABCD中，AC与BD交于点O，E为OD的中点，连接AE并延长交DC于点F", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_524.jpg" }, { "index": 525, "ocr_en": "In parallelogram ABCD, P is a point on side AB. Ray CP intersects the extension of DA at point E.", "ocr_zh": "有平行四边形ABCD，点P是边AB上的一点，射线CP交DA的延长线于点E", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_525.jpg" }, { "index": 526, "ocr_en": "In square ABCD, AC and BD intersect at point O. The angle bisector of angle ACB intersects AB and BD at points M and N, respectively.", "ocr_zh": "有正方形ABCD，连接AC与BD相交于点O，角ACB的角平分线分别交AB、BD于M、N两点", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_526.jpg" }, { "index": 527, "ocr_en": "In triangle ABC, angle BAD is a right angle. Fold the triangle so that point B coincides with point C at a point E on BC. The fold line is auxiliary line DE. Connect restored line BD.", "ocr_zh": "在三角形ABC中，角BAD为直角，将其折叠使B点与C点重合在BC上有一点E，连接辅助线DE为其折痕，连接复置县BD", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_527.jpg" }, { "index": 528, "ocr_en": "In triangle ABC, angle A is a right angle. M is the midpoint of BC. E and F are on AB and AC respectively, and ME is perpendicular to MF.", "ocr_zh": "三角形ABC中，角A为直角，M是BC的中点，E，F分别在AB，AC上，ME与MF相互垂直于点M", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_528.jpg" }, { "index": 529, "ocr_en": "In square ABCD, the bisector of angle BAC intersects BC at E. Draw EF perpendicular to AC at point F, and draw FG perpendicular to AB at point G.", "ocr_zh": "在正方形ABCD中，角BAC的平分线交BC于E，作EF垂直于AC\n于点F，作FG垂直于AB于点G", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_529.jpg" }, { "index": 530, "ocr_en": "There is a quadrilateral ABCD.", "ocr_zh": "有四边形ABCD", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_530.jpg" }, { "index": 531, "ocr_en": "You have a right triangle ACD. Fold triangle ABC so that point B coincides with point A, and the fold line is DE.", "ocr_zh": "有直角三角形ACD，将三角形ABC折叠，使点B与点A重合，折痕为DE", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_531.jpg" }, { "index": 532, "ocr_en": "In quadrilateral ABCD, AC is perpendicular to BD, and AC and BD intersect at point O.", "ocr_zh": "四边形ABCD中，AC垂直于BD于点O，AC与BD交于O点", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_532.jpg" }, { "index": 533, "ocr_en": "In triangle ABC, P is a point inside the triangle. Connect AP, BP, and CP.", "ocr_zh": "有三角形ABC，P为三角形ABC内一点，连接AP、BP、CP", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_533.jpg" }, { "index": 534, "ocr_en": "Start with a base **square** (labeled E in the figure). Then, from its top two vertices, extend outwards to construct **triangles** where the segments connecting these two vertices serve as the hypotenuses. On the other two sides of these triangles, build two new **squares**. On the top segments of these two squares, draw new triangles using these segments as hypotenuses. From the other two sides of the left triangle, construct squares A and B. From the other two sides of the right triangle, construct squares C and D.", "ocr_zh": "有一个基础的正方形（图中的E）开始；然后，在其顶部两个顶点分别向外延伸，以这两个顶点线段为斜边构建三角形，另外两个边向外构建两个新的正方形，并在这两个正方形的顶上线段作为斜边分别绘制新的三角形，左边三角形另外两边为边长构建正方形A和B，右边三角形另外两边构建正方形C和D", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_534.jpg" }, { "index": 535, "ocr_en": "In triangle ACD, B is a point on CD. Connect AB. E is a point on AB. Both triangle ABD and triangle BCE are isosceles triangles.", "ocr_zh": "三角形ACD中，B为CD上一点，连接AB，E为AB上一点，三角形ABD和三角形BCE都是等腰三角形", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_535.jpg" }, { "index": 536, "ocr_en": "In triangle ACD, B is a point on CD. Connect AB. E is a point on AB. Both triangle ABD and triangle BCE are isosceles triangles.", "ocr_zh": "在直角三角形ABC中，角ACB为90度，作AB的垂直平分线DE交BC的延长线于点E", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_536.jpg" }, { "index": 537, "ocr_en": "In isosceles triangle ABC, AB = AC. D is a point on BC. Connect AD, and AD is perpendicular to BC.", "ocr_zh": "等腰三角形ABC中，AB=AC，D为BC上一点，连接AD且AD垂直于BC于点D", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_537.jpg" }, { "index": 538, "ocr_en": "There is a large square ABCD. Inside this large square, there are four congruent right-angled triangles: ADF, ABG, BCH, and CDE. The right-angled vertices of these four triangles form the four vertices of a smaller square EFGH.", "ocr_zh": "有一个大正方形 ABCD，在大正方形的内部有四个全等的直角三角形ADF、直角三角形ABG、直角三角形BCH、直角三角形CDE，四个三角形的直角顶点构成小正方形EFGH的四个顶点", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_538.jpg" }, { "index": 539, "ocr_en": "Consider a square with side length $a+b$. Inside it, there is a smaller square with side length c. The vertices of the smaller square lie exactly on the sides of the larger square. The empty space around the smaller square forms four right-angled triangles, each with legs of length a and b.", "ocr_zh": "有一边长为a+b的正方形，其内有一边长为c的小正方形，小正方形的各个顶点恰好在大正方形的各个边上，周边的空白恰好行成四个直角边长分别为a和b的直角三角形", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_539.jpg" }, { "index": 540, "ocr_en": "You have a right trapezoid ABCD, where angles ADC and ECB are right angles. E is a point on CD. Connect AE and BE. The lengths of AE and BE are $c$. The lengths of BC and DE are $b$. The lengths of CE and AD are $a$.", "ocr_zh": "有直角梯形ABCD，角ADC和角ECB分别为直角，E为CD上一点，连接AE、BE，AE、BE长为c，BC、DE长为b，CE、AD长为a", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_540.jpg" }, { "index": 541, "ocr_en": "In right triangle ACB, point D is on AB. Connect CD.", "ocr_zh": "有直角三角形ACB，点D在AB上，连接CD", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_541.jpg" }, { "index": 542, "ocr_en": "The top-left image shows right triangle ABC, with angle ABC being a right angle and angle CAB being 30 degrees. The bottom-left image shows triangle ABC, where D is a point on AB, CD is connected and perpendicular to AB, and there is a point on AC connected to B such that the segment forms the angle bisector of angle ABC. The middle image shows triangle ABC, with point D on AB, and CD connected. The right image shows triangle ACB, with point D on AB, and CD connected and perpendicular to AB.", "ocr_zh": "左上图为直角三角形ABC，角ABC为直角，角CAB为30度；左下图为三角形ABC，D为AB上一点，连接CD，CD垂直于AB于点D，AC上有一点，连接该点于点B连成线段恰好构成角ABC的角平分线；中图为三角形ABC，点D为AB上一点，连接CD；右图为三角形ACB，点D为AB上一点，连接CD且CD垂直于AB于点D", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_542.jpg" }, { "index": 543, "ocr_en": "In triangle ABC, D is a point on AB. Connect CD, and CD is perpendicular to AB. There is a point on AC such that the segment connecting this point to B is the angle bisector of angle ABC.", "ocr_zh": "有三角形ABC，D为AB上一点，连接CD，CD垂直于AB于点D，AC上有一点，连接该点于点B连成线段恰好构成角ABC的角平分线", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_543.jpg" }, { "index": 544, "ocr_en": "In right triangle ACB, angle ACB is a right angle. Point D is on BC. Connect AD. Point E is on AB. Connect auxiliary line DE. Angle CAD is Angle 1, angle DAB is Angle 2, and Angle 1 equals Angle 2. DE is perpendicular to AB.", "ocr_zh": "有直角三角形ACB,角ACB为直角，点D为BC上一点，连接AD，点E为AB上一点，连接辅助线DE，角CAD为角1，角DAB为角2，且角1与角2相等，DE与AB相互垂直于点E", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_544.jpg" }, { "index": 545, "ocr_en": "In right triangle ACB, angle ACB is a right angle. Point D is on BC. Connect AD. Point E is on AB. Connect auxiliary line DE. Angle CAD is Angle 1, angle DAB is Angle 2, and Angle 1 equals Angle 2. DE is perpendicular to AB.", "ocr_zh": "有直角三角形ACB，角ACB为直角，分别以AC、AB、BC为直径绘制半圆AC、半圆AB、半圆BC，半圆AB切其余两个半圆得到阴影部分", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_545.jpg" }, { "index": 546, "ocr_en": "The left and middle images show two trees of different heights. The right image shows a right trapezoid ABCD, where angle ABC is a right angle and angle DCB is a right angle. Point E is on AB. Connect auxiliary line ED.", "ocr_zh": "左图和中图为两科高度不同的树，右图为直角梯形ABCD，角ABC为直角，角DCB为直角，点E为AB上一点，连接辅助线ED", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_546.jpg" }, { "index": 547, "ocr_en": "In triangle ABC, point D is on AB. Connect CD.", "ocr_zh": "有三角形ABC，点D为AB上一点，连接CD", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_547.jpg" }, { "index": 548, "ocr_en": "On the Cartesian coordinate system xOy, there is a parabola that intersects the x-axis at point A on the negative x-axis, at point B on the positive x-axis, and intersects the y-axis at point C. Its vertex is point D.", "ocr_zh": "在平面直角坐标系xOy上有一抛物线与x轴在x的负半轴交于点A、在x的正半轴交于点B，与y轴交于点C，顶点为点D ", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_548.jpg" }, { "index": 549, "ocr_en": "On the Cartesian coordinate system xOy, there are points A(-3,0), C(1,0), and G($0,3\\sqrt{3}$). Connect CG. P is a point inside triangle ACG. Connect PA, PC, and PG. Construct equilateral triangle APR on the left side of AP, and equilateral triangle AGQ on the left side of AG. Connect QR.", "ocr_zh": "在平面直角坐标系xOy上，有点A（-3,0）、C(1,0）、G（0,3√3），连接CG，P为三角形ACG内一点，连接PA、PC、PG，分别以AP、AG为边，在其左侧作等边三角形APR，等边三角形AGQ，连接QR", "class": "Analytic Geometry", "difficulty": 3, "image_path": "fig_549.jpg" }, { "index": 550, "ocr_en": "In the bottom-left corner of the diagram, two straight line segments originate from the same vertex, forming a large angle. This large angle is divided into two parts by a ray emitted from the same vertex: the bottom angle, closer to the horizontal line, is 45 degrees, and the upper angle, closer to the slanted line on the left, is 30 degrees. Inside the 45-degree angle, there is a vertical line segment perpendicular to the horizontal line. Inside the 30-degree angle, there is a slanted line perpendicular to the ray.", "ocr_zh": "在图的左下角，有两条直线段从同一个顶点发出，形成一个大角。这个大角被一条从相同顶点发出的射线分成了两个部分，底部靠近水平线的角是 45度，上面靠近左侧斜线的角是 30 度，45度角内有一垂直线段与水平线相互垂直，30度角有一斜线与射线相互垂直", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_550.jpg" }, { "index": 551, "ocr_en": "In the bottom-left corner of the diagram, two straight line segments originate from the same vertex, forming a large angle. This large angle is divided into two parts by a ray emitted from the same vertex: the bottom angle, closer to the horizontal line, is 45 degrees, and the upper angle, closer to the slanted line on the left, is 30 degrees. Inside the 45-degree angle, there is a vertical line segment perpendicular to the horizontal line and extending upwards to intersect a horizontal line within the 30-degree angle. Inside the 30-degree angle, there is a slanted line perpendicular to the ray.", "ocr_zh": "在图的左下角，有两条直线段从同一个顶点发出，形成一个大角。这个大角被一条从相同顶点发出的射线分成了两个部分，底部靠近水平线的角是45度，上面靠近左侧斜线的角是 30 度，45度角内有一垂直线段与水平线相互垂直且向上延长与30度角内水平线相交，30度角有一斜线与射线相互垂直", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_551.jpg" }, { "index": 552, "ocr_en": "In the bottom-left corner of the diagram, two straight line segments originate from the same vertex, forming a large angle. This large angle is divided into two parts by a ray emitted from the same vertex: the bottom angle, closer to the horizontal line, is 45 degrees, and the upper angle, closer to the slanted line on the left, is 30 degrees. Inside the 45-degree angle, there is a vertical line segment perpendicular to the horizontal line, extending upwards to intersect a horizontal line within the 30-degree angle. This horizontal line then extends to the left, intersecting a perpendicular line segment that originates from the common vertex. Inside the 30-degree angle, there is a slanted line perpendicular to the ray.", "ocr_zh": "在图的左下角，有两条直线段从同一个顶点发出，形成一个大角。这个大角被一条从相同顶点发出的射线分成了两个部分，底部靠近水平线的角是45度，上面靠近左侧斜线的角是 30 度，45度角内有一垂直线段与水平线相互垂直且向上延长与30度角内水平线相交然后向左延长该水平线与左侧一过从顶点触发的垂线段相交，30度角有一斜线与射线相互垂直", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_552.jpg" }, { "index": 553, "ocr_en": "An angle O of 45 degrees is given, with the horizontal line as the first side and another line as the second side, both originating from common vertex O. Then, a point A is chosen on the second side. From point A, a perpendicular line is drawn to the exterior of the angle to point B, such that angle AOB is 30 degrees, thus constructing a right triangle ABO with OA as the common side and a 30-degree angle. Subsequently, perpendicular lines are drawn from points A, B, and O to the coordinate axes, forming a rectangle OCDE.", "ocr_zh": "有一个角O为45度的角，其中水平线为第一条边，另一条边为第二条边，两条边以O点为公共顶点。接着，在第二条边上选取一点A，从A点向角的外部作一条垂线至点B，使得角AOB 为30度，从而构造出一个以OA为公共边、含30度角的直角三角形三角形ABO。随后，分别过A、B、O三点作坐标轴方向的垂线，构成一个矩形OCDE", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_553.jpg" }, { "index": 554, "ocr_en": "The left image shows an angle O of 45 degrees, with the horizontal line as the first side and another line as the second side, both originating from common vertex O. The middle image shows an angle O of 45 degrees, with the horizontal line as the first side and another line as the second side, both originating from common vertex O. Then, a point A is chosen on the second side. From point A, a perpendicular line is drawn to the exterior of the angle to point B, such that angle AOB is 30 degrees, thus constructing a right triangle ABO with OA as the common side and a 30-degree angle. The right image shows an angle O of 45 degrees, with the horizontal line as the first side and another line as the second side, both originating from common vertex O. Then, a point A is chosen on the second side. From point A, a perpendicular line is drawn to the exterior of the angle to point B, such that angle AOB is 30 degrees, thus constructing a right triangle ABO with OA as the common side and a 30-degree angle. Subsequently, perpendicular lines are drawn from points A, B, and O to the coordinate axes, forming a rectangle OCDE.", "ocr_zh": "左图为有一个角O为45度的角，其中水平线为第一条边，另一条边为第二条边，两条边以O点为公共顶点；中图为有一个角O为45度的角，其中水平线为第一条边，另一条边为第二条边，两条边以O点为公共顶点。接着，在第二条边上选取一点A，从A点向角的外部作一条垂线至点B，使得角AOB 为30度，从而构造出一个以OA为公共边、含30度角的直角三角形三角形ABO；右图有一个角O为45度的角，其中水平线为第一条边，另一条边为第二条边，两条边以O点为公共顶点。接着，在第二条边上选取一点A，从A点向角的外部作一条垂线至点B，使得角AOB 为30度，从而构造出一个以OA为公共边、含30度角的直角三角形三角形ABO。随后，分别过A、B、O三点作坐标轴方向的垂线，构成一个矩形OCDE", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_554.jpg" }, { "index": 555, "ocr_en": "There is an angle O, where the horizontal line is the first side and another line is the second side, both originating from the common vertex O.", "ocr_zh": "有一个角O，其中水平线为第一条边，另一条边为第二条边，两条边以O点为公共顶点", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_555.jpg" }, { "index": 556, "ocr_en": "There is an angle O, with the horizontal line as the first side and another line as the second side, both originating from common vertex O. Then, a point A is selected on the second side. From point A, a perpendicular line is drawn to the exterior of the angle to point B, such that angle AOB is 30 degrees, thus constructing a right triangle ABO with OA as the common side and a 30-degree angle.", "ocr_zh": "有一个角O，其中水平线为第一条边，另一条边为第二条边，两条边以O点为公共顶点。接着，在第二条边上选取一点A，从A点向角的外部作一条垂线至点B，使得角AOB 为30度，从而构造出一个以OA为公共边、含30度角的直角三角形三角形ABO", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_556.jpg" }, { "index": 557, "ocr_en": "The left diagram shows an angle O. The middle diagram shows an angle O of 45 degrees, with a point A taken on one of its sides. A ray passing through point B is drawn inside angle O such that angle AOB is 30 degrees, and BA is perpendicular to OA at point A. The right diagram also shows an angle O of 45 degrees, with a point A taken on one of its sides. A ray passing through point B is drawn inside angle O such that angle AOB is 30 degrees, and BA is perpendicular to OA at point A. Then, perpendicular lines are drawn from points A, B, and O to form a rectangle OCDE based on these three points.\n", "ocr_zh": "左图为有一个角O；中图为有一个角O为45度的角，然后在其中一边上取一点A，并在角O的内部作一条经过点 B 的射线，使得角AOB为30度，BA与OA垂直于点A；右图有一个角O为45度的角，然后在其中一边上取一点A，并在角O的内部作一条经过点 B 的射线，使得角AOB 为 30度，BA与OA垂直于点A。接着，分别以 A、B、O 三点为基准作出坐标轴的垂线，构成一个矩形OCDE", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_557.jpg" }, { "index": 558, "ocr_en": "As shown in the figure, there is a rectangle EDCO. Point A is on CD. Draw AB perpendicular to OA at point A, with point B on ED. Connect OB and OA. ∠EBO = 75°, ∠DAB = 45°, ∠AOC = 45°, ∠OAC = 45°, ∠BOA = 30°, and ∠EBO = 75°. The lengths of sides OC and AC are √3, OE is √3 + 1, EB is √3 − 1, and AD and BD are both 1.\n", "ocr_zh": "如图有矩形EDCO，点A在CD上，过点A作AB垂直OA于点A，点B在ED上，连接OB、OA，角EBO为75度、角DAB为45度、角AOC为45度、角OAC为45度、角BOA为30度、角EBO为75度，边OC、AC长为√3，边OE长为√3+1，边EB长为√3-1.边AD、BD长为1", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_558.jpg" }, { "index": 559, "ocr_en": "First, construct an angle O of 45 degrees. Then, take a point A on one of its sides. From point A, draw a ray inside angle O such that angle AOB is 30 degrees, forming a right triangle ABO. Next, draw perpendicular lines to the coordinate axes from points A, B, and O, which will form a rectangle OCDE.", "ocr_zh": "先作一个角O为45度的角，然后在其中一边上取一点A，并在角O的内部作一条经过点 B 的射线，且AB与OA垂直于点A，使得角AOB 为 30度。接着，分别以 A、B、O 三点为基准作出坐标轴的垂线，构成一个矩形OCDE", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_559.jpg" }, { "index": 560, "ocr_en": "There is an **angle O** with a **horizontal line** as its first side and another line as its second side, both sharing **point O** as their common vertex.", "ocr_zh": "有一个角O，其中水平线为第一条边，另一条边为第二条边，两条边以O点为公共顶点", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_560.jpg" }, { "index": 561, "ocr_en": "There is a 45-degree angle O. Inside angle O, there is a point B. Construct ray OB. Take point A on the second side of the angle such that AB is perpendicular to OA at point A.\n", "ocr_zh": "有一个45度角O，角O内部有一点B，构建射线OB，在第二条线上取点A，AB与OA垂直于点A", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_561.jpg" }, { "index": 562, "ocr_en": "There is a 45-degree angle O. Inside angle O, there is a point B. Construct ray OB. Take point A on the second side of the angle such that AB is perpendicular to OA at point A. Draw perpendicular lines from points A, B, and O to the coordinate axes to form a rectangle OCDE. The lengths of OC and AC are √3, OE is √3 + 1, BE is √3 − 1, and both AD and BD are 1.\n", "ocr_zh": "有一个45度角O，角O内部有一点B，构建射线OB，在第二条线上取点A，AB与OA垂直于点A，过点A、B、O三点分别作坐标轴的垂线，围成一个矩形OCDE。OC、AC长为√3，OE长为√3+1，BE长为√3-1，AD、BD长为1", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_562.jpg" }, { "index": 563, "ocr_en": "There are two nested right triangles. In the right triangle on the right, the legs are further subdivided: one vertical leg is divided into segments of lengths 3 and 5, and one horizontal leg is divided into segments of lengths 4 and 5. There are four angles: α, β, and their double angles, 2α and 2β.\n", "ocr_zh": "有两个互相嵌套的直角三角形，右侧直角三角形的直角边被进一步细分，其中一条垂直边被分为长度为3和5的两段，另一条水平边则被分为长度为4和5的两段，有四个角度，分别为 α、β、以及它们的二倍角 2α 和 2β", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_563.jpg" }, { "index": 564, "ocr_en": "Rectangle EFCD is given. Point B is on ED, and point A is on CD. Connect AB, FB, and AF. AB and AF are perpendicular to each other. The lengths of BE and EF are 5, BD is 1, AC is 3, AD is 2, and FC is 6.", "ocr_zh": "矩形EFCD，点B在ED上、点A在CD上，连接AB、FB、AF，AB与AF相互垂直于点A。BE、EF长为5，BD长为1，AC长为3，AD长为2，FC长为6", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_564.jpg" }, { "index": 565, "ocr_en": "Rectangle EOCD is given. Point B is on ED, and point A is on CD. Connect AB, OB, and AO. AB and AO are perpendicular to each other. The lengths of CO and EO are 4, BD is 1, AD and AC are 2, and BE is 3.", "ocr_zh": "矩形EOCD，点B在ED上、点A在CD上，连接AB、OB、AO，AB与AO相互垂直于点A。CO、EO长为4，BD长为1，AD、AC长为2，BE长为3", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_565.jpg" }, { "index": 566, "ocr_en": "Rectangle ABCD has side lengths AB = 3k and AD = 7k. Point E is on side BC, dividing it into BE = 6k and EC = k. Point F is on side CD, dividing it into DF = k and FC = 2k. Connect segments AE, AF, and EF. ∠BAE = α and ∠EAF = β. AE is perpendicular to EF at point E.\n", "ocr_zh": "矩形 ABCD，其边长 AB为3k，AD为7k。点 E 位于边 BC 上，将 BC 分成 BE=6k 和 EC=k 两部分。点 F 位于边 CD 上，将 CD 分成 DF=k 和 FC=2k 两部分。连接线段 AE、AF 和 EF，角BAE=α 和 角EAF=β， AE 垂直于 EF于点E ", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_566.jpg" }, { "index": 567, "ocr_en": "In the Cartesian coordinate system xOy, there is a triangle ABO in the first quadrant, with angle ABO being 90 degrees.", "ocr_zh": "平面直角坐标系xOy中，在第一象限有三角形ABO，角ABO为90度", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_567.jpg" }, { "index": 568, "ocr_en": "On the Cartesian coordinate system xOy, points G and F are on the x-axis, and point D is on the y-axis. Auxiliary lines DE, OF, and EF are connected to form rectangle DOFE. Point A is on DE, point G is on OF, and point C is on EF. Connect OA, AC, and OC. Point B is on OC. Connect AB, auxiliary line AC, auxiliary line BG, auxiliary line BC, OB, and OA. AB is perpendicular to OC. Angle ACB is 45 degrees. The lengths of sides OD and AE are 4. The lengths of DA, EC, and CF are 2. The length of OF is 6.", "ocr_zh": "在平面直角坐标徐xOy上，点G、F在x轴上，点D在y轴上，连接辅助线DE、OF、EF构成矩形DOFE，点A在DE上、点G在OF上、点C在EF上，连接OA、AC、OC，点B在OC上，连接AB、辅助线AC、辅助线BG、辅助线BC、OB、OA，AB与OC相互垂直于点B，角ACB为45度，边OD、AE为4，DA、EC、CF长为2、OF长为6", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_568.jpg" }, { "index": 569, "ocr_en": "In the Cartesian coordinate system xOy, point A is on the positive x-axis, point B is on the positive y-axis, and point C is in the first quadrant. Connect CB, AB, and AC to form triangle ABC.", "ocr_zh": "在平面直角坐标系xOy上，点A在x正半轴，点B在y正半轴，点C在第一象限上，连接CB、AB、AC构建三角形ABC", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_569.jpg" }, { "index": 570, "ocr_en": "In the Cartesian coordinate system xOy, point D is on the positive y-axis, point F is on the positive x-axis, and point E is in the first quadrant, forming rectangle DOFE. Point A is on DE, and point B is on EF. Connect OA, OB, and AB. The length of DA is 2, the length of OD is 4, and the length of OA is $2\\sqrt{5}$.", "ocr_zh": "在平面直角坐标系xOy上，点D在y正半轴上，点F在x正半轴上，点E在第一象限上，构成矩形DOFE，点A在DE上，点B在EF上，连接OA、OB、AB，DA长为2，OD长为4，OA长为2√5", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_570.jpg" }, { "index": 571, "ocr_en": "In the Cartesian coordinate system xOy, points A and D are on the positive x-axis. Points C and P are on the negative y-axis. Connect line PD. Point B is on line PD and is located in the fourth quadrant. Connect points A, C, and B to form triangle ABC.", "ocr_zh": "在平面直角坐标系xOy上，点A、D在x的正半轴上，点C、P在y的负半轴上，连接直线PD，点B在直线PD上且位于第四象限内，连接点A、点C、点B构建三角形ABC", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_571.jpg" }, { "index": 572, "ocr_en": "The left image shows **triangle ABC** in the Cartesian coordinate system xOy, with point A on the positive x-axis, point B on the positive y-axis, and point C in the first quadrant, connected by CB, AB, and AC. The right image shows **rectangle FOED** in the Cartesian coordinate system xOy, with points B and F on the positive y-axis, points A and E on the positive x-axis, and point D in the first quadrant. Point G is on FD. BD and AB are connected. Point C is on BD, and AC is connected. Auxiliary lines CG and AD are connected. Angle BAD is a right angle, and angle DBA is 60 degrees. The length of OA is 4, AE is $3\\sqrt{3}$, OB is 3, BF is $4\\sqrt{3}-3$, FD is $4+3\\sqrt{3}$, and ED is $4\\sqrt{3}$.", "ocr_zh": "左图为在平面直角坐标系xOy上，点A在x正半轴，点B在y正半轴，点C在第一象限上，连接CB、AB、AC构建三角形ABC；右图为在平面直角坐标系xoy上，点B、F在y正半轴上，点A、E在x的正半轴上，点D在第一象限内，构建矩形FOED。点G在FD上。连接BD、AB，点C在BD上，连接AC，连接辅助线CG、辅助线AD，角BAD为直角，角DBA为60度。OA长为4，AE长为3√3，OB长为3，BF长为4√3-3，FD长为4+3√3，ED长为4√3", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_572.jpg" }, { "index": 573, "ocr_en": "In the Cartesian coordinate system xOy, there is an inverse proportional function graph passing through points E and P. Connect OE and OP.", "ocr_zh": "在平面直角坐标系xOy上，有一反比例函数图像经过点E、点P，连接OE、OP", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_573.jpg" }, { "index": 574, "ocr_en": "In the Cartesian coordinate system xOy, point C is on the positive y-axis, and point A is on the positive x-axis. An inverse proportional function graph passes through points E and P. Connect OE and OP. Point B is on the inverse proportional function graph. Construct rectangle OABC. Connect auxiliary line EF, where EF is perpendicular to OE.", "ocr_zh": "在平面直角坐标系xOy上，点C在y的正半轴上，点A在x的正半轴上，有一反比例函数图像经过点E、点P，连接OE、OP，点B在反比例函数图像上，构建矩形OABC，连接辅助线EF，EF与OE相互垂直于点E", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_574.jpg" }, { "index": 575, "ocr_en": "In the Cartesian coordinate system xOy, points A, H, and D are on the positive x-axis. Points C, N, and P are on the negative y-axis. There is a point M in the fourth quadrant. Rectangle ONMD is constructed. Line PD is connected. AC, AM, and CM are connected. Point B is on line PD and in the fourth quadrant. AB and HB are connected. AC is perpendicular to AM, BH is perpendicular to OD, and angle ACB is 60 degrees. The length of side OC is $2\\sqrt{3}$, DM is $4\\sqrt{3}$, OA is 4, and HD is 6.", "ocr_zh": "在平面直角坐标系xoy上，点A、H、D在x的正半轴上，点C、N、P在y的负半轴上，有一点M在第四象限内，构建矩形ONMD，连接直线PD，连接AC、AM、CM，点B在直线PD上且在第四象限内，连接AB、HB。AC垂直于AM于点A，BH垂直于OD于点H，角ACB为60度。边OC为2√3，边DM为4√3，边OA长为4，边HD长为6", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_575.jpg" }, { "index": 576, "ocr_en": "In the Cartesian coordinate system xOy, points F, B, and P1 are on the positive x-axis, and points A and P2 are on the negative x-axis. A downward-opening parabola passes through points A and B, with its vertex D in the first quadrant at coordinates (1, 4). A tangent line to the parabola passes through points P1 and D, and extends through point H in the second quadrant. Connect auxiliary lines DP2 and DF.", "ocr_zh": "在平面直角坐标系xoy上，点F、B、P1在x的正半轴上，点A、P2在x的负半轴上，过点A、B作抛物线开口向下，点D为顶点且在第一象限内，点D坐标为（1，4），过点P1、D作抛物线的切线在第二象限内过点H。连接辅助线DP2、DF。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_576.jpg" }, { "index": 577, "ocr_en": "In the Cartesian coordinate system xOy, point B is on the positive x-axis, and point A is on the negative x-axis. A downward-opening parabola passes through points A and B and intersects the y-axis at point C. Point D is the vertex of the parabola and is located in the first quadrant.", "ocr_zh": "在平面直角坐标系xoy上，点B在x的正半轴，点A在x的负半轴上，过点A、B作抛物线开口向下，过y轴交于点C，点D为抛物线顶点且在第一象限内", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_577.jpg" }, { "index": 578, "ocr_en": "In rectangle EFAD, point B is on ED, and point C is on AD. Connect BC, BF, and CF. Angle CFA is Angle 1, angle BFC is Angle 2, angle EBF is $\\alpha$, and angle FBC is a right angle. EF is 4, EB is 1, BD is 8, FA is 9, AC is 2, and DC is 2.", "ocr_zh": "在矩形EFAD中，点B在ED上，点C在AD上，连接BC、BF、CF，角CFA为角1，角BFC为角2，角EBF为角α，角FBC为直角。EF为4，EB为1，BD为8，FA为9，AC为2，DC为2", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_578.jpg" }, { "index": 579, "ocr_en": "In the Cartesian coordinate system xOy, points F and B are on the positive x-axis, and points A, H, and P are on the negative x-axis. Point P has coordinates (-17, 0). A downward-opening parabola passes through points A and B, with its vertex D in the first quadrant. Point G is in the second quadrant, forming rectangle DGHF. Auxiliary lines DF and PD are connected, with PD intersecting GH at point E. Connect auxiliary line EO and connect OD. Angle GDO is α. The lengths of sides GE and EH are 2. The length of OH is 8, OF is 1, DF is 4, GD is 9, and PH is 9.", "ocr_zh": "在平面直角坐标系xoy上，点F、B在x的正半轴上，点A、H、P在x的负半轴上，点P坐标为（-17，0），抛物线开口向下经过点A和点B，顶点为D在第一象限内，点G在第二象限内，构建矩形DGHF，连接辅助线DF、PD与GH交于点E，连接辅助线EO，连接OD，角GDO为角α。边GE、EH长为2，边OH为8，OF为1，DF为4，GD为9，PH为9", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_579.jpg" }, { "index": 580, "ocr_en": "In the Cartesian coordinate system xOy, points F, B, P are on the positive x-axis, with P at (19, 0). Points A, G are on the negative x-axis. A downward-opening parabola passes through points A and B, with its vertex at D. Point E is in the first quadrant, and point H is in the second quadrant, forming rectangle HGFE. Connect auxiliary lines AH and HP. HP intersects EF at point D. Connect auxiliary lines DF and AD. The length of side HG is 8, AG is 16, AF is 2, DF is 4, HE is 18, and ED is 4.", "ocr_zh": "在平面直角坐标系xoy上，点F、B、P在x的正半轴上，P点坐标为（19，0），点A、G在x的负半轴上，有抛物线过点A、点B开口向下，顶点为D点，点E在第一象限内,点H在第二象限内，构成矩形HGFE，连接辅助线AH、辅助线HP，HP与EF相交于点D，连接辅助线DF、AD。边HG长为8，AG长为16，AF长为2，DF长为4，HE长为18，ED长为4", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_580.jpg" }, { "index": 581, "ocr_en": "In the Cartesian coordinate system xOy shown in the figure above, point A is on the negative x-axis, point C is on the positive x-axis, and point G is on the positive y-axis. A line parallel to the x-axis passes through point G into the second quadrant to point Q, forming triangle QGA. Connect CG. P is a point inside triangle ACG. Connect PA, PC, and PG. Construct equilateral triangle APR to the left using AP and equilateral triangle AGQ to the left using AG as sides. Connect QR. In the Cartesian coordinate system xOy shown in the figure below, point A is on the negative x-axis, point C is on the positive x-axis, and point G is on the positive y-axis. A line parallel to the x-axis passes through point G into the second quadrant to point Q, forming triangle QGA. Connect CG. P is a point inside triangle ACG. Connect PA, PC, and PG. Construct equilateral triangle APR to the left using AP and equilateral triangle AGQ to the left using AG as sides. Connect QR. Mark the area of triangle PGA in light green and the area of triangle QRA in emerald green.", "ocr_zh": "在上图平面直角坐标系xoy中，有点A在x负半轴上，点C在x正半轴上，点G在y的正半轴上，过点G平行于x轴在第二象限内至点Q，构建三角形QGA，连接CG，P为三角形ACG内一点，连接PA、PC、PG，分别以AP、AG为边在左侧作等边三角形APR，等边三角形AGQ，连接QR；在下图平面直角坐标系xoy中，有点A在x负半轴上，点C在x正半轴上，点G在y的正半轴上，过点G平行于x轴在第二象限内至点Q，构建三角形QGA，连接CG，P为三角形ACG内一点，连接PA、PC、PG，分别以AP、AG为边在左侧作等边三角形APR，等边三角形AGQ，连接QR。将三角形PGA的范围内标注浅绿色，三角形QRA内标注翠绿色", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_581.jpg" }, { "index": 582, "ocr_en": "In the Cartesian coordinate system xOy, point C is on the positive x-axis, points A and H are on the negative x-axis, and point G is on the positive y-axis. A line parallel to the x-axis passes through point G into the second quadrant to point Q. Connect triangle QGA. Connect QC and CG. Points R and P1 are on QC, with point Q inside triangle AGQ and point P1 inside triangle GAC. Connect AR, AP1, and auxiliary line HP1. Point A is at (-3, 0), point C is at (1, 0), and point G is at (0, 5).", "ocr_zh": "在平面直角坐标系xoy内，点C在x的正半轴上，点A、H在x的负半轴上，点G在y的正半轴上，过点G平行于x轴在第二象限内至点Q，连接三角形QGA，连接QC、CG，点R、P1在QC上且点Q在三角形AGQ内、点P1在三角形GAC内，连接AR、AP1、辅助线HP1，点A坐标为（-3，0），点C坐标为（1，0），点G坐标为（0，5）", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_582.jpg" }, { "index": 583, "ocr_en": "In the left figure, in the Cartesian coordinate system xOy, point C is on the positive x-axis, points A and H are on the negative x-axis, and point G is on the positive y-axis. A line parallel to the x-axis passes through point G into the second quadrant to point Q. Connect triangle QGA, QC, and CG. Points R and P1 are on QC, with point Q inside triangle AGQ and point P1 inside triangle GAC. Connect AR, AP1, and auxiliary line HP1. Point A is at (-3, 0), point C is at (1, 0), and point G is at (0, 5).\n\n---\n\nIn the right figure, there is a quadrilateral ABCD. Point C is on FE, and point D is on AE. Connect BC, CD, and BD. The length of side BF is $7\\sqrt{3}$, FC is 7, CE is 9, DE is $3\\sqrt{3}$, AD is $4\\sqrt{3}$, and AB is 16. Angle ABC is 60 degrees.", "ocr_zh": "左图在平面直角坐标系xoy内，点C在x的正半轴上，点A、H在x的负半轴上，点G在y的正半轴上，过点G平行于x轴在第二象限内至点Q，连接三角形QGA，连接QC、CG，点R、P1在QC上且点Q在三角形AGQ内、点P1在三角形GAC内，连接AR、AP1、辅助线HP1，点A坐标为（-3，0），点C坐标为（1，0），点G坐标为（0，5）；右图有四边形ABCD，点C在FE上，点D在AE上，连接BC、CD、BD，边BF长为7√3，FC长为7，CE长为9，DE长为3√3，AD长为4√3，AB长为16，角ABC为60度", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_583.jpg" }, { "index": 584, "ocr_en": "In the Cartesian coordinate system xOy, point C is on the positive x-axis, points A and H are on the negative x-axis, and point G is on the positive y-axis. A line parallel to the x-axis passes through point G into the second quadrant to point Q. Connect triangle QGA, QC, and CG. Points R and P are on QC, with point Q inside triangle AGQ and point P1 inside triangle GAC. Connect AR, AP1, and auxiliary line HP. Point A is at (-3, 0), point C is at (1, 0), and point G is at (0, 5). Point E is in the first quadrant, and point F is in the second quadrant, forming rectangle FEDH. Connect PE and CE. Angle PCE is 90 degrees, and angle EPC is 60 degrees.", "ocr_zh": "在平面直角坐标系xoy内，点C在x的正半轴上，点A、H在x的负半轴上，点G在y的正半轴上，过点G平行于x轴在第二象限内至点Q，连接三角形QGA，连接QC、CG，点R、P在QC上且点Q在三角形AGQ内、点P1在三角形GAC内，连接AR、AP1、辅助线HP，点A坐标为（-3，0），点C坐标为（1，0），点G坐标为（0，5）。有点E在第一象限，点F在第二象限，构建矩形FEDH,连接PE、CE，角PCE为90度，角EPC为60度", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_584.jpg" }, { "index": 585, "ocr_en": "In the Cartesian coordinate system xOy, point A is on the negative x-axis, and point B is on the positive x-axis. An upward-opening parabola passes through points A and B and intersects the negative y-axis at point C. Connect to form triangle ABC.", "ocr_zh": "在平面直角坐标系xoy上，点A在x的负半轴上，点B在x的正半轴上，有开口向上的抛物线过点A、点B与y轴负半轴相交于点C，连接构建三角形ABC", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_585.jpg" }, { "index": 586, "ocr_en": "In the Cartesian coordinate system xOy, point A is on the negative x-axis, and points B and Q are on the positive x-axis. A parabola passes through points A and B and intersects the negative y-axis at point C. Point M is in the third quadrant, and point N is in the fourth quadrant, forming rectangle AMNB which intersects the parabola at point E. Connect AC and AE. P2 is a point on the parabola in the first quadrant. Connect AP2. A perpendicular line is drawn from point Q into the fourth quadrant, intersecting the parabola at point P. Angle EAQ is 45 degrees. The length of AM is 5, ME is 3, CN is 5, and AP2 is 8.", "ocr_zh": "在平面直角坐标系xoy上，点A在x的负半轴上，点B、Q在x的正半轴上，作抛物线过点A、点B与y的负半轴相交于点C，点M在第三象限，点N在第四象限，构建矩形AMNB与抛物线相交于点E，连接AC、AE,P2为抛物线在第一象限内一点，连接AP2，过点Q向第四象限作垂线与抛物线相交于点P，角EAQ为45度。AM长为5，ME长为3，CN长为5，AP2长为8", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_586.jpg" }, { "index": 587, "ocr_en": "In the Cartesian coordinate system xOy, point B is on the negative x-axis, and point C is on the positive x-axis. An upward-opening parabola passes through points B and C, intersecting the negative y-axis at point A. Connect points A, B, and C to form triangle ABC.", "ocr_zh": "在平面直角坐标系xoy内，点B在x的负半轴上，点C在x的正半轴上，过点B、点C作开口向上的抛物线交y的负半轴与点A，连接点A、B、C构建三角形ABC", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_587.jpg" }, { "index": 588, "ocr_en": "In the Cartesian coordinate system xOy, point A is on the negative x-axis, and point C is on the positive x-axis. An upward-opening parabola intersects the negative y-axis at point B. Point D is on OC. A line parallel to the y-axis is drawn through point D.", "ocr_zh": "在平面直角坐标系xoy上，点A在x的负半轴上，点C在x轴的正半轴上，作开口向上的抛物线与y轴负半轴相交于点B，点D为OC上一点，过点D作y轴的平行线", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_588.jpg" }, { "index": 589, "ocr_en": "There is a square ABCD. Extend CB to point E. Connect AE. Point F is below the square. Connect points E, C, and F to form triangle CEF.", "ocr_zh": "有正方形ABCD，延长CB至点E，连接AE，点F在正方形下，连接点E、C、F构建三角形CEF", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_589.jpg" }, { "index": 590, "ocr_en": "Square ABCD, point E is the midpoint of side BC, angle AEF is 90 degrees, and EF intersects the bisector of the exterior angle of the square, CF, at point F.", "ocr_zh": "正方形ABCD，点E是边BC的中点，角AEF为90度，且EF交正方形外角的平分线CF于点F", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_590.jpg" }, { "index": 591, "ocr_en": "Square ABCD, point E is a point on the extension of side BC, angle AEF is 90 degrees, and EF intersects CF, the bisector of the exterior angle of the square, at point F.", "ocr_zh": "有正方形ABCD，点E是边BC的延长线上一点，角AEF为90度，且EF交正方形外角的平分线CF于点F", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_591.jpg" }, { "index": 592, "ocr_en": "In quadrilateral ABCD, which is a square, point E is on side BC. Angle AEF is 90 degrees, and AE = EF.", "ocr_zh": "有四边形ABCD是正方形，点E是边BC上一点，角AEF为90度，AE=EF\n", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_592.jpg" }, { "index": 593, "ocr_en": "In quadrilateral ABCD, which is a square, point E is the midpoint of side BC. Angle AEF is 90 degrees, and AE = EF.", "ocr_zh": "有四边形ABCD是正方形，点E是边BC的中点，角AEF为90度，AE=EF", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_593.jpg" }, { "index": 594, "ocr_en": "In quadrilateral ABCD, which is a square, point E is on the extension of side CB. Angle AEF is 90 degrees, and AE = EF.", "ocr_zh": "有四边形ABCD是正方形，点E是边CB的延长线上一点，角AEF为90度，AE=EF", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_594.jpg" }, { "index": 595, "ocr_en": "In quadrilateral ABCD, which is a square, point E is on the extension of side BC. Angle AEF is 90 degrees, and AE = EF.", "ocr_zh": "有四边形ABCD是正方形，点E是边BC的延长线上一点，角AEF为90度，AE=EF", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_595.jpg" }, { "index": 596, "ocr_en": "In quadrilateral ABCD, which is a square, point E is the midpoint of side BC. AE = EF, and EF intersects CF, the bisector of the exterior angle of the square, at point F.", "ocr_zh": "有四边形ABCD是正方形，点E是边BC的中点，AE=EF，且EF交正方形外角的平分线CF于点F，", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_596.jpg" }, { "index": 597, "ocr_en": "In quadrilateral ABCD, which is a square, point E is on the extension of side BC. AE = EF, and EF intersects CF, the bisector of the exterior angle of the square, at point F.", "ocr_zh": "有四边形ABCD是正方形，点E是边BC的延长线上一点，AE=EF，且EF交正方形外角的平分线CF于点F", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_597.jpg" }, { "index": 598, "ocr_en": "In quadrilateral ABCD, which is a square, point E is the midpoint of side BC. Angle AEF is 90 degrees, and EF intersects CF, the bisector of the exterior angle of the square, at point F.", "ocr_zh": "有四边形ABCD是正方形，点E是边BC的中点，角AEF为90度，且EF交正方形外角的平分线CF于点F", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_598.jpg" }, { "index": 599, "ocr_en": "In quadrilateral ABCD, which is a square, point E is on side BC. Angle AEF is 90 degrees, and EF intersects CF, the bisector of the exterior angle of the square, at point F.", "ocr_zh": "有四边形ABCD是正方形，点E是边BC上一点，角AEF为90度，且EF交正方形外角的平分线CF于点F", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_599.jpg" }, { "index": 600, "ocr_en": "In quadrilateral ABCD, which is a square, point E is on the extension of side CB. AE = EF, and EF intersects CF, the bisector of the exterior angle of the square, at point F.", "ocr_zh": "有四边形ABCD 是正方形，点E是边CB 的延长线上一点，AE=EF，且EF交正方形外角的平分线CF于点F", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_600.jpg" }, { "index": 601, "ocr_en": "In quadrilateral ABCD, which is a square, point E is on the extension of side CB. Angle AEF is 90 degrees, and EF intersects CF, the bisector of the exterior angle of the square, at point F.", "ocr_zh": "有四边形ABCD是正方形，点E是边CB的延长线上一点，角AEF为90度，且EF交正方形外角的平分线CF于点F", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_601.jpg" }, { "index": 602, "ocr_en": "In quadrilateral ABCD, which is a square, point E is on the extension of side BC. Angle AEF is 90 degrees, and EF intersects CF, the bisector of the exterior angle of the square, at point F.", "ocr_zh": "有四边形ABCD是正方形，点E是边BC的延长线上一点，角AEF为90度，且EF交正方形外角的平分线CF于点F", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_602.jpg" }, { "index": 603, "ocr_en": "In the left figure, there is a circle O. Line segment AB is a diameter and intersects chord CD perpendicularly at point E. Points A, B, C, and G are on the circle, forming an inscribed quadrilateral ABCG. Additionally, there is an external point F, which forms a right-angled triangle FBC with points B and C on the circle. H is a point on CG. Connect OH, CO, OD, auxiliary line CA, auxiliary line DA, and auxiliary line DG. There is an external point N, which forms a right-angled triangle ADN with points A and D, with angle AND being a right angle. DM is perpendicular to CA. Mark triangle AMD in blue and triangle AND in yellow.\n\n---\n\nIn the middle figure, there is a circle O. Line segment AB is a diameter and intersects chord CD perpendicularly at point E. Points A, B, C, and G are on the circle, forming an inscribed quadrilateral ABCG. Additionally, there is an external point F, which forms a right-angled triangle FBC with points B and C on the circle. H is a point on CG. Connect OH, CO, OD, auxiliary line CA, auxiliary line DA, and auxiliary line DG. There is an external point N, which forms a right-angled triangle ADN with points A and D, with angle AND being a right angle. DM is perpendicular to CA. Mark triangle CMD in blue and triangle GND in blue.\n\n---\n\nIn the right figure, there is a circle O. Line segment AB is a diameter and intersects chord CD perpendicularly at point E. Points A, B, C, and G are on the circle, forming an inscribed quadrilateral ABCG. Additionally, there is an external point F, which forms a right-angled triangle FBC with points B and C on the circle. H is a point on CG. Connect OH, CO, OD, auxiliary line CA, auxiliary line DA, and auxiliary line DG. There is an external point N, which forms a right-angled triangle ADN with points A and D, with angle AND being a right angle. DM is perpendicular to CA. Mark triangle HGD in purple and triangle AND in grayish-blue.", "ocr_zh": "左图有圆O，线段AB为直径，并与弦 CD 垂直相交于点 E。圆周上分布着 A,B,C,G 四点，它们共同构成了一个内接四边形 ABCG。此外，圆外还存在一个点 F，与圆上的 B,C 两点共同构成了一个直角三角形 FBC，H为CG上的一点，连接OH、CO、OD，连接辅助线CA、辅助线DA、辅助线DG、圆外有一点N，与点A、D、N构成直角三角形ADN，角AND为直角，DM与CA相互垂直于点M。将三角形AMD标记为蓝色、三角形AND标记为黄色；中图有圆O，线段AB为直径，并与弦 CD 垂直相交于点 E。圆周上分布着 A,B,C,G 四点，它们共同构成了一个内接四边形 ABCG。此外，圆外还存在一个点 F，与圆上的 B,C 两点共同构成了一个直角三角形 FBC，H为CG上的一点，连接OH、CO、OD，连接辅助线CA、辅助线DA、辅助线DG、圆外有一点N，与点A、D、N构成直角三角形ADN，角AND为直角，DM与CA相互垂直于点M。将三角形CMD标记为蓝色、三角形GND标记为蓝色；右图有圆O，线段AB为直径，并与弦 CD 垂直相交于点 E。圆周上分布着 A,B,C,G 四点，它们共同构成了一个内接四边形 ABCG。此外，圆外还存在一个点 F，与圆上的 B,C 两点共同构成了一个直角三角形 FBC，H为CG上的一点，连接OH、CO、OD，连接辅助线CA、辅助线DA、辅助线DG、圆外有一点N，与点A、D、N构成直角三角形ADN，角AND为直角，DM与CA相互垂直于点M。将三角形HGD标记为紫色、三角形AND标记为灰蓝色", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_603.jpg" }, { "index": 604, "ocr_en": "There is a circle O. Line segment AB is a diameter and intersects chord CD perpendicularly at point E. Points A, B, C, and G are distributed on the circumference, forming an inscribed quadrilateral ABCG. Additionally, there is a point F outside the circle, which forms a right-angled triangle FBC with points B and C on the circle. H is a point on CG. Connect OH, CO, and OD.", "ocr_zh": "有圆O，线段AB为直径，并与弦 CD 垂直相交于点 E。圆周上分布着 A,B,C,G 四点，它们共同构成了一个内接四边形 ABCG。此外，圆外还存在一个点 F，与圆上的 B,C 两点共同构成了一个直角三角形 FBC，H为CG上的一点，连接OH、CO、OD", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_604.jpg" }, { "index": 605, "ocr_en": "In the left figure, there is a circle O. Line segment AB is a diameter and intersects chord CD perpendicularly at point E. Points A, B, C, and G are on the circle, forming an inscribed quadrilateral ABCG. Additionally, there is an external point F, which forms a right-angled triangle FBC with points B and C on the circle. H is a point on CG. Connect OH, CO, OD, auxiliary line CA, auxiliary line DA, and auxiliary line DG. Mark triangle AGD in green and triangle ACD in gray.\n\n---\n\nIn the middle figure, there is a circle O. Line segment AB is a diameter and intersects chord CD perpendicularly at point E. Points A, B, C, and G are on the circle, forming an inscribed quadrilateral ABCG. Additionally, there is an external point F, which forms a right-angled triangle FBC with points B and C on the circle. H is a point on CG. Connect OH, CO, OD, auxiliary line CA, auxiliary line DA, and auxiliary line DG. There is an external point N, which forms a right-angled triangle ADN with points A and D, with angle AND being a right angle. DM is perpendicular to CA. Mark triangle AMD in blue and triangle AND in yellow.\n\n---\n\nIn the right figure, there is a circle O. Line segment AB is a diameter and intersects chord CD perpendicularly at point E. Points A, B, C, and G are on the circle, forming an inscribed quadrilateral ABCG. Additionally, there is an external point F, which forms a right-angled triangle FBC with points B and C on the circle. H is a point on CG. Connect OH, CO, OD, auxiliary line CA, auxiliary line DA, and auxiliary line DG. There is an external point N, which forms a right-angled triangle ADN with points A and D, with angle AND being a right angle. DM is perpendicular to CA. Mark triangle CDM in blue and triangle GND in blue.", "ocr_zh": "左图有圆O，线段AB为直径，并与弦 CD 垂直相交于点 E。圆周上分布着 A,B,C,G 四点，它们共同构成了一个内接四边形 ABCG。此外，圆外还存在一个点 F，与圆上的 B,C 两点共同构成了一个直角三角形 FBC，H为CG上的一点，连接OH、CO、OD，连接辅助线CA、辅助线DA、辅助线DG。将三角形AGD标记为绿色、三角形ACD标记为灰色；中图有圆O，线段AB为直径，并与弦 CD 垂直相交于点 E。圆周上分布着 A,B,C,G 四点，它们共同构成了一个内接四边形 ABCG。此外，圆外还存在一个点 F，与圆上的 B,C 两点共同构成了一个直角三角形 FBC，H为CG上的一点，连接OH、CO、OD，连接辅助线CA、辅助线DA、辅助线DG、圆外有一点N，与点A、D、N构成直角三角形ADN，角AND为直角，DM与CA相互垂直。将三角形AMD标记为蓝色、三角形AND标记为黄色；右图有圆O，线段AB为直径，并与弦 CD 垂直相交于点 E。圆周上分布着 A,B,C,G 四点，它们共同构成了一个内接四边形 ABCG。此外，圆外还存在一个点 F，与圆上的 B,C 两点共同构成了一个直角三角形 FBC，H为CG上的一点，连接OH、CO、OD，连接辅助线CA、辅助线DA、辅助线DG、圆外有一点N，与点A、D、N构成直角三角形ADN，角AND为直角，DM与CA相互垂直。将三角形CDM标记为蓝色、三角形GND标记为蓝色", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_605.jpg" }, { "index": 606, "ocr_en": "There is a circle O. Line segment AB is a diameter and intersects chord CD perpendicularly at point E. Points A, B, C, and G are distributed on the circumference, forming an inscribed quadrilateral ABCG. Additionally, there is an external point F, which forms a right-angled triangle FBC with points B and C on the circle. Connect BD. H is a point on CG. Connect DH passing through the center O. Connect auxiliary lines DG and DA. Angle AOD is Angle 2, Angle DCH is Angle 1, Angle FCE is Angle 4, and Angle BCE is Angle 3.", "ocr_zh": "有圆O，线段AB为直径，并与弦 CD 垂直相交于点 E。圆周上分布着 A,B,C,G 四点，它们共同构成了一个内接四边形 ABCG。此外，圆外还存在一个点 F，与圆上的 B,C 两点共同构成了一个直角三角形 FBC,连接BD，H为CG上一点，连接DH过圆心O，连接辅助线DG、辅助线DA。角AOD为角2，角DCH为角1,角FCE为角4，角BCE为角3", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_606.jpg" }, { "index": 607, "ocr_en": "There is a triangle GAD. Extend DA to a point, and connect this point to G such that the angle formed by this point, G, and A is equal to angle DGA, both being Angle 3. The length of side GD is $4\\sqrt{13}$, and the length of side AD is $a$.", "ocr_zh": "有三角形GAD，延长DA至一点并连接该点与点G使得该点与点G和点A所成的夹角与角DGA相等，同为角3，边GD长为4√13，边AD长为a", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_607.jpg" }, { "index": 608, "ocr_en": "There is a triangle GAD. Extend GA to point T. Connect auxiliary line DT. Angle G is Angle 1, Angle GAD is Angle 4, Angle ADT is Angle 2, and Angle T is Angle 3. The lengths of DG and DT are $4\\sqrt{13}$, and the length of GA is 3.", "ocr_zh": "有三角形GAD，延长GA至点T，连接辅助线DT，角G为角1，角GAD为角4，角ADT为角2，角T为角3.DG、DT长为4√13 ，GA长为3", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_608.jpg" }, { "index": 609, "ocr_en": "There is a triangle GAD. Point T is on GD. Connect auxiliary line AT. Angle TAD is Angle a, and Angle G is Angle a. The length of GD is $4\\sqrt{13}$, and the length of AD is $b$.", "ocr_zh": "有三角形GAD，点T为GD上一点，连接辅助线AT，角TAD为角a，角G为角a，GD长为4√13 ， AD长为b", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_609.jpg" }, { "index": 610, "ocr_en": "In the upper left figure within the rectangular coordinate plane xoy, point A lies on the positive y-axis and point B lies on the positive x-axis. Connect AB.\n\nIn the upper right figure within the rectangular coordinate plane xoy, point A lies on the positive y-axis and point B lies on the positive x-axis. Connect AB.\n\nIn the middle left figure within the rectangular coordinate plane xoy, points A, G, and P lie on the positive y-axis, and point B lies on the positive x-axis. There is a point H in the first quadrant that shares the same y-coordinate as point G and the same x-coordinate as point B. Connect auxiliary line GH. Connect auxiliary line PB, which intersects GH at point K. Connect AK, and AK is perpendicular to PB. The length of GK is 4 − 2m, the length of KH is 2m, GA is m, and HB is m + 3. The coordinates of point P are (0, 8), and the length of segment HB is equal to 2(4 − 2m).\n\nIn the middle right figure within the rectangular coordinate plane xoy, point B lies on the positive x-axis, and points A and P lie on the positive y-axis. Connect auxiliary line PB. Point K is on PB. Connect auxiliary lines AK and AB. Through point K, draw a line parallel to the x-axis that intersects the positive y-axis. Angle KAB is a right angle. Segment OB has length 4, and segment AO has length 3. The coordinates of point K are (3/2, 5). The equation of the line passing through points B and P (denoted as l\\_BP) is y = −2x + 8.\n\nIn the lower left figure within the rectangular coordinate plane xoy, points A and P lie on the positive y-axis, point B lies on the positive x-axis, and point H lies on the negative x-axis. There is a point G in the second quadrant. Connect GH. There is a point on GH connected to points P and B such that the angles formed are right angles. The length of GP is 2t. Assuming that the length of segment GP is 2t, the length of segment HB can be expressed as 2t + 4. The ratio of lengths KH to BH is 4:3. Solving this, the variable t is found to be 6/5, and the coordinates of point P are determined to be (0, 8).\n\nIn the lower right figure within the rectangular coordinate plane xoy, point B lies on the positive x-axis, and points A and P lie on the positive y-axis. Through point P, draw a line parallel to the x-axis. It intersects the perpendicular from point B to the y-axis at point H in the first quadrant and intersects in the second quadrant at point G. There is another point K in the second quadrant. Points G, P, and K form a right triangle. In the first quadrant, there is a right triangle PHB. Assuming that the length of GP is m, and with GP = m, the coordinates of point K are (−m, 2m). The variable m is found to be 4, and the coordinates of point P are determined to be (0, 8).\n", "ocr_zh": "左上图在平面直角坐标系xoy中，点A在y轴正半轴上，点B在x轴正半轴上，连接AB；右上图在平面直角坐标系xoy中，点A在y轴正半轴上，点B在x轴正半轴上，连接AB；左中图在平面直角坐标系xoy中，点A、G、P在y正半轴上，点B在x轴正半轴上，第一象限有一点H与点G共y、与点B共x，连接辅助线GH，连接辅助线PB与GH相交于点K，连接AK，AK与PB相互垂直。GK长为4-2m,KH长为2m，GA长为m，HB长为m+3，点 P 的坐标是 (0, 8)，线段 HB 的长度等于 2(4−2m)，；右中图在平面直角坐标系xoy上，点B在x正半轴上，点A、P在y正半轴上，连接辅助线PB，点K为PB上的一点，连接辅助线AK、连接AB，过点K与x轴平行与y正半轴相交，角KAB为直角，边OB长为4，边AO长为3，点 K 的坐标是(3/2,5)，经过点 B 和点 P 的直线（通常记作 lBP）的方程是 y=−2x+8；左下图在平面直角坐标系xoy中，点A、点P在y正半轴上，点B在x正半轴上，点H在x负半轴上，第二象限内有一点G，连接GH，GH上有一点连接该点与点P、点B且所成角为直角，GP长为2t，假设线段 GP 的长度为 2t，根据 GP 的长度线段 HB 的长度可以表示为 2t+4，线段 KH 与 BH 的比值等于4/3，可以解出变量 t 的值为6/5，进一步确定点 P 的坐标为 (0,8) ；右下图在平面直角坐标系xoy中，点B在x正半轴上，点A、P在y正半轴上，过点P作x轴平行线在第一象限与点B与y轴的垂线相交于点H，在第二象限交于点G，第二象限另有一点K，点G、P、K为直角三角形，在第一象限有直角三角形PHB，假设线段 GP 的长度为 m，在 GP=m 的情况下，点 K 的坐标为 (-m, 2m)，得出变量 m 的值为 4，确定点 P 的坐标为 (0,8)", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_610.jpg" }, { "index": 611, "ocr_en": "In the left figure within the rectangular coordinate plane xoy, points A and P lie on the positive y-axis, point B is on the positive x-axis, and point H is on the negative x-axis. There is a point G in the second quadrant. Connect GH, and on GH there is a point connected to points P and B, forming right angles. The length of GP is 2t. Assuming that the length of segment GP is 2t, the length of segment HB can be expressed as 2t + 4. The ratio of the lengths of segments KH to BH is 4:3. Solving this, the variable t is found to be 6/5, and the coordinates of point P are determined to be (0, 8).\n\nIn the right figure within the rectangular coordinate plane xoy, point B lies on the positive x-axis, and points A and P lie on the positive y-axis. Through point P, draw a line parallel to the x-axis, intersecting the perpendicular from point B to the y-axis at point H in the first quadrant, and intersecting in the second quadrant at point G. There is another point K in the second quadrant. Points G, P, and K form a right triangle. In the first quadrant, there is a right triangle PHB. Assuming the length of GP is m, then with GP = m, the coordinates of point K are (-m, 2m). The variable m is found to be 4, and the coordinates of point P are determined to be (0, 8).\n", "ocr_zh": "左图在平面直角坐标系xoy中，点A、点P在y正半轴上，点B在x正半轴上，点H在x负半轴上，第二象限内有一点G，连接GH，GH上有一点连接该点与点P、点B且所成角为直角，GP长为2t，假设线段 GP 的长度为 2t，根据 GP 的长度线段 HB 的长度可以表示为 2t+4，线段 KH 与 BH 的比值等于4/3，可以解出变量 t 的值为6/5，进一步确定点 P 的坐标为 (0,8) ；右图在平面直角坐标系xoy中，点B在x正半轴上，点A、P在y正半轴上，过点P作x轴平行线在第一象限与点B与y轴的垂线相交于点H，在第二象限交于点G，第二象限另有一点K，点G、P、K为直角三角形，在第一象限有直角三角形PHB，假设线段 GP 的长度为 m，在 GP=m 的情况下，点 K 的坐标为 (-m, 2m)，得出变量 m 的值为 4，确定点 P 的坐标为 (0,8)", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_611.jpg" }, { "index": 612, "ocr_en": "In the Cartesian coordinate system xOy, points A and P are on the positive y-axis, point B is on the positive x-axis, and point H is on the negative x-axis. There is a point G in the second quadrant. Connect GH. There is a point on GH that connects to points P and B, forming a right angle. The length of GP is $2t$.", "ocr_zh": "在平面直角坐标系xoy中，点A、点P在y正半轴上，点B在x正半轴上，点H在x负半轴上，第二象限内有一点G，连接GH，GH上有一点连接该点与点P、点B且所成角为直角，GP长为2t，假设线段 GP 的长度为 2t，根据 GP 的长度线段 HB 的长度可以表示为 2t+4，线段 KH 与 BH 的比值等于4/3，可以解出变量 t 的值为6/5，进一步确定点 P 的坐标为 (0,8)", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_612.jpg" }, { "index": 613, "ocr_en": "In the rectangular coordinate plane xoy, point B lies on the positive x-axis, and points P, A, and H lie on the positive y-axis. Connect AB. Point G lies on AB. Connect auxiliary line HG, which is parallel to the x-axis. Connect auxiliary lines GP and PB. The length of HP is 2, and OB is 4. Let PO = 2m. The length of segment HG is m. It is deduced that the coordinates of point G are (m, 2m + 2). Substituting point G into line AB yields the value of m as 4/11, and the coordinates of point P are (0, 8/11).\n", "ocr_zh": "在平面直角坐标系xoy中，有点B在x正半轴上，点p、点A、点H在y正半轴上，连接AB，点G在AB上，连接辅助线HG与x轴平行，连接辅助线Gp、辅助线pB。Hp长为2，OB长为4，设 PO=2m，线段 HG 的长度等于 m，推导出点 G 的坐标为 (m, 2m+2)，把G带入lAB 求得m的值为4/11，点P的坐标为（0，8/11）\n​", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_613.jpg" }, { "index": 614, "ocr_en": "In the rectangular coordinate plane xoy, points P and A lie on the positive y-axis, and point B lies on the positive x-axis. In the first quadrant, point H shares the same y-coordinate as point A and the same x-coordinate as point B. In the second quadrant, there are points G and K forming a right trapezoid GKBH. Connect AK and AB. Angle KAB is a right angle. The coordinates of point K are (−1.5, 1). The equation of line BK is y = −(2/11)x + 8/11, and the coordinates of point P are (0, 8/11).\n", "ocr_zh": "在平面直角坐标系xoy内，点p、点A在y正半轴上，点B在x正半轴上，第一象限有点H与点A共y、与点B共x,在第二象限有点G、K，构成直角梯形GKBH，连接AK、AB，角KAB为直角，点 K 的坐标是 (-1.5, 1)，lBK 的方程为y=-(2/11)x+(8/11)，点P的坐标为（0，8/11）", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_614.jpg" }, { "index": 615, "ocr_en": "In the rectangular coordinate plane xoy, points P and A lie on the positive y-axis, point B lies on the positive x-axis, and point H lies on the negative y-axis. In the second quadrant, there is a point G sharing the same y-coordinate as point A, and another point K. Connect AB, AG, AK, GK, KB, and KH. Segment GA is denoted as $a$, and segment KH is $2a$. Angle AKB is a right angle. Given $AG = a$, the length of segment KH can be derived as $2a$. Solving yields $a = 4/5$, the coordinates of point K are (−5/2, 4/5), and the coordinates of point P are (0, 8/11).\n", "ocr_zh": "在平面直角坐标系xoy上，点p、点A在y正半轴上，点A在x正半轴上，点H在y负半轴上，在第二象限有一点G与点A共y，另有一点K，连接AB、AG、AK、GK、KB、KH。边GA为a,边KH为2a,角AKB为直角，设 AG=a，线段 KH 的长度可以推导为 2a，求得a=4/5，K 点的坐标是 (−5/2,4/5)，P 点坐标为(0,8/11)", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_615.jpg" }, { "index": 616, "ocr_en": "In the Cartesian coordinate system xOy, point B is on the positive x-axis, and points P, G, and A are on the positive y-axis. Point H is in the first quadrant, sharing its y-coordinate with G and its x-coordinate with B. Connect auxiliary lines GH, HB, and PB. Connect AB, which intersects GH at point K. Connect PK. Angle PKB is a right angle. The length of side GK is $m$, KH is $4-m$, and HB is $2m$.", "ocr_zh": "在平面直角坐标系xoy上，有点B在x正半轴上，点p、G、A在y正半轴上，点H在第一象限与G共y、与B共x，连接辅助线GH、HB、PB,连接AB与GH交于点K，连接pK，角pKB为直角。边GK为m,KH为4-m，边HB为2m", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_616.jpg" }, { "index": 617, "ocr_en": "In the Cartesian coordinate system xOy, point B is on the positive x-axis, and points A, P, and P2 are on the positive y-axis. Connect PB. There is a point K on PB, with coordinates (3/2, 5). Extend BP2 into the second quadrant to point K'. A line parallel to the x-axis passes through point K and intersects the y-axis. Angle KAB is a right angle.", "ocr_zh": "在平面直角坐标系xoy中，点B在x正半轴上，点A、P、P2在y正半轴上，连接PB,PB上有一点K，点K坐标为（3/2，5），连接BP2延长在第二象限至点K'，过点K作x轴平行线交于y轴，角KAB为直角", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_617.jpg" }, { "index": 618, "ocr_en": "In the Cartesian coordinate system xOy, points P and A are on the positive y-axis, and point B is on the positive x-axis. Connect PB and PH. PH is perpendicular to AB. The length of PO is $3-5t$, AP is $5t$, PH is $4t$, AH is $3t$, and HB is $8t$.", "ocr_zh": "在平面直角坐标系xoy上，点p、点A在y正半轴上，点B在x正半轴上，连接pB、pH,pH垂直于AB于点H。pO长为3-5t，Ap长为5t，pH长为4t，AH长为3t，HB长为8t", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_618.jpg" }, { "index": 619, "ocr_en": "In the left figure, in the Cartesian coordinate system xOy, point B is on the positive x-axis, and points P, G, and A are on the positive y-axis. Point H is in the first quadrant, sharing its y-coordinate with G and its x-coordinate with B. Connect auxiliary lines GH, HB, and PB. Connect AB, which intersects GH at point K. Connect PK. Angle PKB is a right angle. The length of side GK is $m$, KH is $4-m$, and HB is $2m$.\n\n---\n\nIn the right figure, in the Cartesian coordinate system xOy, points P and A are on the positive y-axis, and point B is on the positive x-axis. Connect PB and PH. PH is perpendicular to AB. The length of PO is $3-5t$, AP is $5t$, PH is $4t$, AH is $3t$, and HB is $8t$.", "ocr_zh": "左图在平面直角坐标系xoy上，有点B在x正半轴上，点p、G、A在y正半轴上，点H在第一象限与G共y、与B共x，连接辅助线GH、HB、PB,连接AB与GH交于点K，连接pK，角pKB为直角。边GK为m,KH为4-m，边HB为2m；右图在平面直角坐标系xoy上，点p、点A在y正半轴上，点B在x正半轴上，连接pB、pH,pH垂直于AB于点H。pO长为3-5t，Ap长为5t，pH长为4t，AH长为3t，HB长为8t", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_619.jpg" }, { "index": 620, "ocr_en": "In the Cartesian coordinate system xOy, points C and Q1 are on the positive y-axis, points D and B are on the positive x-axis, point A is on the negative x-axis, and point Q2 is on the negative y-axis. Connect PQ1, AC, CD, PQ2, and OQ2. Point Q2' is on line segment PQ2.", "ocr_zh": "在平面直角坐标系xoy中，点C、Q1在y正半轴上，点D、点B在x正半轴上，点A在x负半轴上，点Q2在y负半轴上。连接PQ1、AC、CD、PQ2、OQ2，点Q2'在线段PQ2上", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_620.jpg" }, { "index": 621, "ocr_en": "In the Cartesian coordinate system xOy, points Q and C are on the positive y-axis, points D and B are on the positive x-axis, and point A is on the negative x-axis. Connect PQ, PA, and CD. PA intersects the y-axis at point C.", "ocr_zh": "在平面直角坐标系xoy中，点Q、C在y正半轴上，点D、点B在x正半轴上，点A在x负半轴上，连接PQ、PA、CD，PA与y轴交于点C", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_621.jpg" }, { "index": 622, "ocr_en": "In the Cartesian coordinate system xOy, points D and B are on the positive x-axis, point A is on the negative x-axis, and point C is on the positive y-axis. Connect to form triangle ACD. Extend AC into the first quadrant to point P.", "ocr_zh": "在平面直角坐标系xoy中，点D、点B在x正半轴上，点A在x负半轴上，点C在y正半轴上，连接构成三角形ACD，延长AC在第一象限至点p", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_622.jpg" }, { "index": 623, "ocr_en": "The left figure shows a right triangle AOB in the Cartesian coordinate system xoy, with point A on the positive y-axis and point B on the positive x-axis. The right figure shows triangle AOB in the Cartesian coordinate system xoy, with points B and C on the positive x-axis and point A on the positive y-axis. Auxiliary lines AC and BC are drawn.", "ocr_zh": "左图在平面直角坐标系xoy内，点A在y正半轴上，点B在x正半轴上，构成直角三角形AOB；右图在平面直角坐标系xoy中，点B、点C在x正半轴上，点A在y正半轴上，连接构成三角形AOB，连接辅助线AC、辅助线BC", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_623.jpg" }, { "index": 624, "ocr_en": "The figure shows a right triangle AOB in the Cartesian coordinate system xoy, with point A on the positive y-axis and point B on the positive x-axis. In the Cartesian coordinate system xoy, point A is on the positive y-axis, points B and C are on the positive x-axis, forming right triangle AOB. Auxiliary line AC is drawn. The length of AC is $x$, the length of OC is $8-x$, and the length of AO is $4$.", "ocr_zh": "图在平面直角坐标系xoy内，点A在y正半轴上，点B在x正半轴上，构成直角三角形AOB；右图在平面直角坐标系xoy中，点A在y正半轴，点B、点C在x正半轴上，构建直角三角形AOB、连接辅助线AC，AC长为x，OC长为8-x，AO长为4", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_624.jpg" }, { "index": 625, "ocr_en": "In the Cartesian coordinate system xoy, point A is on the positive y-axis, and point B is on the positive x-axis, forming a right triangle AOB.", "ocr_zh": "在平面直角坐标系xoy内，点A在y正半轴上，点B在x正半轴上，构成直角三角形AOB", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_625.jpg" }, { "index": 626, "ocr_en": "Given triangle ABC, extend BC to point D. Connect AD. There is a point E on AD. Connect BE, which intersects side AC at point F.", "ocr_zh": "有三角形ABC，延长BC至点D，连接AD，AD上有一点E，连接BE与边AC交于点F", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_626.jpg" }, { "index": 627, "ocr_en": "In triangle ABC, point D is on BC, and point E is on AB. Connect CE and connect AD. AD is perpendicular to BC.", "ocr_zh": "在三角形ABC上，点D在BC上，点E在AB上，连接CE，连接AD，AD与BC相互垂直于点D", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_627.jpg" }, { "index": 628, "ocr_en": "Given square ABCD, point G is on AB, and point E is on BC. Connect AE. Point F is on AE. Connect GF and connect FD.", "ocr_zh": "有正方形ABCD，点G在AB上，点E在BC上，连接AE，点F在AE上，连接GF，连接FD", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_628.jpg" }, { "index": 629, "ocr_en": "Given a tetrahedron with vertex A and base BCD.", "ocr_zh": "有四面体，顶点为A，底面为BCD", "class": "Solid Geometry", "difficulty": 1, "image_path": "fig_629.jpg" }, { "index": 630, "ocr_en": "Given right triangle ABC, point E is on BC, point D is on AC. Connect AE and ED. Angle ABC is 90 degrees.", "ocr_zh": "有直角三角形ABC，点E在BC上，点D在AC上，连接AE、ED，角ABC为90度", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_630.jpg" }, { "index": 631, "ocr_en": "Given a right triangle ABC, with angle ABC equal to 90 degrees. Point E is on BC, and point D is on AC. Connect AE and ED.", "ocr_zh": "有直角三角形ABC，点E在BC上，点D在AC上，连接AE、ED，角ABC为90度", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_631.jpg" }, { "index": 632, "ocr_en": "Given a right triangle ABC, with angle ABC equal to 90 degrees. Point E is on BC, and point D is on AC. Connect AE and ED.", "ocr_zh": "有直角三角形ABC，点E在BC上，点D在AC上，连接AE、ED，角ABC为90度", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_632.jpg" }, { "index": 633, "ocr_en": "In triangle ABC, extend CA to point E, and extend BA to point D. Connect EB and connect CD.", "ocr_zh": "在三角形ABC，延长CA至点E，延长BA至点D，连接EB，连接CD", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_633.jpg" }, { "index": 634, "ocr_en": "Given an isosceles trapezoid ADCB, point F is on CD, and point E is on BC. Connect AF and AE. Rotate triangle ADF clockwise around point A by 120 degrees to obtain triangle ABF′. Construct the circumcircle O of triangle AEF′. Draw an auxiliary perpendicular line AG from point A intersecting BE at point G. Draw an auxiliary perpendicular line OH from point O intersecting AE at point H. Connect auxiliary lines AF′, OA, and OF′.", "ocr_zh": "有等腰梯形ADCB，点F在CD上，点E在BC上，连接AF、AE，将三角形ADF绕A点顺时针旋转120度，得三角形ABF′，作AEF′的外接圆圆O，从点A作辅助垂线AG交BE于点G，从点O作辅助垂涎OH与AE交于点H，连接辅助线AF'、OA、OF‘", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_634.jpg" }, { "index": 635, "ocr_en": "Given an isosceles trapezoid ADCB, with point F on CD and point E on BC. Connect AF and AE", "ocr_zh": "有等腰梯形ADCB，点F在CD上，点E在BC上，连接AF、AE", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_635.jpg" }, { "index": 636, "ocr_en": "Given triangle ABC, construct its circumcircle O. Points H and D are on BC. Connect AD, auxiliary line OA, auxiliary line OC, and auxiliary line OB. Draw a perpendicular line OE from point O to AD, intersecting at point E. Draw a perpendicular line OH from point O to BC, intersecting BC at point H. AD is perpendicular to BC.", "ocr_zh": "有三角形ABC，作该三角形外接圆圆O，点H、点D在BC上，连接AD、辅助线OA、辅助线OC、辅助线OB，过点O向AD作垂线OE交于点E,从点O向BC作垂线OH交BC于点H， AD于与BC相互垂直于点D", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_636.jpg" }, { "index": 637, "ocr_en": "There is a set of parallel lines. Point A is on one line, and points B, C, D are on the other line. Connect AB, AD, AC, BD, CD, and BC. Draw the circumcircle O of triangle ABC.", "ocr_zh": "有一组平行线，其中一条线上有点A，另外一条线上有点B、点C、点D，连接AB、AD、AC、BD、CD、BC，作三角形ABC外接圆圆0", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_637.jpg" }, { "index": 638, "ocr_en": "Given triangle ABC, point D is on AB. Connect AD. Construct the circumcircle O of triangle ABC. Connect auxiliary lines OA, OB, and OC. Draw a perpendicular line OH from point O to AB, intersecting AB at point H.", "ocr_zh": "有三角形ABC,点D在AB上，连接AD。作三角形ABC外接圆圆O，连接辅助线OA、辅助线OB、辅助线OC，过点O作垂线OH与AB相交于点H", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_638.jpg" }, { "index": 639, "ocr_en": "In quadrilateral ABCD, point E is on AB, and point F is on AD. Connect CF, CE, and AC. Angle D is a right angle, angle B is a right angle, and CD = CB.", "ocr_zh": "在四边形ABCD中，点E在AB上，点F在AD上，连接CF、CE、AC，角D为直角、角B为直角，CD=CB", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_639.jpg" }, { "index": 640, "ocr_en": "In triangle ABC, point D is on AB. Connect CD, and CD is perpendicular to AB.", "ocr_zh": "在三角形ABC中，点D在AB上，连接CD且CD垂直于AB于点D", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_640.jpg" }, { "index": 641, "ocr_en": "In quadrilateral ABCD, point E is on AB, and point F is on AD. Connect CF, CE, and AC. Angle D is a right angle, angle B is a right angle, and CD = CB. Find a point H on AB such that AH = HC. Extend AB to G such that BG = FD. Connect CG. Construct the circumcircle O of triangle CEG. Connect auxiliary lines OC, OK, OG, OE, and CH.", "ocr_zh": "在四边形ABCD中，点E在AB上，点F在AD上，连接CF、CE、AC，角D为直角、角B为直角，CD=CB，在AB上找一点H，使AH=HC。延长AB至G，使BG=FD，连CG，作三角形CEG的外接圆圆O，连接辅助线OC、OK、OG、OE、CH", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_641.jpg" }, { "index": 642, "ocr_en": "Given an equilateral triangle ABC, with point P as an interior point.", "ocr_zh": "有等边三角形ABC，点P为其内一点", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_642.jpg" }, { "index": 643, "ocr_en": "Given an equilateral triangle ABC, point P is an interior point. Points P1 and P2 are located on the left and right sides of the triangle, respectively. Connect P1 and P2, intersecting AB at point M and AC at point N. Connect auxiliary lines PP1 and PP2.", "ocr_zh": "有等边三角形ABC，点P为其内一点，点P1、点P2分别位于三角形左右两侧，连接P1、P2分别与AB、AC相交于点M、点N，连接辅助线PP1、辅助线PP2", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_643.jpg" }, { "index": 644, "ocr_en": "Given an equilateral triangle ABC, point P is an interior point. Points P1 and P2 are located on the left and right sides of the triangle, respectively. Connect P1 and P2, intersecting AB at point M and AC at point N. Connect auxiliary lines PP1 and PP2. Draw circles with center A and radii AP1 and AP2. Connect AP, MP, and NP. Draw AD perpendicular to P1P2 from point A, intersecting at point D.", "ocr_zh": "有等边三角形ABC，点P为其内一点，点P1、点P2分别位于三角形左右两侧，连接P1、P2分别与AB、AC相交于点M、点N，连接辅助线PP1、辅助线PP2，以点A为圆心，AP1、AP2为半径作圆，连接AP、MP、NP，过点A作AD垂直于P1P2于点D", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_644.jpg" }, { "index": 645, "ocr_en": "In triangle AMN, points H and D are on MN. Draw AD perpendicular to MN from point A, intersecting at point D. Construct the circumcircle O of triangle AMN. Connect auxiliary lines OM, ON, and OA. Draw an auxiliary line OH perpendicular to MN from point O, intersecting MN at point H.", "ocr_zh": "在三角形AMN中，点H、点D在MN上，过点A作AD垂直于MN于点D，作三角形AMN的外接圆圆O，连接辅助线OM、辅助线ON、辅助线OA，过点O作辅助线OH垂直于MN于点H", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_645.jpg" }, { "index": 646, "ocr_en": "In quadrilateral ABCD, angle C is a right angle. Point M is on BC, and point N is on CD. Connect AN, AM, and MN.", "ocr_zh": "有四边形ABCD，角C为直角，点M在BC上，点N在CD上，连接AN、AM、MN", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_646.jpg" }, { "index": 647, "ocr_en": "In quadrilateral ABCD, AB = AD. Point F is on CD, and point E is on BC. Connect AF and AE.", "ocr_zh": "在四边形ABCD中，AB=AD，点F在CD上，点E在BC上，连接AF、AE", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_647.jpg" }, { "index": 648, "ocr_en": "In quadrilateral ABCD, point F is on CD, and point E is on BC. Connect AF, AE, and EF.", "ocr_zh": "在四边形ABCD中，点F在CD上，点E在BC上，连接AF、AE、EF", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_648.jpg" }, { "index": 649, "ocr_en": "In rhombus ABCD, point E is on AB, and point F is on BC. Connect ED, EF, and DF. Angle EDF is a right angle.", "ocr_zh": "在菱形ABCD中，点E在AB上，点F在BC上，连接ED、EF、DF，角EDF为直角", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_649.jpg" }, { "index": 650, "ocr_en": "In triangle ABC, extend CB to D such that BD = AB. Extend BC to E such that CE = AC. Connect auxiliary lines AD and AE to form triangle ADE. Circle O is its circumcircle.", "ocr_zh": "有三角形ABC，延长CB至D，使得BD=AB，延长BC至E，使得CE=AC，连接辅助线AD、辅助线AE，构成三角形ADE，圆O为其外接圆", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_650.jpg" }, { "index": 651, "ocr_en": "Given triangle ABC, construct the excircle O of triangle ABC tangent to side BC from below. Connect auxiliary lines OA, OB, and OC. Draw OD perpendicular to BC at point D. Extend AB to intersect the excircle at point E, and extend AC to intersect the excircle at point F. OE is perpendicular to AE, and OF is perpendicular to AF.\n", "ocr_zh": "有三角形ABC，作三角形ABC的旁切圆O在BC下方，连接辅助线OA、辅助线OB、辅助线OC，过点O作OD垂直于BC于点D，延长AB与旁切圆相交于点E，延长AC与旁切圆相交于点F，OE垂直于AE，OF垂直于AF", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_651.jpg" }, { "index": 652, "ocr_en": "In triangle ABC, extend BA to D such that AD = AC. Then BD = BA + AC. Connect auxiliary line CD, forming triangle BCD, and construct its circumcircle O.", "ocr_zh": "有三角形ABC，在BA的延长线上截取AD=AC，则BD=BA+AC，连接辅助线CD，构成三角形BCD并作其外接圆圆O", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_652.jpg" }, { "index": 653, "ocr_en": "Given triangle ABC, construct the excircle O of triangle ABC tangent to side AB from below. Connect auxiliary lines OA, OB, and OC. Draw OD perpendicular to BC at point D. Extend AB to intersect the excircle at point E, and extend AC to intersect the excircle at point F. OE is perpendicular to AE, and OF is perpendicular to AF.\n", "ocr_zh": "有三角形ABC，作三角形ABC的旁切圆O在AB下方，连接辅助线OA、辅助线OB、辅助线OC，过点O作OD垂直于BC于点D，延长AB与旁切圆相交于点E，延长AC与旁切圆相交于点F，OE垂直于AE，OF垂直于AF", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_653.jpg" }, { "index": 654, "ocr_en": "In triangle ABO, construct its circumcircle Q. Connect auxiliary lines BQ and OQ. Draw QH perpendicular to AB from point Q, intersecting at point H. Draw OD perpendicular to AB from point O, intersecting at point D.", "ocr_zh": "有三角形ABO，作其外接圆圆Q，连接辅助线BQ、OQ，过点Q作QH垂直于AB于点H，过点O作OD垂直于AB于点D", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_654.jpg" }, { "index": 655, "ocr_en": "In triangle ABC, point D is on AC. Connect AD, and AD is perpendicular to AC.", "ocr_zh": "有三角形ABC，点D在AC上,连接AD且AD垂直于AC于点D", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_655.jpg" }, { "index": 656, "ocr_en": "In triangle ABC, extend AC to D such that CD = CB. Then AC + CB = AD. Connect auxiliary line BD. Construct the circumcircle O of triangle ABD. Connect auxiliary lines OA and OB.", "ocr_zh": "有三角形ABC，在AC的延长线上截取CD=CB，则AC+CB=AD，连接辅助线BD，作三角形ABD的外接圆O，连接辅助线OA、辅助线OB", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_656.jpg" }, { "index": 657, "ocr_en": "Given triangle ABC, construct the excircle O of triangle ABC located to the upper right of side AC. From point B, draw BD perpendicular to AC at point D. Connect auxiliary line BO. Let point G be the intersection of triangle ABC and the excircle. Connect OG, with OG perpendicular to AC. Extend BA to intersect the excircle at point E, and extend BC to intersect the excircle at point F. Connect auxiliary lines OE and OF. OE is perpendicular to BE at point E, and OF is perpendicular to BF at point F.\n", "ocr_zh": "有三角形ABC，作三角形ABC的旁切圆O在AC右上方，过点B作BD垂直于AC于点D，连接辅助线BO，点G为三角形ABC与旁切圆的交点，连接OG，OG垂直于AC，延长BA与旁切圆相交于点E，延长BC与旁切圆相交于点F，连接辅助线OE、OF，OE与BE相互垂直于点E，OF与BF相互垂直于点F", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_657.jpg" }, { "index": 658, "ocr_en": "In triangle ABC.", "ocr_zh": "有三角形ABC", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_658.jpg" }, { "index": 659, "ocr_en": "In triangle OBC, construct its circumcircle OQ. Draw QH perpendicular to BC from Q, intersecting at H. Connect BQ and OQ. Draw QH perpendicular to BC from point Q, intersecting at point H. Draw OF perpendicular to BC from point O, intersecting at point F.", "ocr_zh": "有三角形OBC，作三角形OBC的外接圆OQ，过Q作QH垂直于BC于H，连接BQ、OQ，过点Q作QH垂直于BC于点H，过点O作OF垂直于BC于点F", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_659.jpg" }, { "index": 660, "ocr_en": "The upper figure shows circle O, with points A and B on circle O. Point P is a moving point on the major arc AB. Connect PA, PB, and AB. AC is perpendicular to AP, intersecting line PB at point C. The lower figure shows circle O, with points A and B on circle O. Point P is a moving point on the major arc AB. Connect PA, PB, and AB. AC is perpendicular to AP, intersecting line PB at point C. Construct the circumcircle D of triangle ABC. Connect auxiliary lines AD and BD.", "ocr_zh": "上图有圆O，点A、点B为圆O上的点，点P为优弧AB上一点，连接PA、PB、AB, AC垂直于AP于点A交直线PB于点C；下图有圆O，点A、点B为圆O上的点，点P为优弧AB上一点，连接PA、PB、AB, AC垂直于AP于点A交直线PB于点C，作三角形ABC外接圆D，连接辅助线AD、辅助线BD", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_660.jpg" }, { "index": 661, "ocr_en": "The upper figure shows an isosceles right triangle ABC, with angle BAC equal to 90 degrees and AB = AC. Point D is on AC. Connect BD. A circle with AD as its diameter intersects BD at point E. Connect CE. The lower figure shows an isosceles right triangle ABC, with angle BAC equal to 90 degrees and AB = AC. Point D is on AC. Connect BD. A circle with AD as its diameter intersects BD at point E. Connect CE. Draw circle O with AB as its diameter. Circle O intersects the other circle at points A and E. Connect auxiliary line CO, which intersects BD at point E.", "ocr_zh": "上图有等腰直角三角形ABC，角BAC为90度，AB=AC，点D在AC上，连接BD，以AD为直径的圆交BD于点E，连接CE；下图有等腰直角三角形ABC，角BAC为90度，AB=AC，点D在AC上，连接BD，以AD为直径的圆交BD于点E，连接CE，以AB为直径作圆O，圆O与另一个圆相交于点A和点E，连接辅助线CO与BD相交于点E", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_661.jpg" }, { "index": 662, "ocr_en": "In the Cartesian coordinate system xoy, point A is on the positive x-axis, point M is on the negative x-axis, and points C and N are on the positive y-axis. Connect lines MC and MN. Point B is in the first quadrant. There is a square OABC. Connect AN. Line AN intersects MC at point P.", "ocr_zh": "在平面直角坐标系xoy内，点A在x正半轴上，点M在x负半轴上，点C、点N在y正半轴上，连接直线MC、MN，点B在第一象限内，有正方形OABC，连接AN，直线AN与MC相交于点P", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_662.jpg" }, { "index": 663, "ocr_en": "In right triangle ABC, angle ACB is a right angle. Point D is a moving point on BC. Construct circle O with CD as its diameter. Connect AD, which intersects circle O at point E. Connect BE.", "ocr_zh": "有直角三角形ABC，角ACB为直角，点D为BC上的点，以CD为直径构建圆O，连接AD与圆O交于点E，连接BE", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_663.jpg" }, { "index": 664, "ocr_en": "In the left diagram, triangles ABC and EFG are both equilateral triangles with point D being the intersection of sides BC and EF. Lines AG and CF are drawn, intersecting at point M. The right diagram builds upon the left one by marking point O as the midpoint of AC, then constructing auxiliary lines AD, DG, BO, and OM, and finally drawing an auxiliary circle O centered at point O with radius equal to the length of OA.", "ocr_zh": "左图中，三角形ABC和三角形EFG均为等边三角形，点D为BC和EF的交点，连接AG和CF，直线AG和CF相交于点M。\n右图在左图的基础上，取AC的中点O，作辅助线AD、DG、BO和OM，以点O为圆心，OA的长度为半径作辅助圆O。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_664.jpg" }, { "index": 665, "ocr_en": "In the Cartesian coordinate system xOy, point A is located on the positive x-axis, point B on the negative x-axis, and point C on the positive y-axis. Connecting AC and BC forms triangle ABC. Point P serves as the circumcenter of triangle ABC (the intersection point of the perpendicular bisectors of AB, AC, and BC). Using P as the center and PC as the radius, an auxiliary circle is drawn, which precisely passes through points A, B, and C. Point E is an arbitrary point on segment AB, with auxiliary lines PE, PA, PB, and PC constructed. Additionally, an auxiliary line PF is drawn perpendicular to the y-axis, meeting it at foot point F.", "ocr_zh": "在直角坐标系xOy中，点A位于x轴正半轴，点B位于x轴负半轴，点C位于y轴正半轴，连接AC和BC得到三角形ABC。P为三角形ABC的外心（AB、AC和BC垂直平分线的交点），以点P为圆心，PC的长度为半径画辅助圆，恰好经过点A、B和C。E为线段AB上一点，作辅助线PE、PA、PB和PC，作辅助线PF垂直于Y轴，垂足为F。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_665.jpg" }, { "index": 666, "ocr_en": "In the Cartesian coordinate system xOy, point A is located on the positive x-axis, point B on the negative x-axis, and point C on the negative y-axis. Connecting AC and BC forms triangle ABC. Point P serves as the circumcenter of triangle ABC (the intersection point of the perpendicular bisectors of AB, AC, and BC). Using P as the center and PC as the radius, an auxiliary circle is drawn, which precisely passes through points A, B, and C. Point E is an arbitrary point on segment AB, with auxiliary lines PE, PA, PB, and PC constructed. Additionally, an auxiliary line PF is drawn perpendicular to the y-axis, meeting it at foot point F.", "ocr_zh": "在直角坐标系xOy中，点A位于x轴正半轴，点B位于x轴负半轴，点C位于y轴负半轴，连接AC和BC得到三角形ABC。P为三角形ABC的外心（AB、AC和BC垂直平分线的交点），以点P为圆心，PC的长度为半径画辅助圆，恰好经过点A、B和C。E为线段AB上一点，作辅助线PE、PA、PB和PC，作辅助线PF垂直于Y轴，垂足为F。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_666.jpg" }, { "index": 667, "ocr_en": "In the Cartesian coordinate system xOy, point G is a point on the y-axis. A circle G is drawn with point G as its center. Circle G intersects the x-axis at points A and B, and intersects the y-axis at points C and D. Point E is a point on circle G, and segment AE is drawn. Point F is a point on segment AE, and segment CF is drawn.", "ocr_zh": "在直角坐标系xOy中，点G为y轴上一点，以点G为圆心画圆得到圆G，圆G与x轴交于A、B两点，与y轴交于C、D两点，点E为圆G上一点，连接AE。点F为AE上一点，连接CF。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_667.jpg" }, { "index": 668, "ocr_en": "Given triangle ABC, draw a ray AM through point A parallel to BC. Let point P be a point on the ray AM, and point O be the intersection point of the perpendicular bisectors of AP, AC, and PC. With point O as the center and OP as the radius, draw circle O. Connect BP to intersect circle O at point D, and then connect AD.", "ocr_zh": "已知三角形ABC，过点A作射线AM平行于BC，点P为射线AM上一点，点O为AP、AC和PC垂直平分线的交点，以点O为圆心，OP的长度为半径作圆O，连接BP交圆O为点D，连接AD。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_668.jpg" }, { "index": 669, "ocr_en": "Given triangle ABC, let point D be a point inside the triangle. Connect AD and CD, and let point O be the intersection point of the perpendicular bisectors of the three segments AD, AC, and CD. Using O as the center and the length of AO as the radius, draw circle O. Let E be the intersection point of BC and circle O. Connect BD, and let P be the intersection point of line BD and circle O.", "ocr_zh": "已知三角形ABC，点D为三角形ABC内一点，连接AD和CD，点O为三条边AD、AC和CD垂直平分线的交点，以点O为圆心，线段AO的长度为半径作圆O，BC与圆O的交点为点E，连接BD，直线BD与圆O的交点为点P。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_669.jpg" }, { "index": 670, "ocr_en": "Given square ABCD, let points E and F be two points on side AD. Connect BD, and let CF intersect BD at point G. Connect AG, and let BE intersect AG at point H. Finally, connect DH.", "ocr_zh": "已知正方形ABCD，点E、F为边AD上两点，连接BD，连接CF交BD与点G，连接AG，连接BE交AG与点H，连接DH。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_670.jpg" }, { "index": 671, "ocr_en": "Given a circle O, the line segment AB is the diameter of circle O in the horizontal direction. Point C is a point on circle O, located above AB and closer to point B. Point P lies on the minor arc BC. Connect AP and BC, and point D is the midpoint of AP. Connect CD.", "ocr_zh": "已知圆O，线段AB为圆O沿水平方向的直径。点C为圆O上一点，位于AB上方且更靠近点B。点P位于劣弧BC上，连接AP和BC，点D为AP的中点，连接CD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_671.jpg" }, { "index": 672, "ocr_en": "In the Cartesian coordinate system xOy, points A and B are located on the positive x-axis, with point B to the right of point A. Point M is the midpoint of AB. A circle M is drawn with M as the center and AM as the radius. Starting from point O, a ray OF is drawn, intersecting circle M at points E and F, where point E is closer to point A than point F. Point C is a point on the arc AB, and point D is a point on EF. Segment CD is then connected.", "ocr_zh": "在直角坐标系xOy中，点A和点B为x轴正半轴上的点，点B在点A的右侧。点M为AB的中点，以点M为圆心，AM的长度为半径作圆M，从点O出发作一条射线OF，射线OF与圆M相交于点E和点F，点E比点F更靠近点A。点C为弧AB上一点，点D为EF上一点，连接CD。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_672.jpg" }, { "index": 673, "ocr_en": "Given a semicircle O with point O as the center and segment BC as the diameter. Point A is the midpoint of arc BC. Point P lies on arc AC. Segments AB, AC, AP, and BP are drawn. Point D is a point on BP, and segments AD and CD are connected.", "ocr_zh": "已知半圆O，点O为圆心，线段BC为直径，点A为弧BC的中点，点P为弧AC上一点，连接AB、AC、AP和BP，点D为BP上一点，连接AD和CD。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_673.jpg" }, { "index": 674, "ocr_en": "Given an equilateral triangle ABC, point D is on side BC, and point E is on side AC. Connect AD and BE, which intersect at point P. Then, connect CP.", "ocr_zh": "已知等边三角形ABC，点D为边BC上一点，点E为边AC上一点，连接AD和BE，AD和BE相交于点P，连接CP。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_674.jpg" }, { "index": 675, "ocr_en": "Given triangle ABC, point D lies on side BC, and point E lies on side AC. Connect AD, BE, and DE.", "ocr_zh": "已知三角形ABC，点D为边BC上一点，点E为边AC上一点，连接AD、BE和DE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_675.jpg" }, { "index": 676, "ocr_en": "Given a plane angle MON, point P is a point on the angle bisector of angle MON. A perpendicular PA is drawn from point P to OM, with foot of perpendicular at point A. A perpendicular PB is drawn from point P to ON, with foot of perpendicular at point B.", "ocr_zh": "已知平面角MON，P是角MON的角平分线上一点。过点P作PA垂直于OM，垂足为点A。过点P作PB垂直于ON，垂足为点B。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_676.jpg" }, { "index": 677, "ocr_en": "In the left diagram, given a plane angle ABC, point P lies on the angle bisector of angle ABC. A perpendicular PF is drawn from point P to AB, with the foot of the perpendicular at point F. A perpendicular PE is drawn from point P to BC, with the foot of the perpendicular at point E. Angles BFP and BEP are both right angles.\nIn the middle diagram, given a plane angle ABC, point P lies on the angle bisector of angle ABC. Point F is a point on BA, and point E is a point on BC, with BF and BE being equal in length. Segments PF and PE are connected. Angles BPF and BPE are both obtuse angles.\nIn the right diagram, given a plane angle ABC, point P lies on the angle bisector of angle ABC. Point F is a point on BA, and point E is a point on BC. Segments PF and PE are connected. Angles BPF and BPE are both right angles.", "ocr_zh": "在左图中，已知平面角ABC，P是角ABC的角平分线上一点。过点P作PF垂直于AB，垂足为点F。过点P作PE垂直于BC，垂足为点E。角BFP和角BEP均为直角。\n在中图中，已知平面角ABC，P是角ABC的角平分线上一点。点F为BA上一点，点E为BC上一点，BF与BE长度相等，连接PF和PE。角BPF和角BPE均为钝角。\n在右图中，已知平面角ABC，P是角ABC的角平分线上一点。点F为BA上一点，点E为BC上一点，连接PF和PE。角BPF和角BPE均为直角。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_677.jpg" }, { "index": 678, "ocr_en": "Given a plane angle ABC, point P lies on the angle bisector of angle ABC. Points F and E are located on BA and BC respectively, with segments PF and PE connected. Both angles BPF and BPE are right angles.", "ocr_zh": "已知平面角ABC，P是角ABC的角平分线上一点。点F为BA上一点，点E为BC上一点，连接PF和PE。角BPF和角BPE均为直角。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_678.jpg" }, { "index": 679, "ocr_en": "Given triangle ABC, AD is the angle bisector of angle CAB, and AD intersects BC at point D.", "ocr_zh": "已知三角形ABC，AD为角CAB的角平分线，AD与BC相交于点D。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_679.jpg" }, { "index": 680, "ocr_en": "Given angle A, points B and C lie on the two sides of angle A, respectively. Connect B and C to form segment BC. AP is the angle bisector of angle BAC, and BP and CP are connected. The angle between the extension of AB and BP is denoted as angle 1, the angle between BC and BP as angle 2, the angle between BC and CP as angle 3, and the angle between the extension of AC and CP as angle 4.", "ocr_zh": "已知角A，点B和点C分别位于角A的两条边上。连接BC。AP为角BAC的角平分线，连接BP和CP。AB的延长线与BP的夹角记作角1，BC与BP的夹角记作角2，BC与CP的夹角记作角3，AC的延长线与CP的夹角记作角4。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_680.jpg" }, { "index": 681, "ocr_en": "Given angle A, points B and C lie on the two sides of angle A, respectively. Connect B and C to form segment BC. AP is the angle bisector of angle BAC, and BP and CP are connected. The angle between the extension of AB and BP is denoted as angle 1, the angle between BC and BP as angle 2, the angle between BC and CP as angle 3, and the angle between the extension of AC and CP as angle 4. Let point E be a point on BC, point F be a point on the ray AB, and point G be a point on the ray AC. Construct auxiliary lines PE, PF, and PG", "ocr_zh": "已知角A，点B和点C分别位于角A的两条边上。连接BC。AP为角BAC的角平分线，连接BP和CP。AB的延长线与BP的夹角记作角1，BC与BP的夹角记作角2，BC与CP的夹角记作角3，AC的延长线与CP的夹角记作角4。设点E为BC上一点，点F为射线AB上一点，点G为射线AC上一点，做辅助线PE、PF和PG。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_681.jpg" }, { "index": 682, "ocr_en": "Given triangle ABC, extend BC to point D, making angle ACD an exterior angle of triangle ABC. CP is the angle bisector of angle ACD. Connect AP and BP.", "ocr_zh": "已知三角形ABC，延长BC至点D，角ACD为三角形ABC的外角。CP为角ACD的角平分线，连接AP和BP。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_682.jpg" }, { "index": 683, "ocr_en": "Given quadrilateral ABCD where BC is longer than AB. Connect BD. Let point E be a point on line AB, and point F be a point on line BC. Construct auxiliary lines DE and DF.", "ocr_zh": "已知四边形ABCD，BC比AB更长，连接BD。点E为直线AB上一点，点F为直线BC上一点，做辅助线DE和DF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_683.jpg" }, { "index": 684, "ocr_en": "Given triangle ABC, extend side BC to point D, making ∠ACD the exterior angle of triangle ABC. Let CP be the angle bisector of ∠ACD. Connect points AP and BP. Extend BA, then construct auxiliary line PF perpendicular to BA (extended) with foot F. Construct PM perpendicular to AC with foot M. Finally, construct PN perpendicular to BD with foot N.", "ocr_zh": "已知三角形ABC，延长BC至点D，角ACD为三角形ABC的外角。CP为角ACD的角平分线，连接AP和BP。延长BA，过点P作辅助线PF垂直于BA，垂足为F。过点P作辅助线PM垂直于AC，垂足为M。过点P作辅助线PN垂直于BD，垂足为N。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_684.jpg" }, { "index": 685, "ocr_en": "Given circle O with point P on the circle and point A outside the circle. Line AO is extended to intersect circle O at points P₁ and P₂.", "ocr_zh": "已知圆O，点P为圆O上一点，点A为圆O外一点，连接AO并延长，交圆O与点P1和P2。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_685.jpg" }, { "index": 686, "ocr_en": "Given rhombus ABCD, construct circle A centered at point A and circle B centered at point B, such that the two circles are externally tangent. Let P be a point on side CD, E be a point on circle A, and F be a point on circle B. Connect segments PE and PF.", "ocr_zh": "已知菱形ABCD，以点A为圆心作圆A，以点B为圆心作圆B，圆A和圆B外切。点P、E和F分别为CD、圆A和圆B上的点，连接PE和PF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_686.jpg" }, { "index": 687, "ocr_en": "In the rectangular coordinate system xOy, there exists a rectangle OACB where point A lies on the positive x-axis, point B lies on the positive y-axis, and point C is the rectangle's upper-right vertex. Point P is located on segment BC. A horizontal dashed line is drawn leftward from P, intersecting the y-axis at point B. Segment OP is constructed, and point B' is the mirror image of B about line OP. Finally, segments OB' and PB' are drawn.", "ocr_zh": "在直角坐标系xOy中，有一矩形OACB，点A为x轴正半轴上一点，点B为y轴正半轴上一点，点C是矩形的右上顶点。点P为线段BC上一点，从点P向左作水平虚线，交y轴于点B，连接OP，点B’为点B关于直线OP的镜像点，连接OB’和PB’。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_687.jpg" }, { "index": 688, "ocr_en": "Given circle O with diameter AB, points M and N on the circle, and point P on AB.", "ocr_zh": "已知圆O，AB为圆O的直径，点M和点N是圆O上的点，点P为直径AB上的点。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_688.jpg" }, { "index": 689, "ocr_en": "Given circle O with diameter AB, points C and D on the circle, point P on AB, with segments CP and DP drawn.", "ocr_zh": "已知圆O，AB为圆O的直径，点C和点D为圆O上的点，点P为直径AB上的点，连接CP和DP。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_689.jpg" }, { "index": 690, "ocr_en": "Given rhombus ABCD with points M on AD and N on AB. Segment MN is drawn, and point A' is the reflection of A across MN. Segments MA', NA', and CA' are connected, with AM and AN shown as dashed lines.", "ocr_zh": "已知菱形ABCD，点M和点N分别是边AD和AB上的点，连接MN，点A’为点A关于直线MN的镜像点，连接MA’、NA’和CA’。AM和AN改为虚线表示。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_690.jpg" }, { "index": 691, "ocr_en": "Given quadrilateral ABCD with right angle at A, point P on AD, and point Q inside the quadrilateral. Segments BQ, BP, PQ, CQ, and DQ are drawn.", "ocr_zh": "已知四边形ABCD，角A为直角，点P为AD上一点，点Q为四边形ABCD内一点，连接BQ、BP、PQ、CQ和DQ。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_691.jpg" }, { "index": 692, "ocr_en": "Given points A and O₁, connect segment AO₁. Construct auxiliary circle O₁ centered at O₁. Let P be a point on circle O₁. Connect segments AP and PO₁. Take point O₂ on segment AO₁, and let M be the midpoint of AP. Connect MO₂. Then, construct auxiliary circle O₂ centered at O₂ with radius equal to the length of MO₂。", "ocr_zh": "已知点A和点O₁，连接AO₁，以点O₁为圆心作辅助圆O₁，点P为圆O₁上一点，连接AP和PO₁，O₂为AO₁上一点，M为AP的中点，连接MO₂。以点O₂为圆心，MO₂长度为半径，作辅助圆O₂。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_692.jpg" }, { "index": 693, "ocr_en": "Given right-angled triangle ABC with the right angle at C. Let D be a point on AC, and D’ be a point outside the triangle such that AD = AD’. Construct auxiliary circle A centered at A with radius AD. Connect BD’, and let F be the midpoint of BD’. Draw segment CF. Extend BC to point B’, where C is the midpoint of BB’. Finally, construct auxiliary line B’D’.", "ocr_zh": "已知三角形ABC，角C为直角，点D为AC上一点，D’为三角形ABC外一点，连接AD’，AD与AD’长度相等，以点A为圆心，AD的长度为半径，作辅助圆A。连接BD’，点F为BD’的中点，连接CF。延长BC至点B’，点C是BB’的中点，作辅助线B’D’。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_693.jpg" }, { "index": 694, "ocr_en": "Given right triangle ABC with the right angle at C, point D on AC, and point D' outside the triangle. Segments AD' and BD' are drawn. Points G and F lie on AB and BD' respectively, with segment CF connected. Auxiliary line FG is constructed, and auxiliary circle G is drawn with center at G and radius equal to FG.", "ocr_zh": "已知三角形ABC，角C为直角，点D为AC上一点，D’为三角形ABC外一点，连接AD’和BD’。点G和F分别为AB和BD’上的点，连接CF。作辅助线FG，以点G为圆心，FG的长度为半径，作辅助圆G。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_694.jpg" }, { "index": 695, "ocr_en": "Given triangle ABC with point D on AC and point D' outside the triangle. Segments AD' and BD' are drawn. Point F is on BD', and segment CF is connected.", "ocr_zh": "已知三角形ABC，点D为AC上一点，D’为三角形ABC外一点，连接AD’和BD’。点F为BD’上一点，连接CF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_695.jpg" }, { "index": 696, "ocr_en": "Given semicircle O with AB as its diameter, point P lies on the semicircle, while points C and D are located outside the semicircle and above AB. Segments AC, CP, PD, and BD are connected, where AC is perpendicular to AB with foot at point A, and BD is perpendicular to AB with foot at point B.", "ocr_zh": "已知半圆O，AB为半圆O的直径。点P为半圆O上一点，点C和D为半圆外的点，且均位于AB上方。连接AC、CP、PD和BD，AC与AB垂直，垂足为点A，BD与AB垂直，垂足为点B。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_696.jpg" }, { "index": 697, "ocr_en": "Given equilateral triangle ABC, a semicircle is constructed externally on diameter AC. Point P lies on the semicircle, segment BP is drawn, and point M is taken on BP.", "ocr_zh": "已知等边三角形ABC，以AC为直径向三角形ABC外侧方向作半圆，点P为半圆上一点，连接BP，点M为BP上一点。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_697.jpg" }, { "index": 698, "ocr_en": "Given circle O with point A on the circle and point P inside the circle. Segment AP is extended to intersect the circle again at B. Segment AO is extended to intersect the circle at C. Segment BC is then drawn.", "ocr_zh": "已知圆O，点A为圆O上一点，点P为圆O内一点，连接AP并延长交圆O于点B。连接AO并延长，交圆O于点C。连接BC。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_698.jpg" }, { "index": 699, "ocr_en": "Given circle O with diameter AB and point C on the circle. Circle C is drawn with center at C. Point P lies on AB, and segment PM is tangent to circle C at point M.", "ocr_zh": "已知圆O，AB为圆O的直径，点C为圆O上一点，以C为圆心作圆C。点P为AB上一点，PM于圆C相切，切点为M。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_699.jpg" }, { "index": 700, "ocr_en": "Given triangle ABC with point D on AC. Segment BD is drawn, and point M is taken on BD. Segment CM is then connected.", "ocr_zh": "已知三角形ABC，点D为AC上一点，连接BD，点M为BD上一点，连接CM。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_700.jpg" }, { "index": 701, "ocr_en": "Given semicircle O with AB as its diameter, point P lies on the semicircle, while points C and D are located outside the semicircle and above AB. Segments AC, CP, PD, and BD are connected, satisfying that AC is perpendicular to AB with foot at point A, and BD is perpendicular to AB with foot at point B. Auxiliary lines CD, OP, and OC are drawn, with OC intersecting the semicircle O at point P₁. Point M is selected on CD, and auxiliary line PM is constructed to complete the geometric configuration.", "ocr_zh": "已知半圆O，AB为半圆O的直径。点P为半圆O上一点，点C和D为半圆外的点，且C和D均位于AB上方，连接AC、CP、PD和BD，满足AC垂直于AB，垂足为点A，BD垂直于AB，垂足为点B。作辅助线CD、OP和OC，OC交半圆O于点P1。点M为CD上一点，作辅助线PM。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_701.jpg" }, { "index": 702, "ocr_en": "Given circle O with points A and B on its circumference, segment AB is drawn. On the major arc AB, there exist three distinct points, each denoted as C. For each point C, segments AC and BC are constructed.", "ocr_zh": "已知圆O，点A和B为圆O上的点，连接AB。优弧AB上有三个点，均记作点C。对每个点C，连接AC和BC。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_702.jpg" }, { "index": 703, "ocr_en": "Given circle O with points A and B on its circumference, segment AB is drawn. Point P is taken on the major arc AB, with segments AP and BP constructed. PB is extended to point C, and segment AC is then drawn.", "ocr_zh": "已知圆O，点A和B为圆O上的点，连接AB。点P为优弧AB上一点，连接AP和BP。延长PB于点C，连接AC。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_703.jpg" }, { "index": 704, "ocr_en": "Given equilateral triangle ABC with point D on BC. Segment AD is drawn. A perpendicular BE is constructed to AD with foot at point E. Segment CE is then connected.", "ocr_zh": "已知等边三角形ABC，点D为BC上一点，连接AD。作BE垂直于AD，垂足为点E。连接CE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_704.jpg" }, { "index": 705, "ocr_en": "Given triangle ABC with point D on AC and point O as the midpoint of AD. Circle O is drawn with center O and radius OA. Segment BD is constructed, intersecting circle O at point E. Segment CE is then connected.", "ocr_zh": "已知三角形ABC，点D为AC上一点，点O为AD的中点，以点O为圆心，OA的长度为半径作圆O。连接BD，BD于圆O交于点E，连接CE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_705.jpg" }, { "index": 706, "ocr_en": "Given sector AOD with center at point O. Point P lies on arc AD, and point Q lies on segment OD. Segments OP and PQ are drawn to form triangle OPQ. Point I is located inside triangle OPQ.", "ocr_zh": "已知扇形AOD，点O为圆心。点P为弧AD上一点，点Q为线段OD上一点，连接OP和PQ得到三角形OPQ。点I为三角形OPQ内一点。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_706.jpg" }, { "index": 707, "ocr_en": "Given sector AOD centered at O, with point P on arc AD and point H on segment OD. Triangle OPH is formed by connecting OP and PH. Point I lies inside triangle OPH. Auxiliary lines IO, IP, ID and IH are drawn. Point O' is the intersection point of the perpendicular bisectors of segments IO, OD and ID. Circle O' is constructed with center O' and radius OO', with auxiliary lines OO' and DO' drawn. Point P' lies on circle O', with auxiliary lines OP' and DP' constructed.", "ocr_zh": "已知扇形AOD，点O为圆心。点P为弧AD上一点，点H为线段OD上一点，连接OP和PH得到三角形OPH。点I为三角形OPH内一点。作辅助线IO、IP、ID和IH，点O’为三条线段IO、OD和ID垂直平分线的交点。以点O’为圆心，OO’的长度为半径画圆O’，作辅助线OO’和DO’。点P’为圆O’上的点，作辅助线OP’和DP’。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_707.jpg" }, { "index": 708, "ocr_en": "Given triangle ABC with point O located to the right of AC, where the circle centered at O passes through points A and C. Segment BO is drawn.", "ocr_zh": "已知三角形ABC，点O位于AC右侧，以点O为圆心的圆过A、C两点，连接BO。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_708.jpg" }, { "index": 709, "ocr_en": "Given isosceles right triangle ABC with right angle at C. Points D and E lie on sides AC and AB respectively. Segment DE is drawn, and square DEFG is constructed externally on DE. Segment AF is then connected.", "ocr_zh": "已知等腰直角三角形ABC，角C为直角。D,E分别为等腰直角三角形ABC的边 AC、AB 上的点，连接DE，以DE为边向外作正方形DEFG，连接AF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_709.jpg" }, { "index": 710, "ocr_en": "Given equilateral triangle ABC with points D on BC and E on AC. Segments AD and BE are drawn, intersecting at point P. Segment CP is then connected.", "ocr_zh": "已知等边三角形ABC，D、E分别为边BC、AC上的点，连接AD和BE，AD和BE相交于点P，连接CP。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_710.jpg" }, { "index": 711, "ocr_en": "In the Cartesian coordinate system xOy, points A and B are located on the positive y-axis with B above A. Point C lies on the positive x-axis. Segments AC and BC are drawn.", "ocr_zh": "直角坐标系xOy中，点A和点B位于Y轴正半轴，点B位于点A上方。点C位于x轴正半轴。连接AC和BC。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_711.jpg" }, { "index": 712, "ocr_en": "In the rectangular coordinate system xOy, points A and B are located on the positive x-axis.", "ocr_zh": "直角坐标系xOy中，点A和点B位于x轴正半轴。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_712.jpg" }, { "index": 713, "ocr_en": "Given right-angled triangle ABC with the right angle at C. Point P lies on AB and point Q lies on AC. Segments PC and PQ are drawn.", "ocr_zh": "已知直角三角形ABC，角C为直角。点P为AB上一点，点Q为AC上一点，连接PC和PQ。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_713.jpg" }, { "index": 714, "ocr_en": "In the left diagram, given the rectangular coordinate system xOy with points A and B on the positive x-axis, the perpendicular bisector of AB is drawn with points O and O' located on this bisector in the first quadrant (O' above O); circle O is constructed centered at O with AB as its chord, tangent to the y-axis at point C, and segment AC is drawn; dashed circle O' is constructed centered at O' with AB as its chord, intersecting the y-axis at point C₁, and segments AC₁ and BC₁ are drawn, while point C₂ on circle O' is connected via segments AC₂ and BC₂. In the right diagram, given the same coordinate system and points A/B on the x-axis, the perpendicular bisector is drawn with O and O' similarly positioned; dashed circle O is constructed centered at O (AB as chord), tangent to the y-axis at C, with segments AC and BC drawn; point P is defined as the x-axis symmetric point of O, and a dashed circle centered at P with radius PA is constructed, tangent to the y-axis at C.", "ocr_zh": "在左图中，已知直角坐标系xOy，点A和点B位于x轴正半轴。作AB的垂直平分线，点O和O’位于AB的垂直平分线上，且均位于第一象限，O’在O的上方。以点O为圆心作圆O，AB为圆O的弦，圆O与y轴相切，切点为C，连接AC。以点O’为圆心作虚线圆O’，AB为圆O’的弦，圆O’与y轴相交与点C1，连接AC1和BC1。C2为圆O’上一点，连接AC2和BC2。\n在右图中，已知直角坐标系xOy，点A和点B位于x轴正半轴。作AB的垂直平分线，点O和O’位于AB的垂直平分线上，且均位于第一象限，O’在O的上方。以点O为圆心作虚线圆O，AB为圆O的弦，圆O与y轴相切，切点为C，连接AC和BC。设点O关于x轴对称的点为点P，以点P为圆心，PA的长度为半径作虚线圆，与y轴相切，切点为点C。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_714.jpg" }, { "index": 715, "ocr_en": "In the rectangular coordinate system xOy, points A and B lie on the positive x-axis. The perpendicular bisector of AB is constructed, with points O and O' located on this bisector in the first quadrant (O' positioned above O). A dashed circle O is drawn centered at point O, using AB as its chord, tangent to the y-axis at point C, with segments AC and BC connected. Point P is defined as the reflection of point O across the x-axis, and a dashed circle is constructed centered at P with radius PA, tangent to the y-axis at point C.", "ocr_zh": "已知直角坐标系xOy，点A和点B位于x轴正半轴。作AB的垂直平分线，点O和O’位于AB的垂直平分线上，且均位于第一象限，O’在O的上方。以点O为圆心作虚线圆O，AB为圆O的弦，圆O与y轴相切，切点为C，连接AC和BC。设点O关于x轴对称的点为点P，以点P为圆心，PA的长度为半径作虚线圆，与y轴相切，切点记作点C。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_715.jpg" }, { "index": 716, "ocr_en": "Given the rectangular coordinate system xOy with points A and B on the positive x-axis, the perpendicular bisector of AB is drawn with points O and O' located on this bisector in the first quadrant (O' above O); circle O is constructed centered at O with AB as its chord, tangent to the y-axis at point C, and segment AC is drawn; dashed circle O' is constructed centered at O' with AB as its chord, intersecting the y-axis at point C₁, and segments AC₁ and BC₁ are drawn, while point C₂ on circle O' is connected via segments AC₂ and BC₂. ", "ocr_zh": "已知直角坐标系xOy，点A和点B位于x轴正半轴。作AB的垂直平分线，点O和O’位于AB的垂直平分线上，且均位于第一象限，O’在O的上方。以点O为圆心作圆O，AB为圆O的弦，圆O与y轴相切，切点为C，连接AC。以点O’为圆心作虚线圆O’，AB为圆O’的弦，圆O’与y轴相交与点C1，连接AC1和BC1。C2为圆O’上一点，连接AC2和BC2。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_716.jpg" }, { "index": 717, "ocr_en": "In the Cartesian coordinate system xOy, points A and B are located on the positive x-axis, with point B positioned to the right of point A. A circle A is drawn with point A as the center and the length of AO as the radius. Point P lies on circle A in the first quadrant, and the segment PB is connected. Point M is located to the right of PB, and by connecting PM and BM, the triangle PMB is formed, which is an equilateral triangle. Finally, the segment AM is connected.", "ocr_zh": "在直角坐标系xOy中，点A和点B是x轴正半轴上的点，点B位于点A右侧。以点A为圆心，AO的长度为半径作圆A，点P为圆A上一点，位于第一象限，连接PB。点M位于PB右侧，连接PM和BM，得到三角形PMB，三角形PMB为等边三角形。连接AM。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_717.jpg" }, { "index": 718, "ocr_en": " In the left diagram, within the rectangular coordinate system xOy where the x-axis is depicted as a dashed line and the y-axis as a solid line, points A and B are positioned on the positive x-axis with B located to the right of A, and segment AB is drawn. A green circle A is constructed with center at point A and radius AO. Point P lies on circle A within the first quadrant, and segment PB is connected. Point M is placed to the right of PB, and by connecting AM, PM, and BM, an equilateral triangle PMB is formed. Point H is situated in the first quadrant, and auxiliary segments AH and BH are drawn to create equilateral triangle ABH. Auxiliary segment AP forms triangle ABP, while auxiliary segment HM yields triangle BHM. Triangle ABP is filled in purple, and triangle BHM is filled in brown.\nIn the middle diagram, using the same coordinate system xOy with the x-axis as a dashed line and y-axis as a solid line, points A and B remain on the positive x-axis with B to the right of A, connected by segment AB. The green circle A is again drawn centered at A with radius AO, and point P is positioned on the circle in the first quadrant. Point H is located in the first quadrant, where auxiliary segment AH is constructed to be equal in length to AB with angle HAB measuring 60 degrees. A green auxiliary circle H is then drawn centered at H with radius AO. Segment AH is extended to intersect circle H at points M1 and M2, where M2 is closer to point A. Point M is selected on circle H to the right of AM1, and auxiliary segment HM is drawn.\nIn the right diagram, maintaining the same xOy coordinate system with dashed x-axis and solid y-axis, points A and B are again placed on the positive x-axis with B to the right of A, connected by AB. The green circle A is constructed centered at A with radius AO, and point P is positioned on the circle in the first quadrant. Point H is situated in the first quadrant, where auxiliary segment BH is drawn equal in length to AB with angle HBA measuring 60 degrees. A green auxiliary circle H is then constructed centered at H with radius AO.", "ocr_zh": "在左图中，已知直角坐标系xOy，x轴为虚线，y轴为实线。点A、B位于x轴正半轴，且B在A右侧，连接AB。以A为圆心，AO为半径作一绿色圆A。点P位于圆A上且在第一象限，连接PB。点M位于PB右侧，连接AM、PM和BM，得到等边三角形PMB。点H位于第一象限，作辅助线AH和BH，得到等边三角形ABH。作辅助线AP得到三角形ABP。作辅助线HM得到三角形BHM。三角形ABP填充紫色，三角形BHM填充棕色。\n在中图中，已知直角坐标系xOy，x轴为虚线，y轴为实线。点A、B位于x轴正半轴，且B在A右侧，连接AB。以A为圆心，AO为半径作一绿色圆A。点P位于圆A上且在第一象限。点H位于第一象限，作辅助线AH，AH与AB等长且角HAB为60度。以H为圆心，AO的长度为半径作绿色辅助圆H。作辅助线AH并延长，与圆H相交于点M1和M2，其中M2更靠近点A。点M位于圆H上，且在AM1右侧，作辅助线HM。\n在右图中，已知直角坐标系xOy，x轴为虚线，y轴为实线。点A、B位于x轴正半轴，且B在A右侧，连接AB。以A为圆心，AO为半径作一绿色圆A。点P位于圆A上且在第一象限。点H位于第一象限，连接辅助线BH，BH与AB等长且角HBA为60度，以H为圆心，AO的长度为半径作绿色辅助圆H。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_718.jpg" }, { "index": 719, "ocr_en": "In the rectangular coordinate system xOy with the x-axis shown as a dashed line and the y-axis as a solid line, point A is located on the positive x-axis. Let point P be positioned between A and O. A green circle O is drawn with center at origin O and radius OP. Point B is a point on circle O lying in the first quadrant. Segment AB is connected, and point C is placed in the first quadrant to the right of AB. By connecting AC and BC, an isosceles right triangle BAC is formed with the right angle at A. The segments AB, AC, and BC are all colored blue.", "ocr_zh": "在直角坐标系xOy中，x轴为虚线，y轴为实线。点A位于x轴正半轴，设有一点P位于AO之间，以O为圆心，OP为半径画一绿色圆O。点B为圆O上位于第一象限的点，连接AB，点C位于第一象限且在AB右侧，连接AC和BC，得到等腰直角三角形BAC，角A为直角。线段AB、AC和BC为蓝色。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_719.jpg" }, { "index": 720, "ocr_en": "Given circle O with AB as its diameter, point C lies on the circumference above diameter AB and is positioned closer to point B. Segment BC is drawn, and using BC as one side, square BCDE is constructed externally to the circle with vertex D located above line AB, after which segment OD is connected to complete the configuration.", "ocr_zh": "已知圆O，AB为圆O的直径，点C在圆O上，位于直径AB上方，且更靠近点B。连接BC，以BC为边在圆O外作正方形BCDE，点D在直线AB的上方。连接OD。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_720.jpg" }, { "index": 721, "ocr_en": "We are given an isosceles triangle ABC with AB = AC. Point E is the midpoint of BC, and point F lies somewhere on AB. The triangle ABC is rotated clockwise about point C to produce a corresponding triangle A₁B₁C, where A₁ is positioned to the upper right of C, and triangle A₁B₁C does not overlap with the original triangle ABC. Point F₁, located on A₁B₁, is the image of point F after the rotation. Finally, segment EF₁ is drawn to connect points E and F₁.", "ocr_zh": "已知等腰三角形ABC，AB与AC等长。点E是BC的中点，点F为AB上一点。将三角形ABC绕点C顺时针旋转得到对应的三角形A1B1C，A1在C的右上方，且三角形A1B1C与三角形ABC无重叠。点F1位于A1B1上，是点F旋转后对应的点，连接EF1。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_721.jpg" }, { "index": 722, "ocr_en": "Given a right-angled triangle ABC with ∠ACB = 90°, where point E is the midpoint of AC, triangle ABC is rotated clockwise about point C by an angle Θ (0° < Θ < 90°) to obtain its corresponding triangle A'B'C. The angle formed by ACA' is denoted as Θ. Point P is then defined as the midpoint of A'B', and segment EP is drawn connecting points E and P, establishing a geometric relationship between the original triangle's midpoint and the rotated triangle's midpoint.", "ocr_zh": "已知三角形ABC，角ACB为90度，点E为AC的中点，将三角形ABC绕点C顺时针旋转得到对应的三角形A'B'C，旋转角在0到90度之间。角ACA'记为Θ。点P为A'B'的中点，连接EP。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_722.jpg" }, { "index": 723, "ocr_en": "In a Cartesian coordinate system, the inverse proportionality function is located in the first and third quadrants. Let point A be on the inverse proportionality function in the first quadrant, connect point A to the origin, and let point B be a point on the x-axis. Connect points A and B such that AB is perpendicular to the x-axis, with point B as the foot of the perpendicular.", "ocr_zh": "在直角坐标系中，反比例函数位于第一和第三象限，设点A位于第一象限的反比例函数上，连接A和原点，设点B为x轴上一点，连接AB，AB垂直于x轴，垂足为点B。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_723.jpg" }, { "index": 724, "ocr_en": "In the Cartesian coordinate system xOy, there is a hyperbola located in the first quadrant, with point A being a point on the hyperbola. Connect point A to the origin O, and draw a perpendicular line from point A to the x-axis, intersecting it at point B. Let point C be a point on the negative half of the x-axis, and connect points A and C.", "ocr_zh": "在直角坐标系xOy中，有一双曲线位于第一象限，点A为双曲线上一点，连接AO，过点A作AB垂直于x轴，垂足为点B。点C为x轴负半轴上一点，连接AC。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_724.jpg" }, { "index": 725, "ocr_en": "In the Cartesian coordinate system xOy, there is a hyperbola located in the first and third quadrants. Point A lies on the hyperbola in the first quadrant, while point B lies on the hyperbola in the third quadrant. Segment AB is drawn, intersecting the x-axis at point M and the y-axis at point N. Point F is a point on the negative x-axis, and point D is a point on the positive y-axis, with both D and F positioned to the left side of line AB. An auxiliary line DF is constructed. The lines AD, BF, and DF are all drawn in blue.", "ocr_zh": "在直角坐标系xOy中，有一双曲线位于第一、三象限，点A位于第一象限的双曲线上，点B位于第三象限的双曲线上。连接AB。直线AB与x轴相交于点M，与y轴相交于点N。F为x轴负半轴上一点，D为y轴正半轴上一点，D和F均位于直线AB左侧。作辅助线DF。AD、BF和DF均为蓝色。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_725.jpg" }, { "index": 726, "ocr_en": "In the Cartesian coordinate system with the horizontal axis as x and the vertical axis as y, there are two hyperbolas located in the first quadrant: the left hyperbola is denoted as y = 1/x, and the right hyperbola is denoted as y = 3/x. Point A lies on the left hyperbola, and point B lies on the right hyperbola. Segment AB is drawn parallel to the x-axis. A perpendicular line is drawn from point A to the x-axis, intersecting it at point D, and another perpendicular line is drawn from point B to the x-axis, intersecting it at point C.", "ocr_zh": "在直角坐标系中，横轴为x，纵轴为y。有两条双曲线位于第一象限，左侧双曲线记为y=1/x，右侧双曲线记为y=3/x。点A位于左侧的双曲线上，点B位于右侧的双曲线上。连接AB，AB与x轴平行。过点A作AD垂直于x轴，垂足为D，过点B作BC垂直于x轴，垂足为C。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_726.jpg" }, { "index": 727, "ocr_en": "In the Cartesian coordinate system xOy, there is a hyperbola located in the first quadrant, with points A and B lying on the hyperbola such that A is positioned above B. The line segment AB is drawn, intersecting the x-axis at point M and the y-axis at point N. Point D is a point on the positive y-axis, and point F is a point on the positive x-axis, both situated below the line AB. The segments AD, BF, and DF are then connected.", "ocr_zh": "在直角坐标系xOy中，有一双曲线位于第一象限。A和B为双曲线上的点，且A在B的上方。连接AB，直线AB与x轴相交于点M，与y轴相交于点N。D为y轴正半轴上一点，F为x轴正半轴上一点，D和F均位于直线AB下方。连接AD、BF和DF。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_727.jpg" }, { "index": 728, "ocr_en": "In the Cartesian coordinate system with the horizontal axis as x, the vertical axis as y, and the origin at O, there exists a hyperbola located in the first quadrant. Three points A₁, A₂, and A₃ are marked on the positive x-axis from left to right. Auxiliary lines parallel to the y-axis are drawn through each of these points, intersecting the hyperbola at points B₁, B₂, and B₃, respectively. From each of these points B₁, B₂, and B₃, auxiliary lines parallel to the x-axis are drawn, intersecting the y-axis at points C₁, C₂, and C₃, respectively. The segments OB₁, OB₂, and OB₃ are then connected.\nThe region enclosed by OB₁, B₁C₁, and OC₁ is filled with horizontal lines. The region bounded by OB₂, A₁B₁, and B₂C₂ is filled with diagonal lines. Lastly, the area enclosed by OB₃, A₂B₂, and B₃C₃ is filled with vertical lines.", "ocr_zh": "在直角坐标系中，横轴为x，纵轴为y，设原点为O，有一双曲线位于第一象限。x轴正半轴有三个点，从左到右依次为A1，A2，A3。过这三个点分别作平行于y轴的辅助线，分别与双曲线相交于B1，B2，B3。分别过点B1，B2，B3作平行于x轴的辅助线，分别与y轴交于点C1，C2，C3。连接OB1，OB2和OB3。OB1、B1C1和OC1所围成的区域用横线填充。OB2、A1B1和B2C2所围成的区域用斜线填充。OB3、A2B2和B3C3所围成的区域用竖线填充。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_728.jpg" }, { "index": 729, "ocr_en": "In the Cartesian coordinate system xOy, there is a hyperbola located in the first quadrant. A straight line intersects the positive y-axis at point N and the positive x-axis at point M, while also intersecting the hyperbola at two points A and B, with point A positioned above point B. From point A, a perpendicular line AC is drawn to the x-axis, meeting it at point C, and another perpendicular line AE is drawn to the y-axis, meeting it at point E. Similarly, from point B, a perpendicular line BF is drawn to the x-axis, meeting it at point F, and another perpendicular line BD is drawn to the y-axis, meeting it at point D. The lines AC and BD intersect at point K.", "ocr_zh": "已知直角坐标系xOy，有一双曲线位于第一象限。一条直线于y轴正半轴相交于点N，与x轴正半轴相交于点M，与双曲线相交于A和B两点，点A在点B上方。过点A分别作AC垂直于x轴，垂足为点C，作AE垂直于y轴，垂足为点E。过点B分别作BF垂直于x轴，垂足为点F，作BD垂直于y轴，垂足为点D。AC与BD相交于K。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_729.jpg" }, { "index": 730, "ocr_en": "In the rectangular coordinate system xOy, there exists a hyperbola located in the first and third quadrants. A straight line intersects the negative x-axis at point A and the positive y-axis at point B, while also intersecting the hyperbola in the first quadrant at point D and in the third quadrant at point C. From point D, a perpendicular DF is drawn to the x-axis with foot point F. From point C, a perpendicular CE is drawn to the y-axis with foot point E. Finally, segments DE and CF are connected.", "ocr_zh": "已知直角坐标系xOy，有一双曲线位于第一、三象限。一条直线与x轴负半轴相交于点A，与y轴正半轴相交于点B，与第一象限的双曲线相交于点D，与第三象限的双曲线相交于点C。过点D作DF垂直于x轴，垂足为F。过点C作CE垂直于y轴，垂足为E。连接DE和CF。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_730.jpg" }, { "index": 731, "ocr_en": "In the Cartesian coordinate system xOy, there exists a hyperbola located in the first quadrant. Points P and C lie on this hyperbola, with P positioned above C. From point P, a perpendicular PA is drawn to the x-axis, intersecting it at point A, while from point C, a perpendicular CD is drawn to the x-axis, intersecting it at point D. The segments PC, OP (connecting origin O to P), and OC (connecting origin O to C) are then constructed.", "ocr_zh": "已知直角坐标系xOy，有一双曲线位于第一象限。点P和C为双曲线上两点，P在C的上方。过点P作PA垂直于x轴，垂足为点A，过点C作CD垂直于x轴，垂足为点D。连接PC、OP和OC。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_731.jpg" }, { "index": 732, "ocr_en": "In the rectangular coordinate system xOy, a hyperbola is located in the first quadrant. A straight line intersects the positive x-axis at point C and meets the hyperbola at two points A and B, with point A positioned above point B. Segments OA (connecting origin O to A) and OB (connecting origin O to B) are then drawn.\n", "ocr_zh": "已知直角坐标系xOy，有一双曲线位于第一象限。一条直线与x轴正半轴相交于点C，与双曲线相交于点A和点B，A在B的上方。连接OA和OB。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_732.jpg" }, { "index": 733, "ocr_en": "In the rectangular coordinate system xOy, a hyperbola is situated in both the first and third quadrants. A straight line passing through the origin O intersects the hyperbola at point A in the first quadrant and at point B in the third quadrant.\n", "ocr_zh": "已知直角坐标系xOy，有一双曲线位于第一、三象限。一条过点O的直线与第一象限的双曲线相交于点A，与第三象限的双曲线相交于点B。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_733.jpg" }, { "index": 734, "ocr_en": "In the rectangular coordinate system xOy, there exists a hyperbola located in both the first and third quadrants. A first straight line passing through the origin O intersects the hyperbola at point A in the first quadrant and point B in the third quadrant. A second straight line through O intersects the hyperbola at point C in the first quadrant (positioned above A) and point D in the third quadrant (positioned below B). The quadrilateral is completed by connecting segments AC, CB, BD and DA.", "ocr_zh": "已知直角坐标系xOy，有一双曲线位于第一、三象限。一条过点O的直线与第一象限的双曲线相交于点A，与第三象限的双曲线相交于点B。另一条过点O的直线与第一象限的双曲线相交于点C，与第三象限的双曲线相交于点D。C在A的上方，D在B的下方。连接AC、CB、BD和DA。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_734.jpg" }, { "index": 735, "ocr_en": "In the rectangular coordinate system xOy, there exists a hyperbola located in the first quadrant. A straight line intersects the positive x-axis at point C and meets the hyperbola at two points A and B, with point A positioned above point B. Segments OA (connecting origin O to A) and OB (connecting origin O to B) are drawn. The coordinates (1, 4) are marked to the right of point A, and (3, m) are marked to the right of point B.", "ocr_zh": "已知直角坐标系xOy，有一双曲线位于第一象限。一条直线与x轴正半轴相交于点C，与双曲线相交于点A和点B，A在B的上方。连接OA和OB。A的右侧标记（1，4），B的右侧标记（3，m）。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_735.jpg" }, { "index": 736, "ocr_en": "In the rectangular coordinate system xOy, there are two hyperbolas both located in the first quadrant. Point P lies on the right hyperbola. From P, a perpendicular PC is drawn to the x-axis (foot C) and a perpendicular PD is drawn to the y-axis (foot D). PD intersects the left hyperbola at point B, and PA (note: assuming PA refers to PC extended or another defined line) intersects the left hyperbola at point A. Segments OB and OA are drawn, forming quadrilateral OAPB which is then filled with purple color.", "ocr_zh": "已知直角坐标系xOy，有两条双曲线均位于第一象限。点P位于右侧双曲线上，过点P作PC垂直于x轴，垂足为C，过点P作PD垂直于y轴，垂足为D。PD与左侧双曲线相交于点B，PA与左侧双曲线相交于点A。连接OB和OA。得到四边形OAPB，将四边形OAPB填充为紫色。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_736.jpg" }, { "index": 737, "ocr_en": "In the rectangular coordinate system xOy established with the lines containing OB and OA as the x-axis and y-axis respectively, where point A lies on the positive y-axis and point B lies on the positive x-axis, forming rectangle AOBC, there exists an inverse proportional function in the first quadrant that intersects AC at point E and BC at point F. Segments OE and OF are then connected. Line segment EF is extended in both directions to form the straight line EF.", "ocr_zh": "已知矩形AOBC，分别以OB、OA所在直线为x轴和y轴建立直角坐标系xOy,A在y轴正半轴，B在x轴正半轴。第一象限有一反比例函数，与AC相交于点E，与BC相交于点F，连接OE和OF。连接EF并将线段EF向两端延长，构成直线EF。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_737.jpg" }, { "index": 738, "ocr_en": "In Figure A, within the rectangular coordinate system xOy, there exists a hyperbola situated in the first and third quadrants, along with a straight line passing through the first, third, and fourth quadrants. In Figure B, within the same coordinate system xOy, a hyperbola is located in the second and fourth quadrants, accompanied by a straight line that traverses the first and third quadrants while also passing through the origin O. Figure C shows the coordinate system xOy containing a hyperbola in the second and fourth quadrants, with a straight line extending through the first, second, and fourth quadrants. Finally, in Figure D, the xOy coordinate system features a hyperbola in the first and third quadrants, crossed by a straight line that runs through the second and fourth quadrants and includes the origin O.", "ocr_zh": "A图中，已知直角坐标系xOy，有一双曲线位于一、三象限，有一条直线经过一、三、四象限。\nB图中，已知直角坐标系xOy，有一双曲线位于二、四象限，有一条直线经过一、三象限和原点O。\nC图中，已知直角坐标系xOy，有一双曲线位于二、四象限，有一条直线经过一、二、四象限。\nD图中，已知直角坐标系xOy，有一双曲线位于一、三象限，有一条直线经过二、四象限和原点O。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_738.jpg" }, { "index": 739, "ocr_en": "In the rectangular coordinate system xOy, there is a hyperbola located in the first and third quadrants. Point A lies on the hyperbola in the first quadrant, while point B lies on the hyperbola in the third quadrant. Segment AB is drawn and passes through the origin O. Point C is situated in the fourth quadrant such that AC is parallel to the y-axis and BC is parallel to the x-axis. Finally, segments AC and BC are connected to complete the figure.", "ocr_zh": "已知直角坐标系xOy，有一双曲线位于一、三象限，点A位于第一象限的双曲线上，点B位于第三象限的双曲线上，连接AB，且AB过原点O。点C位于第四象限，满足AC平行于y轴，BC平行于x轴，连接AC和BC。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_739.jpg" }, { "index": 740, "ocr_en": "In the rectangular coordinate system xOy, there exists a hyperbola located in the first quadrant. Points A on the positive y-axis and C on the positive x-axis, together with point B in the first quadrant, form rectangle OABC. The side AB intersects the hyperbola at point D, and OD is connected. The side BC intersects the hyperbola at point E, which is precisely the midpoint of BC.\n", "ocr_zh": "已知直角坐标系xOy，有一条双曲线位于第一象限，A位于y轴正半轴，C位于x轴正半轴，B位于第一象限，构成矩形OABC。AB与双曲线相交于点D，连接OD，BC与双曲线相交于点E，且E是BC的中点。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_740.jpg" }, { "index": 741, "ocr_en": "In the rectangular coordinate system xOy, there is a hyperbola located in the first quadrant. An isosceles right triangle ABC is positioned in the first quadrant with AC parallel to the y-axis (point C above A), AB parallel to the x-axis (point B to the right of A), and both AC and BC being equal in length and intersecting the hyperbola. Point A has coordinates (1,1). Line segment OA is extended in both directions to form the straight line OA.\n", "ocr_zh": "已知直角坐标系xOy，有一条双曲线位于第一象限。等腰直角三角形ABC位于第一象限，AC平行于y轴，C在A的上方，AB平行于x轴，B在A的右方，AC与BC长度相等且均与双曲线相交。点A的横纵坐标均为1，连接OA并向两端延长得到直线OA。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_741.jpg" }, { "index": 742, "ocr_en": "In the rectangular coordinate system xOy, there exists a hyperbola located in the first and third quadrants. Point A lies on the hyperbola in the first quadrant. A perpendicular AB is drawn from A to the x-axis, meeting it at point B, and segment OA is connected, forming right triangle AOB. Inside triangle AOB, a smaller triangle DEF is constructed with sides parallel to and vertices corresponding in order to those of AOB. Point P is a point on the x-axis to the right of B. A line through P intersects sides AB and OA of the main triangle, as well as the corresponding sides DF and ED of the inner triangle. Point O' is a point on OA located above this line, and point B' is a point to the right of AB positioned between the hyperbola and the line. Finally, segment O'B' is drawn to complete the construction.", "ocr_zh": "已知直角坐标系xOy，有一双曲线位于一、三象限，点A位于第一象限的双曲线上，作AB垂直于x轴，垂足为B，连接OA，得到直角三角形AOB。在三角形AOB内部作一与其对应边平行且顶点顺序一致的三角形，设对应顶点为D、E、F。P为x轴上一点，P在B的右侧。过点P作直线与边AB和OA相交，同时也交于对应的边DF和ED。O'是OA上一点，且位于直线上方，B'为AB右侧一点，且位于双曲线与直线之间，连接O'B'。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_742.jpg" }, { "index": 743, "ocr_en": "In the rectangular coordinate system xOy, there exists a hyperbola located in the first quadrant. Points A and C lie on the hyperbola with A positioned above C, and segment AC connects them. Point B is situated to the right of AC such that AB is parallel to the x-axis and BC is parallel to the y-axis. Segments OC, OA, AB, and BC are then connected, with BC extended to intersect the x-axis. Point B' is the symmetric point of B with respect to line AC, and B' lies on OA. Finally, segment B'C is drawn to complete the figure.", "ocr_zh": "已知直角坐标系xOy，有一双曲线位于第一象限。A和C是双曲线上的点，A在C的上方，连接AC。点B在AC右侧，满足AB平行于x轴，BC平行于y轴。连接OC、OA、AB和BC，延长BC与x轴相交。B'是B关于AC对称的点，且B'在OA上。连接B'C。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_743.jpg" }, { "index": 744, "ocr_en": "In the rectangular coordinate system xOy, there exists a hyperbola located in both the first and third quadrants. A straight line passing through the origin O intersects the hyperbola at point A in the first quadrant and at point B in the third quadrant.", "ocr_zh": "已知直角坐标系xOy，有一双曲线位于一、三象限。一条过原点的直线与第一象限的双曲线相交于点A，与第三象限的双曲线相交于点B。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_744.jpg" }, { "index": 745, "ocr_en": "In the rectangular coordinate system xOy, point B is located in the first quadrant. A perpendicular BA is drawn from B to the x-axis, intersecting it at point A. Segment OB is then connected. In the first quadrant, there exists a hyperbola that passes through point D (the midpoint of OB) and intersects AB at point C. Segment OC is connected. Finally, point E is taken on segment OA, and segment DE is drawn to complete the construction.", "ocr_zh": "已知直角坐标系xOy，点B位于第一象限，过点B作BA垂直于x轴，垂足为点A。连接OB。第一象限有一条双曲线，经过OB的中点D，并与AB相交于点C。连接OC。点E为OA上一点，连接DE。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_745.jpg" }, { "index": 746, "ocr_en": "In the rectangular coordinate system xOy, there exists a hyperbola located in both the first and third quadrants. A straight line passing through the origin O intersects the hyperbola at point A in the first quadrant. Point B is situated on the negative x-axis, and segment AB is drawn to connect these two points.", "ocr_zh": "已知直角坐标系xOy，有一双曲线位于一、三象限。一条过原点的直线与第一象限的双曲线相交于点A，点B在x轴负半轴上，连接AB。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_746.jpg" }, { "index": 747, "ocr_en": "In the rectangular coordinate system xOy, there exists a branch of hyperbola located in the first quadrant. Points A and B lie on this hyperbola, with point A positioned above point B. From point A, a perpendicular AC is drawn to the y-axis, meeting it at point C, while from point B, a perpendicular BD is drawn to the x-axis, meeting it at point D. Finally, segments AB and CD are connected to complete the geometric configuration.", "ocr_zh": "已知直角坐标系xOy，有一支双曲线位于第一象限，点A、B为双曲线上的点，A在B的上方。过点A作AC垂直于y轴，垂足为C，过点B作BD垂直于x轴，垂足为D。连接AB和CD。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_747.jpg" }, { "index": 748, "ocr_en": "In the rectangular coordinate system xOy, there exists a branch of a hyperbola located in the first quadrant, with points A and B lying on this hyperbola such that A is positioned above B. From point A, a perpendicular line AC is drawn to the y-axis, intersecting it at point C, while from point B, a perpendicular line BD is drawn to the x-axis, intersecting it at point D. The segments AB and CD are then connected. Additionally, auxiliary lines are constructed: OA (connecting the origin O to point A), OB (connecting O to point B), BC (connecting points B and C), and AD (connecting points A and D).", "ocr_zh": "已知直角坐标系xOy，有一支双曲线位于第一象限，点A、B为双曲线上的点，A在B的上方。过点A作AC垂直于y轴，垂足为C，过点B作BD垂直于x轴，垂足为D。连接AB和CD。作辅助线OA、OB、BC和AD。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_748.jpg" }, { "index": 749, "ocr_en": "In the rectangular coordinate system xOy, there exists a branch of hyperbola located in the first quadrant. Point E is positioned to the right of this hyperbola in the first quadrant. From point E, a perpendicular ED is drawn to the x-axis, meeting it at point D, and a perpendicular EC is drawn to the y-axis, meeting it at point C. The line EC intersects the hyperbola at point A, while the line ED intersects the hyperbola at point B.", "ocr_zh": "已知直角坐标系xOy，有一支双曲线位于第一象限，点E位于第一象限双曲线右侧，过点E作ED垂直于x轴，垂足为D，过点E作EC垂直于y轴，垂足为C。EC与双曲线相交于点A，ED与双曲线相交于点B。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_749.jpg" }, { "index": 750, "ocr_en": "In the rectangular coordinate system xOy, a branch of hyperbola is located in the first quadrant. Points A and B lie on the hyperbola with A positioned above B. Segment AB is extended to form a straight line that intersects the positive x-axis at point F and the positive y-axis at point E.", "ocr_zh": "直角坐标系xOy中，一支双曲线位于第一象限，点A、B为双曲线上的点，A在B的上方，连接AB并延长，直线AB与x轴正半轴交于点F，与y轴正半轴交于点E。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_750.jpg" }, { "index": 751, "ocr_en": "In the rectangular coordinate system xOy, there exists a hyperbola branch located in the first quadrant. Point E is situated to the right of this hyperbola within the first quadrant. From point E, perpendicular ED is drawn to the x-axis (intersecting at D) and perpendicular EC is drawn to the y-axis (intersecting at C). Line EC intersects the hyperbola at point A, while line ED intersects the hyperbola at point B. Auxiliary lines AB and CD are then constructed to complete the diagram.", "ocr_zh": "已知直角坐标系xOy，有一支双曲线位于第一象限，点E位于第一象限双曲线右侧，过点E作ED垂直于x轴，垂足为D，过点E作EC垂直于y轴，垂足为C。EC与双曲线相交于点A，ED与双曲线相交于点B。作辅助线AB和CD。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_751.jpg" }, { "index": 752, "ocr_en": "In the rectangular coordinate system xOy, there exists a branch of hyperbola located in the first quadrant, with points A and B lying on this hyperbola such that A is positioned above B. From point A, an auxiliary perpendicular line AC is drawn to the y-axis, intersecting it at point C, while from point B, an auxiliary perpendicular line BD is drawn to the x-axis, intersecting it at point D. Segment AB is extended to form a straight line that intersects the positive x-axis at point F and the positive y-axis at point E. Finally, an auxiliary line CD is constructed to complete the geometric configuration.", "ocr_zh": "已知直角坐标系xOy，有一支双曲线位于第一象限，点A、B为双曲线上的点，A在B的上方。过点A作辅助线AC垂直于y轴，垂足为C，过点B作辅助线BD垂直于x轴，垂足为D。连接AB并延长得到直线AB，直线AB与x轴正半轴交于点F，与y轴正半轴交于点E。作辅助线CD。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_752.jpg" }, { "index": 753, "ocr_en": "In the rectangular coordinate system xOy shown in the upper diagram, there is a hyperbola branch located in the first quadrant with points A and B on the curve, where A is positioned above B. From point A, a perpendicular AD is drawn to the x-axis with foot point D, while from point B, a perpendicular BC is drawn to the y-axis with foot point C. Lines BC and AD intersect at point E. The ratio CE:EB = DE:EA is marked on the right side of the first quadrant figure. In the lower diagram within the same coordinate system xOy, a hyperbola exists in both the first and third quadrants, with point A on the first-quadrant branch and point B on the third-quadrant branch. Segment AB connects these points, intersecting the positive y-axis at point E and the negative x-axis at point F. The equality AE = BF is indicated on the left side of the second quadrant figure.", "ocr_zh": "上图中，已知直角坐标系xOy，有一支双曲线位于第一象限，点A、B为双曲线上的点，A在B的上方。过A作AD垂直于x轴，垂足为D。过B作BC垂直于y轴，垂足为C。BC与AD交于点E。第一象限图形右侧标有如下内容：CE:EB=DE:EA。\n下图中，已知直角坐标系xOy，有双曲线位于第一、三象限，第一象限双曲线上有一点A，第三象限双曲线上有一点B。连接AB，AB与y轴正半轴交于点E，与x轴负半轴交于点F。第二象限图形左侧标有如下内容：AE=BF。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_753.jpg" }, { "index": 754, "ocr_en": "In the rectangular coordinate system xOy, there exists a hyperbola branch located in the first quadrant, with points A and B lying on the curve where A is positioned above B. From point A, a perpendicular auxiliary line AE is drawn to the y-axis (intersecting at E), and from point B, a perpendicular auxiliary line BF is drawn to the x-axis (intersecting at F). The auxiliary line EF is then constructed. Points D and C are positioned between O-E and O-F respectively, satisfying the conditions that AB is parallel to CD and AD is parallel to BC. Finally, the segments CD, AD, BC, and AB are connected to complete the geometric configuration.", "ocr_zh": "已知直角坐标系xOy，有一支双曲线位于第一象限，点A、B为双曲线上的点，A在B的上方。过点A作辅助线AE垂直于y轴，垂足为E，过点B作辅助线BF垂直于x轴，垂足为F。作辅助线EF。点D位于OE之间，点C位于OF之间，满足AB平行于CD，AD平行于BC，连接CD、AD、BC和AB。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_754.jpg" }, { "index": 755, "ocr_en": "In the rectangular coordinate system xOy, there exists a hyperbola branch located in the first quadrant, with points A and B lying on the curve where A is positioned vertically above B. Point D is situated on the positive y-axis, while point C lies on the positive x-axis, forming a configuration where AB is parallel to CD and AD is parallel to BC. The quadrilateral is completed by connecting segments CD, AD, BC, and AB.", "ocr_zh": "已知直角坐标系xOy，有一支双曲线位于第一象限，点A、B为双曲线上的点，A在B的上方。点D位于y轴正半轴，点C位于x轴正半轴，满足AB平行于CD，AD平行于BC，连接CD、AD、BC和AB。角DCO与角BCx大小相等，角ADy与角CDO大小相等。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_755.jpg" }, { "index": 756, "ocr_en": "In the rectangular coordinate system xOy, there exists a hyperbola located in both the first and third quadrants, with points A and B positioned on its first-quadrant branch where A lies above B. Segment AB connects these points. Points C and D are situated on the negative y-axis and negative x-axis respectively. Segments CD, AD (extended), and BC (extended) are constructed such that AB is parallel to CD and AD is parallel to BC. The extended AD intersects the positive y-axis at point F, while the extended BC meets the negative y-axis at point E. Segment EF is then drawn.", "ocr_zh": "已知直角坐标系xOy，有双曲线位于第一、三象限，A、B为第一象限双曲线上的点，A在B的上方，连接AB。点C位于y轴负半轴，点D位于x轴负半轴，连接CD、连接AD并延长，连接BC并延长，满足AB平行于CD，AD平行于BC。AD与y轴正半轴交于点F，BC与y轴负半轴交于点E，连接EF。角FDE与角CDE大小相等，角DCF和角ECF大小相等。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_756.jpg" }, { "index": 757, "ocr_en": "In the rectangular coordinate system xOy, a hyperbola is located in both the first and third quadrants, with points A and B lying on its first-quadrant branch where A is positioned above B. The line AB connects these points, intersecting the positive y-axis at point F and the positive x-axis at point E. When connecting BO, the line BO meets the hyperbola's third-quadrant branch at point C. The line AC is then drawn, intersecting the positive y-axis at point G and the negative x-axis at point D. This configuration satisfies two angle conditions: angle ADE equals angle AED, and angle AFG equals angle AGF.", "ocr_zh": "已知直角坐标系xOy，有双曲线位于第一、三象限，A、B为第一象限双曲线上的点，A在B的上方，连接AB，直线AB与y轴正半轴交于点F，与x轴正半轴交于点E。连接BO，直线BO与第三象限的双曲线交于点C，连接AC，直线AC与y轴正半轴交于点G，与x轴负半轴交于点D。角ADE和角AED大小相等，角AFG和角AGF大小相等。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_757.jpg" }, { "index": 758, "ocr_en": "In the rectangular coordinate system xOy, there exists a hyperbola located in both the first and third quadrants, with points A and B lying on its first-quadrant branch where A is positioned above B. The line AB connects these points, intersecting the positive y-axis at point F and the positive x-axis at point E. The line BO (connecting point B to the origin O) intersects the hyperbola's third-quadrant branch at point C. The line AC is then drawn, intersecting the positive y-axis at point G and the negative x-axis at point D. This configuration satisfies two angle conditions: angle ADE equals angle AED, and angle AFG equals angle AGF.\nFrom point A, an auxiliary perpendicular AH is drawn to the y-axis, meeting it at point H, while from point B, an auxiliary perpendicular BJ is drawn to the x-axis, meeting it at point J. Similarly, from point C, an auxiliary perpendicular CI is drawn to the x-axis, intersecting it at point I. Finally, auxiliary lines IH and JH are constructed to complete the geometric figure.", "ocr_zh": "已知直角坐标系xOy，有双曲线位于第一、三象限，A、B为第一象限双曲线上的点，A在B的上方，连接AB，直线AB与y轴正半轴交于点F，与x轴正半轴交于点E。连接BO，直线BO与第三象限的双曲线交于点C，连接AC，直线AC与y轴正半轴交于点G，与x轴负半轴交于点D。角ADE和角AED大小相等，角AFG和角AGF大小相等。过A作辅助线AH垂直于y轴，垂足为H。过B作辅助线BJ垂直于x轴，垂足为J。过C作辅助线CI垂直于x轴，垂足为I。作辅助线IH和JH。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_758.jpg" }, { "index": 759, "ocr_en": "In the rectangular coordinate system xOy, there exists a hyperbola located in both the first and third quadrants. Points A and B lie on the hyperbola in the first quadrant. Segments AO and BO are drawn, with line AO intersecting the hyperbola in the third quadrant at point H, and line BO intersecting the hyperbola in the third quadrant at point C. Line AB is formed by extending segment AB in both directions, while line CH is created by extending segment CH in both directions.\nLine AB intersects the positive y-axis at point F and the positive x-axis at point E. Line CH intersects the negative x-axis at point K and the negative y-axis at point L. Segments FK and EL are then connected.\nLine AC is formed by extending segment AC, intersecting the positive y-axis at point G and the negative x-axis at point D. Line BH is created by extending segment BH, intersecting the positive x-axis at point I and the negative y-axis at point J. Segments GI and DJ are then connected.", "ocr_zh": "直角坐标系xOy中，有一双曲线位于一、三象限。点A、B是第一象限的双曲线上的点，A在B的上方,连接AO、BO，直线AO与第三象限中的双曲线相交于点H，直线BO与第三象限中的双曲线相交于点C，连接AB并向两端延长得到直线AB，连接CH并向两端延长的得到直线CH。直线AB与y轴正半轴交于点F，与x轴正半轴交于点E。直线CH与x轴负半轴交于点K，与y轴负半轴交于点L。连接FK、EL。连接AC并延长得到直线AC，与y轴正半轴交于点G，与x轴负半轴交于点D。连接BH并延长得到直线BH，与x轴正半轴交于点I，与y轴负半轴交于点J。连接GI和DJ。角GFA与FGA大小相等，角BEI和角GDI大小相等。", "class": "Analytic Geometry", "difficulty": 3, "image_path": "fig_759.jpg" }, { "index": 760, "ocr_en": "In the rectangular coordinate system xOy, there exists a branch of hyperbola located in the second quadrant, with points C and D lying on this hyperbola where C is positioned above D. Point B is situated on the positive y-axis while point A lies on the negative x-axis. The quadrilateral is formed by connecting AB, BC, CD and AD, with the specific conditions that AB is parallel to CD and BC is parallel to AD.", "ocr_zh": "直角坐标系xOy中，有一支双曲线位于第二象限，点C和D在双曲线上，C在D的上方，点B在y轴正半轴，点A在x轴负半轴，连接AB、BC、CD和AD，满足AB平行于CD，BC平行于AD。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_760.jpg" }, { "index": 761, "ocr_en": "In the rectangular coordinate system xOy, there exists a branch of hyperbola located in the first quadrant, with points C and D lying on this hyperbola where D is positioned above C. Point B is situated on the negative y-axis while point A lies on the negative x-axis. The quadrilateral is formed by connecting AB, BC, CD and AD, with the specific conditions that AB is parallel to CD and BC is parallel to AD. The line AD intersects the positive y-axis at point E.", "ocr_zh": "直角坐标系xOy中，有一支双曲线位于第一象限，点C和D在双曲线上，D在C的上方，点B在y轴负半轴，点A在x轴负半轴，连接AB、BC、CD和AD，满足AB平行于CD，BC平行于AD。AD与y轴正半轴交于点E。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_761.jpg" }, { "index": 762, "ocr_en": "In the rectangular coordinate system xOy, a branch of hyperbola is located in the first quadrant with points C and D lying on the curve, where D is positioned above C. Point B is situated on the negative y-axis and point A lies on the negative x-axis. The quadrilateral is constructed by connecting AB, BC, CD and AD, maintaining the parallel conditions AB∥CD and BC∥AD. The line AD intersects the positive y-axis at point E. Additionally, from point D, a perpendicular DF is drawn to the x-axis, meeting it at point F.", "ocr_zh": "直角坐标系xOy中，有一支双曲线位于第一象限，点C和D在双曲线上，D在C的上方，点B在y轴负半轴，点A在x轴负半轴，连接AB、BC、CD和AD，满足AB平行于CD，BC平行于AD。AD与y轴正半轴交于点E。过点D作DF垂直于x轴，垂足为F。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_762.jpg" }, { "index": 763, "ocr_en": "In the rectangular coordinate system xOy, there exists a hyperbola located in both the first and third quadrants. Points P and B lie on the hyperbola's first-quadrant branch, with P positioned above B. The line BO (connecting point B to the origin O) intersects the hyperbola's third-quadrant branch at point A. Segments PA and PB are then constructed, with PB extended to intersect the x-axis.", "ocr_zh": "直角坐标系xOy中，有一双曲线位于一、三象限。点P、B是第一象限的双曲线上的点，P在B的上方。连接BO，直线BO与第三象限双曲线交于点A，连接PA，连接PB并延长与x轴相交。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_763.jpg" }, { "index": 764, "ocr_en": "In the rectangular coordinate system xOy, there exists a hyperbola branch located in the second quadrant, with points C and D lying on the curve where C is positioned above D. Point B is situated on the positive y-axis while point A lies on the negative x-axis. The quadrilateral is formed by connecting segments AB, BC, CD and AD, satisfying the parallel conditions AB∥CD and BC∥AD. Additionally, from point C, a perpendicular CE is drawn to the y-axis, intersecting it at point E.", "ocr_zh": "直角坐标系xOy中，有一支双曲线位于第二象限，点C和D在双曲线上，C在D的上方，点B在y轴正半轴，点A在x轴负半轴，连接AB、BC、CD和AD，满足AB平行于CD，BC平行于AD。过点C作CE垂直于y轴，垂足为E。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_764.jpg" }, { "index": 765, "ocr_en": "In the rectangular coordinate system xOy, there exists a hyperbola located in both the first and third quadrants, with points P and B lying on its first-quadrant branch where P is positioned above B. The line BO (connecting point B to the origin O) intersects the hyperbola's third-quadrant branch at point A. The line PA is then drawn, intersecting the x-axis at point M, while the line PB intersects the x-axis at point N. Point Q is a point on the hyperbola between P and B. The extension of QB meets the x-axis at point D, and the line QA intersects the x-axis at point C. This configuration satisfies two angle conditions: angle PMN equals angle PNM, and angle QCN equals angle QDC.", "ocr_zh": "直角坐标系xOy中，有一双曲线位于一、三象限。点P、B是第一象限的双曲线上的点，P在B的上方。连接BO，直线BO与第三象限双曲线交于点A，连接PA，直线PA与x轴相交于点M，连接PB，直线PB与x轴相交于点N。点Q是双曲线上P与B之间的点，连接QB并延长与x轴相交于点D，连接QA，QA与x轴交于点C。角PMN与角PNM大小相等。角QCN与角QDC大小相等。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_765.jpg" }, { "index": 766, "ocr_en": "In triangle ABC, let L be the midpoint of side BC and M be the midpoint of side AC. Segments AL and BM are drawn, intersecting at point G (the centroid of the triangle). Segment CG is then extended to meet side AB at point N. Next, segment GL is extended to point D such that GL = LD. Finally, segments CD and BD are constructed to complete the geometric configuration.", "ocr_zh": "三角形ABC中，L是BC的中点，M是AC的中点，连接AL和BM，AL和BM相交于点G，连接CG并延长交AB于点N。延长GL至点D，GL与LD长度相等。连接CD和BD。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_766.jpg" }, { "index": 767, "ocr_en": "On circle O, there are three points z₁, z₂, and z₃. Segments z₁z₂, z₁z₃, and z₂z₃ are connected to form a triangle inscribed in the circle.", "ocr_zh": "已知圆O上有三个点z1，z2，z3，连接z1z2、z1z3和z2z3。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_767.jpg" }, { "index": 768, "ocr_en": "In triangle ABC, point O is an interior point such that segments OA, OB, and OC are equal in length. Points M, N, and L lie on sides AC, AB, and BC respectively, with segments OM, ON, and OL connected to them. A circle O is drawn with center at point O and radius equal to OA.", "ocr_zh": "已知三角形ABC，点O是三角形ABC内一点，连接OA、OB和OC,且OA、OB和OC长度相等。点M、N、L分别是边AC、AB和BC上的点，连接OM、ON、OL。以点O为圆心，OA为半径作圆O。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_768.jpg" }, { "index": 769, "ocr_en": "In triangle ABC, a perpendicular AD is drawn from vertex A to side BC, meeting it at point D, while a perpendicular BE is drawn from vertex B to side AC, intersecting at point E. The two altitudes AD and BE intersect at point H (the orthocenter of the triangle). Segment CH is then extended to meet side AB at point F. This configuration satisfies two angle conditions: angle BAC equals angle EDC, and angle ACF equals angle ADE.", "ocr_zh": "已知三角形ABC，过点A作AD垂直于BC，垂足为点D。过点B作BE垂直于AC，垂足为E。AD与BE相交于H。连接CH并延长与AB交于点F。角BAC等于角EDC。角ACF等于角ADE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_769.jpg" }, { "index": 770, "ocr_en": "In triangle ABC, three parallel lines are constructed: one through point A parallel to BC, one through B parallel to AC, and one through C parallel to AB. The parallel line through A intersects the parallel line through B at point U, the parallel line through A intersects the parallel line through C at point V, and the parallel line through B intersects the parallel line through C at point W. Connecting these points forms triangle UVW, where points A, B, and C serve as the midpoints of sides VW, UW, and UV respectively. Within the original triangle ABC, three altitudes are drawn: AD perpendicular to BC (meeting at D), BE perpendicular to AC (meeting at E), and CF perpendicular to AB (meeting at F). These three altitudes intersect at a common point H (the orthocenter of the triangle).", "ocr_zh": "已知三角形ABC， 分别过点A、B、C作BC、AC、AB的平行线，BC的平行线与AC的平行线交于点W，BC的平行线与AB的平行线交于点V，AC的平行线与AB的平行线交于点U。连接UV、UW和VW，A、B、C分别是VW、UW、UV的中点。过点A作AD垂直于BC，垂足为点D。过点B作BE垂直于AC，垂足为E。过点C作CF垂直于AB，垂足为F。AD、BE和CF相交于H。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_770.jpg" }, { "index": 771, "ocr_en": "In triangle ABC, the angle bisector of ∠A and the angle bisector of ∠B intersect at point I (the incenter). Segment IC is connected, and from point I, perpendiculars are drawn to sides BC, CA, and AB with feet R, S, and T respectively, where the lengths IR, IT, and IS are equal. A circle I is drawn with center at I and radius equal to IR. The side lengths are given as: BC = a with BR = y and RC = z; AB = c with AT = x and TB = y; AC = b with AS = x and SC = z.", "ocr_zh": "已知三角形ABC，内角A的平分线与内角B的平分线交于点I，连接IC，过点I分别作BC、CA、AB的垂线，其垂足分别是R、S、T，有IR、IT、IS三者长度相等。以I为圆心，IR为半径作圆I。BC长度为a，BR长度为y，RC长度为z，AB长度为c，AT长度为x，TB长度为y，AC长度为b，AS长度为x，SC长度为z。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_771.jpg" }, { "index": 772, "ocr_en": "In triangle ABC, the bisector of the exterior angle at B (formed by the extension of AB and side BC) intersects the bisector of the exterior angle at C (formed by the extension of AC and side CB) at point I₁. Similarly, the bisector of the exterior angle at C (formed by the extension of BC and side CA) intersects the bisector of the exterior angle at A (formed by the extension of BA and side AC) at point I₂, while the bisector of the exterior angle at A (formed by the extension of CA and side AB) intersects the bisector of the exterior angle at B (formed by the extension of CB and side BA) at point I₃. The points I₁, I₂, and I₃ are connected to form triangle I₁I₂I₃ with sides I₁I₂, I₂I₃, and I₁I₃. Lines AI₁, BI₂, and CI₃ are drawn, intersecting at a common point I. From I₁, perpendiculars are dropped to the extension of AC (meeting at S₁) and the extension of AB (meeting at T₁), while a perpendicular to BC meets at R₁. A circle centered at I₁ is drawn tangent to BC (with radius I₁R₁), another circle centered at I₂ is tangent to AC (at S₂) and also tangent to the extensions of BC (at R₂) and BA (at T₂), and a third circle centered at I₃ is tangent to AB (at T₃) and the extensions of CA (at S₃) and CB (at R₃), completing the full excircle configuration of the triangle.", "ocr_zh": "已知三角形ABC，外角B（AB的延长线与线段BC的夹角）的平分线与外角C（AC的延长线与线段CB的夹角）的平分线交于I1。外角C（BC的延长线与线段CA的夹角）的平分线与外角A（BA的延长线与线段AC的夹角）的平分线相交于I2。外角A（CA的延长线与线段AB的夹角）的平分线与外角B（CB的延长线与线段BA的夹角）的平分线相交于I3。连接I1I2、I2I3、I1I3。连接AI1、BI2、CI3，三条线段相交于点I。过I1分别作AC延长线的垂线、AB延长线的垂线，垂足分别为S1和T1。过I1作BC的垂线，垂足为R1。以I1为圆心作与BC相切的圆I1，以I2为圆心作与AC相切的圆I2，切点为S2，圆I2与BC延长线的切点为R2，圆I2与BA延长线切点为T2。以I3为圆心作与AB相切的圆I3，切点为T3，圆I3与CA延长线的切点为S3，圆I3与CB延长线的切点为R3。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_772.jpg" }, { "index": 773, "ocr_en": "In triangle ABC, three altitudes are constructed: AD perpendicular to BC with foot point D, CF perpendicular to AB with foot point F, and BE perpendicular to AC with foot point E.", "ocr_zh": "已知三角形ABC，作AD垂直于BC，垂足为D。作CF垂直于AB，垂足为F。作BE垂直于AC，垂足为E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_773.jpg" }, { "index": 774, "ocr_en": "In triangle ABC, three altitudes are constructed: AD perpendicular to BC (meeting at D), CF perpendicular to AB (meeting at F), and BE perpendicular to AC (meeting at E). Parallel lines are then drawn: through point A parallel to BC, through B parallel to AC, and through C parallel to AB. These parallels intersect pairwise to form points C' (intersection of BC's parallel with AC's parallel), B' (intersection of BC's parallel with AB's parallel), and A' (intersection of AC's parallel with AB's parallel). The auxiliary triangle A'B'C' is completed by connecting segments A'B', A'C', and B'C'.", "ocr_zh": "已知三角形ABC，作AD垂直于BC，垂足为D。作CF垂直于AB，垂足为F。作BE垂直于AC，垂足为E。分别过点A、B、C作BC、AC、AB的平行线，BC的平行线与AC的平行线交于点C'，BC的平行线与AB的平行线交于点B'，AC的平行线与AB的平行线交于点A'。作辅助线A'B'、A'C'和B'C'。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_774.jpg" }, { "index": 775, "ocr_en": "In triangle ABC, AD is drawn perpendicular to BC with foot D, and CF is drawn perpendicular to AB with foot F, where AD and CF intersect at point H. Segment BH is extended to meet AC at point E. An auxiliary circle is constructed with the midpoint of AC as its center and AC as its diameter. The auxiliary line DF is then drawn, with angle ACF labeled as angle 1, angle ADF as angle 2, and angle ABE as angle 3.", "ocr_zh": "已知三角形ABC，作AD垂直于BC，垂足为D。作CF垂直于AB，垂足为F。AD和CF相交于点H。连接BH并延长与AC交于点E。以AC中点为圆心，AC为直径画辅助圆。作辅助线DF。角ACF记作角1，角ADF记作角2，角ABE记作角3。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_775.jpg" }, { "index": 776, "ocr_en": "In triangle ABC, three altitudes are constructed: AD perpendicular to BC (meeting at D), CF perpendicular to AB (meeting at F), and BE perpendicular to AC (meeting at E). Two auxiliary circles are drawn - one with the midpoint of AC as center and AC as diameter, another with the midpoint of AB as center and AB as diameter. Auxiliary lines EF, ED, and DF are then connected. The angles are designated as follows: angle ACF as angle 3, angle ADF as angle 4, angle ADE as angle 1, and angle ABE as angle 2.", "ocr_zh": "已知三角形ABC，作AD垂直于BC，垂足为D。作CF垂直于AB，垂足为F。作BE垂直于AC，垂足为E。以AC中点为圆心，AC为直径画辅助圆。以AB中点为圆心，AB为直径画辅助圆。做辅助线EF、ED、DF。角ACF记作角3，角ADF记作角4，角ADE记作角1，角ABE记作角2。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_776.jpg" }, { "index": 777, "ocr_en": "In triangle ABC, three altitudes are constructed: AD perpendicular to BC with foot D, CF perpendicular to AB with foot F, and BE perpendicular to AC with foot E, all intersecting at point H (the orthocenter). An auxiliary line DE is drawn connecting points D and E. An auxiliary circle F is constructed with center at F and diameter AB. The angles are designated as follows: angle ABE as angle 1, angle ADE as angle 2, angle FHB as angle 3, angle EDC as angle 4, and angle EHC as angle 5.", "ocr_zh": "已知三角形ABC，作AD垂直于BC，垂足为D。作CF垂直于AB，垂足为F。作BE垂直于AC，垂足为E。三条垂线相交于点H。作辅助线DE。以F为圆心，AB为直径作辅助圆F。角ABE记作角1，角ADE记作角2，角FHB记作角3，角EDC记作角4，角EHC记作角5。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_777.jpg" }, { "index": 778, "ocr_en": "In the rectangular coordinate system xDy, point C is located on the positive x-axis with coordinates (c, 0), point B lies on the negative x-axis with coordinates (b, 0), and point A is positioned on the positive y-axis with coordinates (0, a), while point D coincides with the origin at (0, 0), where the length of DC is greater than BD. Segments AB, AC, BC, and AD are connected. From point C, a perpendicular CF is drawn to AB, meeting it at point F. Segment BH is extended to intersect AC at point E.", "ocr_zh": "直角坐标系xDy中，C在x轴正半轴，坐标为（c,0），B在x轴负半轴，坐标为（b,0）,A在y轴正半轴，坐标为（0,a），D的坐标是（0,0），DC长度大于BD。连接AB、AC、BC和AD。过C作CF垂直于AB，垂足为点F。连接BH并延长交AC于点E。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_778.jpg" }, { "index": 779, "ocr_en": "In triangle ABC, AD is drawn perpendicular to BC with foot D, and CF is drawn perpendicular to AB with foot F. The lines CF and AD intersect at point H. Segment BH is extended to meet AC at point E. Vectors are then marked as follows: vector a is placed to the right of HA, vector b to the right of HB, and vector c above HC.", "ocr_zh": "已知三角形ABC，作AD垂直于BC，垂足为D。作CF垂直于AB，垂足为F。CF与AD相交于点H，连接BH并延长，与AC交于点E。在HA右侧标记向量a，在HB右侧标记向量b，在HC上方标记向量c。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_779.jpg" }, { "index": 780, "ocr_en": "In the rectangular coordinate system xOy, point C lies on the positive x-axis, point B lies on the positive y-axis, and point A is located in the first quadrant with its x-coordinate smaller than C's and its y-coordinate larger than B's. Connecting AB, AC, and BC forms triangle ABC. From point A, an auxiliary line AG is drawn perpendicular to the x-axis, intersecting it at point G, while AG meets BC at point D. The triangle ABC is filled with green color, segment AD is drawn in red, and segment OC is drawn in blue.", "ocr_zh": "已知直角坐标系xOy，C在x轴正半轴，B在y轴正半轴，A在第一象限，A的横坐标小于C的横坐标，纵坐标大于B的纵坐标。连接AB、AC、BC得到三角形ABC。过点A作辅助线AG垂直于x轴，垂足为G。AG与BC交于点D。三角形ABC填充绿色，用红色线段连接AD，用蓝色线段连接OC。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_780.jpg" }, { "index": 781, "ocr_en": "In the rectangular coordinate system xOy, point C is positioned on the positive x-axis while point B lies on the positive y-axis, with point A located in the second quadrant where its y-coordinate exceeds that of point B. Connecting segments AB, AC, and BC forms triangle ABC. An auxiliary line AG is drawn perpendicular to the x-axis from point A, intersecting the axis at point G and meeting line BC at point D. The triangle ABC is colored green, with segment AD highlighted in red and segment OC displayed in blue.", "ocr_zh": "已知直角坐标系xOy，C在x轴正半轴，B在y轴正半轴，A在第二象限，A的纵坐标大于B的纵坐标。连接AB、AC、BC得到三角形ABC。过点A作辅助线AG垂直于x轴，垂足为G。AG与直线BC交于点D。三角形ABC填充绿色，用红色线段连接AD，用蓝色线段连接OC。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_781.jpg" }, { "index": 782, "ocr_en": "In the Cartesian coordinate system xOy, point C is situated on the positive x-axis (coordinates (c,0), c>0), point B lies on the positive y-axis (coordinates (0,b), b>0), and point A is located in the first quadrant (coordinates (a_x,a_y) with a_x > c and a_y > b. Connecting points AB, AC, and BC forms triangle ABC.\nA perpendicular AG is drawn from point A to the x-axis, intersecting it at point G (coordinates (a_x,0)), and extended to meet line BC at point D.The triangle ABC is colored green, with segment AD shown in red and segment OC (connecting the origin O(0,0) to point C) displayed in blue.", "ocr_zh": "已知直角坐标系xOy，C在x轴正半轴，B在y轴正半轴，A在第一象限，A的纵坐标大于B的纵坐标。A的横坐标大于C的横坐标。连接AB、AC、BC得到三角形ABC。过点A作辅助线AG垂直于x轴，垂足为G。AG的延长线与直线BC交于点D。三角形ABC填充绿色，用红色线段连接AD，用蓝色线段连接OC。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_782.jpg" }, { "index": 783, "ocr_en": "In the rectangular coordinate system xOy, point C is located on the positive x-axis (coordinates (c,0) where c>0), point B is positioned on the positive y-axis (coordinates (0,b) where b>0), and point A lies in the first quadrant (coordinates (a_x,a_y) where a_x>c and a_y>b). Connecting points AB, AC, and BC forms triangle ABC.\nFrom point A, an auxiliary line AE is drawn perpendicular to the x-axis, intersecting it at point E (coordinates (a_x,0)). From point B, a line parallel to the x-axis is drawn, intersecting AC at point D.", "ocr_zh": "已知直角坐标系xOy，C在x轴正半轴，B在y轴正半轴，A在第一象限，A的纵坐标大于B的纵坐标。A的横坐标大于C的横坐标。连接AB、AC、BC得到三角形ABC。过点A作辅助线AE垂直于x轴，垂足为E。过点B作x轴的平行线与AC交于点D。三角形ABC填充绿色，用红色线段连接AE，用蓝色线段连接BD。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_783.jpg" }, { "index": 784, "ocr_en": "In the rectangular coordinate system xOy, point C is located on the positive x-axis (coordinates (c,0) where c>0), point B is positioned on the positive y-axis (coordinates (0,b) where b>0), and point A lies in the first quadrant (coordinates (a_x,a_y) where a_x>c and a_y>b). Connecting points AB, AC, and BC forms triangle ABC.From point A, an auxiliary line AE is drawn perpendicular to the y-axis, intersecting it at point E (coordinates (0,a_y)). From point C, a line parallel to the y-axis is drawn, intersecting AB at point D (coordinates (c,y_D)).The interior of triangle ABC is filled with green color. Segment CD is drawn in red, and segment AE is drawn in blue.", "ocr_zh": "已知直角坐标系xOy，C在x轴正半轴，B在y轴正半轴，A在第一象限，A的纵坐标大于B的纵坐标。A的横坐标大于C的横坐标。连接AB、AC、BC得到三角形ABC。过点A作辅助线AE垂直于y轴，垂足为E。过点C作y轴的平行线与AB交于点D。三角形ABC填充绿色，用红色线段连接CD，用蓝色线段连接AE。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_784.jpg" }, { "index": 785, "ocr_en": "In the rectangular coordinate system xOy, point C is located on the positive x-axis (coordinates (c,0), c>0), point B is positioned on the positive y-axis (coordinates (0,b), b>0), and point A lies in the first quadrant (coordinates (a_x,a_y) where a_y>b and 0 0). A circle A is drawn centered at point A, intersecting the x-axis at points whose x-coordinates are all positive and intersects circle A at two distinct points in the first quadrant, forming an angle α with the positive y-axis .", "ocr_zh": "直角坐标系xOy中，点A是x轴正半轴上一点，以A为圆心作圆A，圆A与x轴交点的横坐标均大于0。过原点的直线l与圆A有两个在第一象限的交点，直线l与y轴正半轴的夹角为α。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_812.jpg" }, { "index": 813, "ocr_en": "In the rectangular coordinate system xOy, a semicircle is drawn centered at the origin O, intersecting the x-axis at points A (left) and B (right). A parabolic arc is constructed through points A and B, opening upward and symmetric about the y-axis.", "ocr_zh": "直角坐标系xOy中，以O为圆心画上半圆，半圆O与x轴交于A和B两点，B在A右侧。过A、B两点作抛物线段，该抛物线段开口向上且关于y轴对称。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_813.jpg" }, { "index": 814, "ocr_en": "In the rectangular coordinate system xOy, there exists a downward-opening parabolic arc symmetric about the y-axis, intersecting the y-axis at point C and the x-axis at points A (left) and B (right). A straight line y = x + b is drawn such that: its y-intercept lies between origin O and point C, its x-intercept lies between A and O, and it intersects the parabolic arc at a single point in the first quadrant.", "ocr_zh": "直角坐标系xOy中有一抛物线段，该抛物线段开口向下且关于y轴对称，与y轴交点为C，与x轴的交点是 A和B，B在A的右侧。有一直线y=x+b，直线与y轴的交点在O与C之间，与x轴的交点在A与O之间，与抛物线段的交点在第一象限。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_814.jpg" }, { "index": 815, "ocr_en": "In the rectangular coordinate system xOy, there exists an upward-opening parabola with its axis of symmetry located to the right of the y-axis. The parabola intersects the negative y-axis at point A and the positive x-axis at point B, with segment AB connecting these points. Point P lies on the parabola below segment AB, and point Q is positioned on AB such that PQ is parallel to the y-axis. Segments AP and PQ are then connected. Additionally, a straight line y = x + b is drawn with its y-intercept located between points A and the origin O.", "ocr_zh": "直角坐标系xOy中有一开口向上的抛物线，对称轴在y轴右侧，与y轴负半轴交于点A，与x轴正半轴交于点B，连接AB。P位于线段AB下方的抛物线上，点Q在线段AB上，满足PQ平行于y轴。连接AP和PQ。直线y=x+b与y轴的交点在A与O之间。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_815.jpg" }, { "index": 816, "ocr_en": "Given circle O with point P on its circumference and point A located to the left of circle O, segment AP is connected with Q as its midpoint. An auxiliary line AO is drawn, and M is designated as the midpoint of AO. Auxiliary lines MQ and OP are then constructed. Finally, an auxiliary circle M is drawn with center at M and radius equal to MQ.", "ocr_zh": "已知圆O，P是圆O上一点，A是圆0左边一点，连接AP，Q是AP中点，作辅助线AO，M是AO中点，作辅助线MQ、OP。以M为圆心，MQ为半径作辅助圆M。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_816.jpg" }, { "index": 817, "ocr_en": "Given circle O with point P on its circumference and point A located to the left of circle O, segment AP is connected with Q as its midpoint.", "ocr_zh": "已知圆O，P是圆O上一点，A是圆0左边一点，连接AP，Q是AP中点。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_817.jpg" }, { "index": 818, "ocr_en": "Given circle O with point P on its circumference and point A located to the left of circle O, segment AP is connected. Dashed lines OP and OA are drawn to form triangle AOP. This triangle AOP is then rotated 90 degrees counterclockwise about point A to obtain the corresponding triangle AMQ. Finally, an auxiliary circle M is constructed with center at point M and radius equal to MQ.", "ocr_zh": "已知圆O，P是圆O上一点，A是圆0左边一点，连接AP，虚线连接OP和OA，得到三角形AOP，将三角形AOP绕点A逆时针旋转90度，得到对应的三角形AMQ。以M为圆心，MQ为半径作辅助圆M。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_818.jpg" }, { "index": 819, "ocr_en": "Given circle O with point P lying on its circumference and points A, Q located outside the circle, segments AP, AQ, and PQ are connected to form the geometric configuration.", "ocr_zh": "已知圆O，P是圆O上一点，A和Q是圆O外的点，连接AP、AQ和PQ。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_819.jpg" }, { "index": 820, "ocr_en": "Given circle O with point P on its circumference and point A located to the left of the circle, segment AP is connected. Point Q is positioned outside circle O, and segment AQ is drawn such that it is perpendicular to AP (AQ⊥AP).", "ocr_zh": "已知圆O，P是圆O上一点，A是圆0左边一点，连接AP.Q圆O外一点，连接AQ，满足AQ垂直于AP。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_820.jpg" }, { "index": 821, "ocr_en": "In the left diagram, given circle O with point P on its circumference and point A located to the left of the circle, segment AP is connected. Point Q lies outside circle O, and segment AQ is drawn, with angle PAQ labeled as α. The middle diagram shows a right-pointing arrow. In the right diagram, given circle O with point P on its circumference and point A to the left of the circle, segment AP is connected. Dashed lines AO and OP form triangle AOP, which is filled green. A dashed circle M is positioned above and to the left of circle O, with point Q on circle M. Segment AQ is drawn, and dashed lines AM and MQ form triangle AMQ, filled red. Both angle MAO and angle QAP are marked as α.", "ocr_zh": "左图中，已知圆O，P是圆O上一点，A是圆0左边一点，连接AP。Q圆O外一点，连接AQ，角PAQ记作α。\n中图是一个朝右的箭头。\n右图中，已知圆O，P是圆O上一点，A是圆0左边一点，连接AP。虚线连接AO和OP得到三角形AOP，三角形AOP填充绿色。虚线圆M位于圆O左上方，Q是圆M上一点，连接AQ，虚线连接AM和MQ，得到三角形AMQ，三角形AMQ填充红色。角MAO和角QAP均标记α。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_821.jpg" }, { "index": 822, "ocr_en": "Given circle O with point P on its circumference and point A located to the left of the circle, segment AP is connected. Dashed lines AO and OP are drawn to form triangle AOP, which is filled green. A dashed circle M is positioned above and to the left of circle O, with point Q lying on circle M. Segment AQ is connected, and dashed lines AM and MQ form triangle AMQ, satisfying AM⊥AO and AP⊥AQ. Triangle AMQ is filled red.", "ocr_zh": "已知圆O，P是圆O上一点，A是圆0左边一点，连接AP。虚线连接AO和OP得到三角形AOP，三角形AOP填充绿色。虚线圆M位于圆O左上方，Q是圆M上一点，连接AQ，虚线连接AM和MQ，得到三角形AMQ，有AM垂直于AO，AP垂直于AQ。三角形AMQ填充红色。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_822.jpg" }, { "index": 823, "ocr_en": "Given circle O with point P on its circumference and point A located to the left of the circle, segment AP is connected. Point Q is positioned above AP such that connecting AQ and PQ forms an isosceles triangle APQ (with AQ = PQ).", "ocr_zh": "已知圆O，P是圆O上一点，A是圆0左边一点，连接AP。Q是AP上方一点，连接AQ和PQ，得到等腰三角形APQ，AQ与PQ长度相等。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_823.jpg" }, { "index": 824, "ocr_en": "Given circle O with point A located to its left, a dashed line connects AO. Point P is positioned above AO, and segment AP is connected with a dashed line OP. An equilateral triangle APQ is constructed upward using AP as one side. A dashed circle M is drawn to the right of point Q, with dashed lines QM and AM connected. The angle MAO measures 60 degrees.", "ocr_zh": "已知圆O，A是圆O左边一点，虚线连接AO，P是AO上方一点，连接AP，虚线连接OP。以AP为边向上作等边三角形APQ。过点Q向右作虚线圆M，虚线连接QM和AM。角MAO为60度。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_824.jpg" }, { "index": 825, "ocr_en": "Given circle O with point P on its circumference and point A located to the left of the circle, segment AP is connected. Point Q is positioned above AP, and segments AQ and PQ are drawn to complete the geometric configuration.", "ocr_zh": "已知圆O，P是圆O上一点，A是圆0左边一点，连接AP。Q是AP上方一点，连接AQ和PQ，得到等边三角形APQ。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_825.jpg" }, { "index": 826, "ocr_en": "In the rectangular coordinate system xOy, circle P is located in the first quadrant, with point B on the positive x-axis having a greater x-coordinate than P. Point M lies on circle P in the lower left quadrant relative to its center P. Segment MB is connected, and point C is marked as the midpoint of MB. Dashed lines MP and BP are drawn, with point O designated as the midpoint of BP. An auxiliary line OC is constructed and extended to intersect the x-axis at point A.", "ocr_zh": "已知直角坐标系xOy，圆P位于第一象限，B是x轴正半轴上一点，B的横坐标大于P的横坐标。M是圆P上的点，且在圆心P的左下方。连接MB，C是MB的中点。虚线连接MP和BP。BP的中点记作O，做辅助线OC。延长OC与x轴相交于点A。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_826.jpg" }, { "index": 827, "ocr_en": "Given circle O with point P on its circumference and point A located outside and to the left of the circle, segment AP is connected. Dashed lines AO and OP are drawn to form triangle AOP. Point Q is positioned above AP, with segments AQ and PQ connected such that AQ = PQ. Point M is located to the lower right of Q, and a dashed circle centered at M passes through Q. Dashed lines AM, MQ, and OM are connected to form triangle AMQ. Triangle AMQ is filled red, while triangle AOP is filled green.", "ocr_zh": "已知圆O，P是圆O上一点，A是圆O外一点，且在圆O的左侧，连接AP，虚线连接AO和OP，得到三角形AOP。Q是AP上方一点，连接AQ和PQ，且AQ和PQ长度相等。M是Q右下方一点，以M为圆心的虚线圆经过点Q。虚线连接AM、MQ和OM，得到三角形AMQ。三角形AMQ填充红色，三角形AOP填充绿色。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_827.jpg" }, { "index": 828, "ocr_en": "In the rectangular coordinate system xOy, circle P is situated in the first quadrant, with point B positioned on the positive x-axis such that its x-coordinate exceeds that of P. Point M lies on circle P to the right of center P. Segment MB is connected, and point C is marked as the midpoint of MB. Point A is located on the x-axis between the origin O and point B. Finally, segment AC is drawn to complete the construction.", "ocr_zh": "已知直角坐标系xOy，圆P位于第一象限，B是x轴正半轴上一点，B的横坐标大于P的横坐标。M是圆P上的点，且在圆心P的右方。连接MB，C是MB的中点。A是x轴上位于O和B之间的点。连接AC。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_828.jpg" }, { "index": 829, "ocr_en": "In the rectangular coordinate system xOy, circle P is located in the first quadrant, with point B on the positive x-axis having a greater x-coordinate than P. Point M lies on circle P to the right of its center P. Segment MB is connected, with C being the midpoint of MB. Dashed lines MP and BP are drawn, and the midpoint of BP is designated as O' (to distinguish it from the origin O). An auxiliary line O'C is constructed. Point A is positioned on the x-axis between the origin O and point B, and segment AC is connected. Finally, a dashed circle O' is drawn with center at O' and radius equal to O'C.", "ocr_zh": "已知直角坐标系xOy，圆P位于第一象限，B是x轴正半轴上一点，B的横坐标大于P的横坐标。M是圆P上的点，且在圆心P的右方。连接MB，C是MB的中点。虚线连接MP和BP。BP的中点记作O（为避免和坐标原点混淆，以下记作O’），做辅助线O'C。A是x轴上位于O和B之间的点。连接AC。以O'为圆心，O'C为半径作虚线圆O'。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_829.jpg" }, { "index": 830, "ocr_en": "In square ABCD with side CD positioned to the right of side AB, point O is the midpoint of BC. Point E lies inside the square while point F is located outside the square to the right of CD. Segments AE, OE, DE, CF, and DF are then connected to complete the geometric configuration.", "ocr_zh": "已知正方形ABCD，边CD在边AB右侧。O是BC的中点，E是正方形ABCD内的点，F是正方形外位于CD右侧的点，连接AE、OE、DE、CF和DF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_830.jpg" }, { "index": 831, "ocr_en": "In isosceles right triangle ABC with the right angle at C, a semicircle is constructed externally on diameter AB. Point P lies on this semicircle, and segment CP is connected with M as its midpoint. Point O, the midpoint of AB, is connected to C via a dashed line CO. Point D is marked as the midpoint of CO, and a dashed semicircle D is drawn with center at D and radius equal to DM. This semicircle D intersects AC at point E and BC at point F.", "ocr_zh": "已知等腰直角三角形ABC，角ACB是直角，以AB为直径向三角形外作半圆，P在该半圆上。连接CP，M是PC的中点。O是AB中点，虚线连接CO。D是CO中点，以D为圆心，DM的长度为半径作虚线半圆D，半圆D与AC交于点E，与BC交于点F。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_831.jpg" }, { "index": 832, "ocr_en": "In isosceles right triangle ABC with the right angle at C, a semicircle is constructed externally on diameter AB. Point P lies on this semicircle, and segment CP is connected. Point M is then marked as the midpoint of PC.", "ocr_zh": "已知等腰直角三角形ABC，角ACB是直角，以AB为直径向三角形外作半圆，P在该半圆上。连接CP，M是PC的中点。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_832.jpg" }, { "index": 833, "ocr_en": "Given square ABCD with side CD positioned to the right of side AB, point O is the midpoint of BC, and point E lies inside the square with segments AE, OE, DE, CF, and DF connected. Point F is obtained by rotating point E 90 degrees counterclockwise about D. A dashed line OD is drawn, and a dashed circle O is constructed with center at O and radius OE. Point M is placed to the right of DF, and a dashed circle centered at M passes through point F, with dashed lines MF and DM connected to complete the geometric configuration.", "ocr_zh": "已知正方形ABCD，边CD在边AB右侧。O是BC的中点，E是正方形ABCD内的点，连接AE、OE、DE、CF和DF。DE绕D逆时针旋转90度得到DF。虚线连接OD，以O为圆心，OE为半径作虚线圆O。M是DF右侧一点，以M为圆心的虚线圆经过点F。虚线连接MF和DM。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_833.jpg" }, { "index": 834, "ocr_en": "Given square ABCD with side CD positioned to the right of side AB, point O is the midpoint of BC, and point E lies inside the square with segments AE, OE, and DE connected. Point G is obtained by rotating point E 90 degrees counterclockwise about D, and segment CG is drawn. A dashed circle O is constructed with center at O and radius OE. Point M is placed to the right of DF, and a dashed circle centered at M passes through point G. Dashed lines MF, OD, OM, and DM are connected, with OM intersecting circle M at point F. The extension of AD is drawn as a dashed line, and a perpendicular from M to this extension meets it at point V, forming triangle DVM. A perpendicular from O to AD meets it at point W, forming triangle OWD. Triangle DVM is filled green, and triangle OWD is filled red to complete the geometric configuration.", "ocr_zh": "已知正方形ABCD，边CD在边AB右侧。O是BC的中点，E是正方形ABCD内的点，连接AE、OE、DE。DE绕点D逆时针旋转90度，设E旋转轴后的点为G，连接CG。以O为圆心，OE为半径作虚线圆O。M是DF右侧一点，以M为圆心的虚线圆经过点G。虚线连接MF、OD、OM和DM，OM与圆M的交点为F。虚线延长AD，过M作AD延长线的垂线，设垂足为V，得到三角形DVM。过O作AD的垂线，设垂足为W，得到三角形OWD。三角形DVM填充绿色，三角形OWD填充红色。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_834.jpg" }, { "index": 835, "ocr_en": "In square ABCD with side CD positioned to the right of side AB, point O is the midpoint of BC. Point E lies inside the square while point F is located outside the square to the right of CD. Segments AE, OE, DE, CF, and DF are connected. A dashed circle O is then constructed with center at O and radius equal to OE.", "ocr_zh": "已知正方形ABCD，边CD在边AB右侧。O是BC的中点，E是正方形ABCD内的点，F是正方形外位于CD右侧的点，连接AE、OE、DE、CF和DF。以O为圆心，OE为半径作虚线圆O。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_835.jpg" }, { "index": 836, "ocr_en": "In triangle ABC, a square BCDE is constructed externally on side BC, with segments BD and CE drawn to intersect at point O. A dashed circle is then drawn with center at A and radius AC. Within triangle ABC, an isosceles right triangle ABM is constructed using AB as the hypotenuse, with AM and BM shown as dashed lines. Finally, a dashed circle centered at M with radius MO is added to complete the geometric configuration.", "ocr_zh": "已知三角形ABC，以BC为边向三角形外作正方形BCDE。连接BD和CE，二者交于点O。以A为圆心，AC为半径作虚线圆。以AB为斜边向三角形ABC内构造等腰直角三角形ABM，AM和BM为虚线。以M为圆心，MO为半径作虚线圆。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_836.jpg" }, { "index": 837, "ocr_en": "In triangle ABC, a square BCDE is constructed externally on side BC, then segments BD and CE are connected and intersect at point O, followed by connecting AO to complete the geometric configuration.", "ocr_zh": "已知三角形ABC，以BC为边向三角形外作正方形BCDE。连接BD和CE，二者交于点O。连接AO。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_837.jpg" }, { "index": 838, "ocr_en": "In triangle ABC, a square BCDE is constructed externally on side BC, then segments BD and CE are drawn and intersect at point O, and finally a dashed circle centered at A with radius AC is drawn to complete the geometric construction.", "ocr_zh": "已知三角形ABC，以BC为边向三角形外作正方形BCDE。连接BD和CE，二者交于点O。以A为圆心，AC为半径作虚线圆。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_838.jpg" }, { "index": 839, "ocr_en": "In triangle ABC, a square BCDE is constructed externally on side BC. Segments BD and CE are connected, intersecting at point O, and AO is then drawn to form triangle AOB. Point A' is located outside the square to its right, with dashed lines A'O and A'C connected to create triangle COA'. Triangle ABO is filled green, while triangle COA' is filled red to complete the geometric configuration.", "ocr_zh": "已知三角形ABC，以BC为边向三角形外作正方形BCDE。连接BD和CE，二者交于点O。连接AO,得到三角形AOB。 A’是正方形外右侧一点，虚线连接A'O和A’C，得到三角形COA'。三角形ABO填充绿色，三角形COA'填充红色。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_839.jpg" }, { "index": 840, "ocr_en": "In triangle ABC, a square BCDE is constructed externally on side BC, with segments BD and CE drawn to intersect at point O. A dashed circle is then drawn centered at A with radius AC. An isosceles right triangle ABM is constructed inside triangle ABC using AB as its hypotenuse, with AM and BM represented by dashed lines. Segment MO is connected with a dashed line, and another dashed circle is drawn centered at M with radius MO to complete the geometric configuration.", "ocr_zh": "已知三角形ABC，以BC为边向三角形外作正方形BCDE。连接BD和CE，二者交于点O。以A为圆心，AC为半径作虚线圆。以AB为斜边向三角形ABC内构造等腰直角三角形ABM，AM和BM为虚线。虚线连接MO，以M为圆心，MO为半径作虚线圆。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_840.jpg" }, { "index": 841, "ocr_en": "In triangle ABC with point A positioned above side BC, two external points Q₁ (below BC) and Q₂ (above AC) are established. Segments AQ₁, BQ₁, AQ₂, and CQ₂ are connected, followed by a dashed line connecting Q₁ and Q₂ to complete the geometric configuration.", "ocr_zh": "已知三角形ABC，A在BC的上方。Q1和Q2是三角形ABC外的点，Q1在BC下方，Q2在AC上方，连接AQ1、BQ1、AQ2和CQ2。虚线连接Q1Q2。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_841.jpg" }, { "index": 842, "ocr_en": "Given segment BC with point P located closer to B than to C, and point A positioned above BC, segment AP is drawn such that angle APC is acute, with Q as the midpoint of AP. Perpendicular auxiliary lines are then constructed from points A and Q to BC, meeting it at points M and N respectively (the feet of the perpendiculars).", "ocr_zh": "已知线段BC，P是BC上一点，P与B距离更近，A是BC上方一点，连接AP。角APC为锐角。Q是AP的中点。分别过点A和Q作BC的垂线辅助线，垂足分别为M和N。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_842.jpg" }, { "index": 843, "ocr_en": "Given segment BC with point P located closer to B than to C, and point A positioned above BC, segment AP is connected such that angle APC is acute, and point Q is marked as the midpoint of AP.", "ocr_zh": "已知线段BC，P是BC上一点，P与B距离更近，A是BC上方一点，连接AP。角APC为锐角。Q是AP的中点。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_843.jpg" }, { "index": 844, "ocr_en": "Given segment BC with point P located closer to B than to C, and point A positioned above BC, segment AP is connected such that angle APC is acute, while point Q is placed below BC with segments AQ and PQ drawn to satisfy AQ = PQ in length.", "ocr_zh": "已知线段BC，P是BC上一点，P与B距离更近，A是BC上方一点，连接AP。角APC为锐角。Q是BC下方一点，连接AQ和PQ。AQ与PQ长度相等。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_844.jpg" }, { "index": 845, "ocr_en": "Given segment BC with point A on it, dashed lines AB and AC are connected to form triangle ABC. Point M lies inside the triangle while point N is located outside, with dashed lines AN and AM drawn. Segment MN is extended to intersect BC at point O, and angle MOC is labeled as α.", "ocr_zh": "已知线段BC，A是BC上方一点，虚线连接AB和AC，得到三角形ABC，M是三角形ABC内一点，N是三角形外一点，虚线连接AN和AM，连接MN并延长与BC相交，设交点为O，角MOC记作α。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_845.jpg" }, { "index": 846, "ocr_en": "Given segment BC with point P located closer to B than to C, and point A positioned above BC, segment AP is drawn such that angle APC is acute. Point Q is placed to the right of AP, and segment AQ is connected, forming an acute angle PAQ labeled as α. Points M and N are positioned to the right of AP, with segment MN connected such that Q lies on MN. Extending NM to intersect BC at point O, angle NOC is also designated as α.", "ocr_zh": "已知线段BC，P是BC上一点，P与B距离更近，A是BC上方一点，连接AP。角APC为锐角。Q是AP右侧一点，连接AQ，角PAQ是锐角，记作α。M和N是AP右侧的点，连接MN，Q在线段MN上。延长NM与BC相交，设交点为O，角NOC记为α。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_846.jpg" }, { "index": 847, "ocr_en": "Given equilateral triangle ABC with point E on AB, point P on AE, point D on BC, and point F inside the triangle, segments PF, PD, and FD are connected.", "ocr_zh": "已知等边三角形ABC，E是AB上一点，P是AE上一点，D是BC上一点，F是三角形ABC内一点，连接PF、PD和FD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_847.jpg" }, { "index": 848, "ocr_en": "In the rectangular coordinate system xOy, point A is located in the first quadrant and point C in the fourth quadrant, with segment AC perpendicular to the x-axis and intersecting it at point M. A straight line passing through the origin and traversing the second and fourth quadrants intersects AC at point N. Point P lies on segment ON, and point B is positioned in the first quadrant to the right of AC. Segments AB, AP, and BP are then connected to complete the geometric configuration.", "ocr_zh": "直角坐标系xOy中，点A在第一象限，C在第四象限，连接AC，AC垂直于x轴，垂足为M。一条经过原点和第二、第四象限的直线与AC交于点N。P是ON上的点，点B在第一象限，且在AC右侧。连接AB、AP和BP。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_848.jpg" }, { "index": 849, "ocr_en": "In the rectangular coordinate system xOy, point A lies on the negative x-axis, point B on the positive y-axis, and point P₁ in the third quadrant, with dashed lines AP₁ and OP₁ connected to form an equilateral triangle AOP₁. Point P₂ is placed on the positive x-axis, and dashed lines AB and BP₂ are drawn to create another equilateral triangle ABP₂. Finally, segment P₁P₂ is connected and extended in both directions to complete the geometric configuration.", "ocr_zh": "直角坐标系xOy中,A是x轴负半轴上一点，B是y轴正半轴上一点，P1在第三象限，虚线连接AP1和OP1得到等边三角形AOP1。P2是x轴正半轴上一点，虚线连接AB和BP2，得到等边三角形ABP2。虚线连接P1P2并向两端延长。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_849.jpg" }, { "index": 850, "ocr_en": "In the rectangular coordinate system xOy, point A is located on the negative x-axis, point B on the positive y-axis, and point P in the fourth quadrant, with segments AB, AP, and BP connected to form an equilateral triangle ABP, followed by connecting segment OP to complete the geometric configuration.", "ocr_zh": "直角坐标系xOy中,A是x轴负半轴上一点，B是y轴正半轴上一点，P在第四象限，连接AB、AP和BP得到等边三角形ABP。连接OP。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_850.jpg" }, { "index": 851, "ocr_en": "In the rectangular coordinate system xOy, point A lies on the negative x-axis, point B on the positive y-axis, and point P₁ in the third quadrant, with dashed lines AP₁ and OP₁ connected to form an equilateral triangle AOP₁. Point P₂ is placed on the positive x-axis, and dashed lines AB and BP₂ are drawn to create another equilateral triangle ABP₂. Segment P₁P₂ is connected with dashed lines and extended in both directions, while a perpendicular OP is constructed from point O to line P₁P₂, meeting it at point P.", "ocr_zh": "直角坐标系xOy中,A是x轴负半轴上一点，B是y轴正半轴上一点，P1在第三象限，虚线连接AP1和OP1得到等边三角形AOP1。P2是x轴正半轴上一点，虚线连接AB和BP2，得到等边三角形ABP2。虚线连接P1P2并向两端延长。过点O作OP垂直于P1P2，垂足为P。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_851.jpg" }, { "index": 852, "ocr_en": "Given square ABCD with point E on BC, point F on AB, and point G inside the square, segments EF, EG, and FG are connected to form an equilateral triangle FEG, followed by connecting segment CG to complete the geometric configuration.", "ocr_zh": "已知正方形ABCD，点E在BC上，点F在AB上，G是正方形ABCD内一点，连接EF、EG、FG得到等边三角形FEG。连接CG。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_852.jpg" }, { "index": 853, "ocr_en": "In the rectangular coordinate system xOy, there is a hyperbola located in the second and fourth quadrants, with point A on the hyperbola in the second quadrant. Segment AO is connected and intersects the hyperbola in the fourth quadrant at point B. Point C lies in the first quadrant, and segments AC and BC are drawn such that AC and BC are equal in length.", "ocr_zh": "直角坐标系xOy中,有一双曲线位于二、四象限。A在第二象限的双曲线上，连接AO与第四象限的双曲线交于点B。C在第一象限，连接AC和BC，AC和BC长度相等。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_853.jpg" }, { "index": 854, "ocr_en": "Given square ABCD with point E on BC, point G₁ inside the square, and point G₂ on CD, dashed lines BG₁, EG₁, AE, EG₂, and AG₂ are connected to form equilateral triangles BEG₁ and AEG₂. A dashed line G₁G₂ is drawn, and a perpendicular dashed line CH is constructed from point C to G₁G₂, meeting it at point H. Another perpendicular dashed line EF is drawn from point E to CH, intersecting it at point F.", "ocr_zh": "已知正方形ABCD，点E在BC上，G1在正方形内部，G2在CD上，虚线连接BG1、EG1、AE、EG2和AG2，得到等边三角形BEG1和AEG2。虚线连接G1G2，过点C作虚线CH垂直于G1G2，垂足为H。过点E作虚线EF垂直于CH，垂足为F。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_854.jpg" }, { "index": 855, "ocr_en": "In the rectangular coordinate system xOy, there is a hyperbola located in the second and fourth quadrants, with point A on the hyperbola in the second quadrant. Segment AO is connected and intersects the hyperbola in the fourth quadrant at point B. Point C lies in the first quadrant, and segments AC and BC are drawn such that AC and BC are equal in length. Dashed line CO is connected, and auxiliary lines perpendicular to the x-axis are constructed from points C and A, meeting the x-axis at points N and M respectively, forming triangles CON and AOM. Triangle CON is filled green, while triangle AOM is filled red.", "ocr_zh": "直角坐标系xOy中,有一双曲线位于二、四象限。A在第二象限的双曲线上，连接AO与第四象限的双曲线交于点B。C在第一象限，连接AC和BC，AC和BC长度相等。虚线连接CO，分过点C、A作垂直于x轴的辅助线，垂足分别为N和M。得到三角形CON和AOM，三角形CON填充绿色，三角形AOM填充红色。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_855.jpg" }, { "index": 856, "ocr_en": "In the rectangular coordinate system xOy, triangle ABC is located in the second quadrant with point B directly above point A and point C directly to the left of point B, forming a right angle at B. Point P lies on segment AB, and point Q is positioned in the first quadrant, with segments PQ and OQ connected to complete the geometric configuration.", "ocr_zh": "直角坐标系xOy中,三角形ABC在第二象限，B在A正上方，C在B的正左方，角B是直角。P是AB上一点，Q在第一象限，连接PQ和OQ。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_856.jpg" }, { "index": 857, "ocr_en": "\nGiven triangle ABC, with point A below BC. An isosceles triangle BCD is constructed upward with BC as the base, where the lengths of CD and BD are equal. Connect AD and extend it to point P such that the lengths of AD and PD are equal, then connect BP.", "ocr_zh": "已知三角形ABC，A在BC下方。以BC为底边向上构造等腰三角形BCD，CD与BD长度相等。连接AD并延长至点P，满足AD与PD长度相等，连接BP。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_857.jpg" }, { "index": 858, "ocr_en": "Given triangle ABC, with point A below BC. An isosceles triangle BCD is constructed upward with BC as the base, where the lengths of CD and BD are equal. Connect AD and extend it to point P such that the lengths of AD and PD are equal, then connect BP. Draw an auxiliary circle A with A as the center and AC as the radius, and this circle intersects AB at point E. M is a point inside triangle ABD; connect AM and BM with dashed lines, where the lengths of AM and BM are equal and angle AMB is a right angle. Draw an auxiliary circle M with M as the center and MD as the radius.", "ocr_zh": "已知三角形ABC，A在BC下方。以BC为底边向上构造等腰三角形BCD，CD与BD长度相等。连接AD并延长至点P，满足AD与PD长度相等，连接BP。以A为圆心AC为半径作辅助圆A，圆A与AB交于点E。M是三角形ABD内一点，虚线连接AM和BM，满足AM与BM长度相等且角AMB是直角。以M为圆心，MD为半径作辅助圆M。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_858.jpg" }, { "index": 859, "ocr_en": "Given triangle ABC, with point A below BC. An isosceles triangle BCD is constructed upward with BC as the base, where the lengths of CD and BD are equal. Connect AD and extend it to point P such that the lengths of AD and PD are equal, then connect BP. Draw an auxiliary circle A with A as the center and AC as the radius, and this circle intersects AB at point E. M is a point inside triangle ABD; connect AM and BM with dashed lines, where the lengths of AM and BM are equal and angle AMB is a right angle. Draw an auxiliary circle M with M as the center and MD as the radius. Extend AM to point N such that the lengths of AM and MN are equal, connect BN with a dashed line, and draw an auxiliary circle N with N as the center and NM as the radius.", "ocr_zh": "已知三角形ABC，A在BC下方。以BC为底边向上构造等腰三角形BCD，CD与BD长度相等。连接AD并延长至点P，满足AD与PD长度相等，连接BP。以A为圆心AC为半径作辅助圆A，圆A与AB交于点E。M是三角形ABD内一点，虚线连接AM和BM，满足AM与BM长度相等且角AMB是直角。以M为圆心，MD为半径作辅助圆M。延长AM至点N，使得AM与MN长度相等，虚线连接BN。以N为圆心NM为半径作辅助圆N。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_859.jpg" }, { "index": 860, "ocr_en": "Given angle MON with point P lying on its angle bisector, point A is placed on ray OM and point B on ray ON such that OA equals OB in length. Segment AP is connected, and dashed line BP is drawn to complete the geometric configuration.", "ocr_zh": "已知角MON，P是角MON角平分线上一点。A是射线OM上一点，B是射线ON上一点，满足OA与OB等长。连接AP，虚线连接BP。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_860.jpg" }, { "index": 861, "ocr_en": "In triangle ABC, the angle bisector of angle A is drawn to intersect side BC at point D, with point P selected on AD and segments BP and CP connected to complete the geometric configuration.", "ocr_zh": "已知三角形ABC，作角A的平分线与BC交于点D。P是AD上一点，连接BP和CP。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_861.jpg" }, { "index": 862, "ocr_en": "In triangle ABC, side BA is extended to point F on the extension line, the angle bisector of angle FAC is drawn to intersect the extension of BC at point D, with point P selected on AD and segment BP connected to complete the geometric configuration.", "ocr_zh": "已知三角形ABC，延长BA，设F是BA延长线上一点，作角FAC的平分线与BC的延长线交于点D。P是AD上一点，连接BP。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_862.jpg" }, { "index": 863, "ocr_en": "In triangle ABC, side BA is extended to point E on the extension line, the angle bisector of angle EAC is drawn to intersect the extension of BC at point D, with point P selected on AD and segments BP, CP, and EP connected to complete the geometric configuration.", "ocr_zh": "已知三角形ABC，延长BA，E是BA延长线上一点，作角EAC的平分线与BC的延长线交于点D。P是AD上一点，连接BP、CP和EP。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_863.jpg" }, { "index": 864, "ocr_en": "In triangle ABC, the angle bisector of angle A is drawn to intersect side BC at point D, with point P selected on AD and segments BP and CP connected, while point E is placed on AC such that AE equals AB in length, and segment PE is drawn to complete the geometric configuration.", "ocr_zh": "已知三角形ABC，作角A的平分线与BC交于点D。P是AD上一点，连接BP和CP。E是AC上一点，AE与AB等长，连接PE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_864.jpg" }, { "index": 865, "ocr_en": "In triangle ABC, the angle bisector of angle C is drawn to intersect side AB at point D, with point E placed on BC and a dashed line DE connected to complete the geometric configuration.", "ocr_zh": "已知三角形ABC，作角C的平分线与AB交于点D，E是BC上一点，虚线连接DE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_865.jpg" }, { "index": 866, "ocr_en": "In triangle ABC, AB equals AC. Take a point D on AC and join BD. Then BD bisects angle ABC", "ocr_zh": "在三角形ABC中，AB等于AC，在AC上取一点D，连接BD，BD平分角ABC", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_866.jpg" }, { "index": 867, "ocr_en": " In triangle ABC, angle BAC measures 108 degrees and AB equals AC. Take a point D on AC and join BD, which bisects angle ABC. Then take a point E on BC such that BE equals BA. Finally, join E to D with a dashed line", "ocr_zh": "在三角形ABC中,角BAC等于108度，AB等于AC，在AC上取一点D，连接BD，BD平分角ABC，在BC上取一点E，使BE=BA ，虚线连接DE", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_867.jpg" }, { "index": 868, "ocr_en": "In triangle ABC, take a point D on AC and join BD such that BD bisects angle ABC. Extend BD to a point E such that DE equals AD. Then join E to C.", "ocr_zh": "在三角形ABC中，在AC上取一点D，连接BD，BD平分角ABC，延长BD至E，使DE等于AD，连接EC", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_868.jpg" }, { "index": 869, "ocr_en": "In triangle ABC, take a point D on AC and join BD such that BD bisects angle ABC. Extend BD to a point E satisfying DE equals AD. Join E to C. Then take a point F on BC such that BF equals BA, and finally join D to F with a dashed line", "ocr_zh": "在三角形ABC中，在AC上取一点D，连接BD，BD平分角ABC，延长BD至E，使DE等于AD，连接EC，在BC上取一点F，使得BF等于BA，虚线连接DF", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_869.jpg" }, { "index": 870, "ocr_en": "Left Diagram:\nIn the plane rectangular coordinate system, there is a point A in the first quadrant with coordinates (2, 3). A circle is drawn with point A as the center, and the circle is tangent to the positive y-axis. There exists a point P on the circle; connect OP. Using OP as a right-angled side, construct an isosceles right triangle POQ (with a right angle at angle QOP, marked with a right angle symbol), where point Q is in the second quadrant. The red dashed line connects OA; using the red dashed line OA as a right-angled side, construct an isosceles right triangle AOA’ (with a right angle at angle AOA’, without a right angle mark), where point A’ is in the second quadrant.\nMiddle Diagram:\nIn the plane rectangular coordinate system, there is a point A in the first quadrant with coordinates (2, 3). A circle is drawn with point A as the center, and the circle is tangent to the positive y-axis. There exists a point P on the circle; connect OP. Using OP as a right-angled side, construct an isosceles right triangle POQ (with a right angle at angle QOP, marked with a right angle symbol), where point Q is in the second quadrant. Connect QA.\nRight Diagram:\nIn the plane rectangular coordinate system, there is a point A in the first quadrant with coordinates (2, 3). A circle is drawn with point A as the center, and the circle is tangent to the positive y-axis. There exists a point P on the circle; connect OP. Using OP as a right-angled side, construct an isosceles right triangle POQ (with a right angle at angle QOP, marked with a right angle symbol), where point Q is in the second quadrant. Connect QA. The red dashed line connects OA; using the red dashed line OA as a right-angled side, construct an isosceles right triangle AOA’ (with a right angle at angle AOA’, without a right angle mark), where point A’ is in the second quadrant. The triangular regions OAP and OQA’ are shaded gray.", "ocr_zh": "左图：在平面直角坐标系中，存在第一象限点A，坐标为（2，3），以点A为圆心画圆，圆与y轴正方向相切，圆上存在一点P，连接OP，以OP为直角边作等腰直角三角形POQ（角QOP为直角，标注直角符号），Q点在第二象限内，红色虚线连接OA，以OA为直角边以红色虚线作等腰直角三角形AOA’（角AOA’为直角，不标注直角符号），A’点在第二象限内\n中间图：在平面直角坐标系中，存在第一象限点A，坐标为（2，3），以点A为圆心画圆，圆与y轴正方向相切，圆上存在一点P，连接OP，以OP为直角边作等腰直角三角形POQ（角QOP为直角，标注直角符号），Q点在第二象限内，连接QA\n右图：在平面直角坐标系中，存在第一象限点A，坐标为（2，3），以点A为圆心画圆，圆与y轴正方向相切，圆上存在一点P，连接OP，以OP为直角边作等腰直角三角形POQ（角QOP为直角，标注直角符号），Q点在第二象限内，连接QA，红色虚线连接OA，以OA为直角边以红色虚线作等腰直角三角形AOA’ （角AOA’为直角，不标注直角符号），A’点在第二象限内，三角形区域OAP和三角形区域OQA‘作灰色阴影", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_870.jpg" }, { "index": 871, "ocr_en": "Left Figure\nIn the plane rectangular coordinate system, there is a point A in the first quadrant with coordinates (2, 3). A circle is drawn with point A as the center, and the circle is tangent to the positive y-axis. There is a point P on the circle. Connect OP, and use OP as a right-angled side to construct an isosceles right triangle POQ (with the right angle at OPQ, marked by a gray right-angle symbol). Point Q is in the first quadrant and inside the circle. The red dashed line connects OA, and an isosceles right triangle AOA’ is constructed with OA as a right-angled side (with the right angle at OAA’, not marked by a right-angle symbol), where point A’ is in the first quadrant.\nMiddle Figure\nIn the plane rectangular coordinate system, there is a point A in the first quadrant with coordinates (2, 3). A circle is drawn with point A as the center, and the circle is tangent to the positive y-axis. There is a point P on the circle. Connect OP, and use OP as a right-angled side to construct an isosceles right triangle POQ (with the right angle at OPQ, marked by a gray right-angle symbol). Point Q is in the fourth quadrant. Connect QA with a red solid line.\nRight Figure\nIn the plane rectangular coordinate system, there is a point A in the first quadrant with coordinates (2, 3). A circle is drawn with point A as the center, and the circle is tangent to the positive y-axis. There is a point P on the circle. Connect OP, and use OP as a right-angled side to construct an isosceles right triangle POQ (with the right angle at OPQ, marked by a gray right-angle symbol). Point Q is in the first quadrant and inside the circle. The red dashed line connects OA, and an isosceles right triangle AOA’ is constructed with OA as a right-angled side (with the right angle at OAA’, not marked by a right-angle symbol), where point A’ is in the first quadrant. The triangular regions OAP and OQA’ are shaded in yellow.", "ocr_zh": "左图：在平面直角坐标系中，存在第一象限点A，坐标为（2，3），以点A为圆心画圆，圆与y轴正方向相切，圆上存在一点P，连接OP，以OP为直角边作等腰直角三角形POQ（角OPQ为直角，标注灰色直角符号），Q点在第一象限且位于圆内，红色虚线连接OA，以OA为直角边用红色虚线作等腰直角三角形AOA’（角OAA’为直角，不标注直角符号），A’点在第一象限内\n中间图：在平面直角坐标系中，存在第一象限点A，坐标为（2，3），以点A为圆心画圆，圆与y轴正方向相切，圆上存在一点P，连接OP，以OP为直角边作等腰直角三角形POQ（角OPQ为直角，标注灰色直角符号），Q点在第四象限内，用红色实线连接QA\n右图：在平面直角坐标系中，存在第一象限点A，坐标为（2，3），以点A为圆心画圆，圆与y轴正方向相切，圆上存在一点P，连接OP，以OP为直角边作等腰直角三角形POQ（角OPQ为直角，标注灰色直角符号），Q点在第一象限且位于圆内，红色虚线连接OA，以OA为直角边用红色虚线作等腰直角三角形AOA’ （角OAA’为直角，不标注直角符号），A’点在第一象限内，三角形区域OAP和三角形区域OQA‘作黄色阴影", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_871.jpg" }, { "index": 872, "ocr_en": "In the plane rectangular coordinate system, there is a point A in the first quadrant with coordinates (2, 3). A circle is drawn with point A as the center, and the circle is tangent to the positive y-axis. There is a point P on the circle. Connect OP, and use OP as a right-angled side to construct an isosceles right triangle POQ (with the right angle at OPQ, marked by a gray right-angle symbol). Point Q is in the first quadrant and inside the circle. The red dashed line connects OA, and an isosceles right triangle AOA’ is constructed with OA as a right-angled side (with the right angle at OAA’, not marked by a right-angle symbol), where point A’ is in the first quadrant. The triangular regions OAP and OQA’ are shaded in yellow.", "ocr_zh": "在平面直角坐标系中，存在第一象限点A，坐标为（2，3），以点A为圆心画圆，圆与y轴正方向相切，圆上存在一点P，连接OP，以OP为直角边作等腰直角三角形POQ（角OPQ为直角，标注灰色直角符号），Q点在第一象限且位于圆内，红色虚线连接OA，以OA为直角边用红色虚线作等腰直角三角形AOA’ （角OAA’为直角，不标注直角符号），A’点在第一象限内，三角形区域OAP和三角形区域OQA‘作黄色阴影", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_872.jpg" }, { "index": 873, "ocr_en": "In the plane rectangular coordinate system, there is a point A in the first quadrant with coordinates (2, 3). A circle is drawn with point A as the center, and the circle is tangent to the positive y-axis. There is a point P on the circle. Connect OP, and use OP as a right-angled side to construct an isosceles right triangle POQ (with the right angle at OPQ, marked by a gray right-angle symbol). Point Q is in the first quadrant and inside the circle. The red dashed line connects OA, and an isosceles right triangle AOA’ is constructed with OA as a right-angled side (with the right angle at OAA’, not marked by a right-angle symbol), where point A’ is in the first quadrant.", "ocr_zh": "在平面直角坐标系中，存在第一象限点A，坐标为（2，3），以点A为圆心画圆，圆与y轴正方向相切，圆上存在一点P，连接OP，以OP为直角边作等腰直角三角形POQ（角OPQ为直角，标注灰色直角符号），Q点在第一象限且位于圆内，红色虚线连接OA，以OA为直角边用红色虚线作等腰直角三角形AOA’（角OAA’为直角，不标注直角符号），A’点在第一象限内", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_873.jpg" }, { "index": 874, "ocr_en": "In the plane rectangular coordinate system, there is a point A in the first quadrant with coordinates (2, 3). A circle is drawn with point A as the center, and the circle is tangent to the positive y-axis. There exists a point P on the circle; connect OP. Using OP as a right-angled side, construct an isosceles right triangle POQ (with a right angle at angle QOP, marked with a right angle symbol), where point Q is in the second quadrant. The red dashed line connects OA; using the red dashed line OA as a right-angled side, construct an isosceles right triangle AOA’ (with a right angle at angle AOA’, without a right angle mark), where point A’ is in the second quadrant.", "ocr_zh": "在平面直角坐标系中，存在第一象限点A，坐标为（2，3），以点A为圆心画圆，圆与y轴正方向相切，圆上存在一点P，连接OP，以OP为直角边作等腰直角三角形POQ（角QOP为直角，标注直角符号），Q点在第二象限内，红色虚线连接OA，以OA为直角边以红色虚线作等腰直角三角形AOA’（角AOA’为直角，不标注直角符号），A’点在第二象限内", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_874.jpg" }, { "index": 875, "ocr_en": "In the plane rectangular coordinate system, there is a point A in the first quadrant with coordinates (2, 3). A circle is drawn with point A as the center, and the circle is tangent to the positive y-axis. There is a point P on the circle. Connect OP, and use OP as a right-angled side to construct an isosceles right triangle POQ (with the right angle at OPQ, marked by a gray right-angle symbol). Point Q is in the fourth quadrant. Connect QA with a red solid line.", "ocr_zh": "在平面直角坐标系中，存在第一象限点A，坐标为（2，3），以点A为圆心画圆，圆与y轴正方向相切，圆上存在一点P，连接OP，以OP为直角边作等腰直角三角形POQ（角OPQ为直角，标注灰色直角符号），Q点在第四象限内，用红色实线连接QA", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_875.jpg" }, { "index": 876, "ocr_en": "In the plane rectangular coordinate system, draw the hyperbola y equals 4/x. Take a point A on the first-quadrant branch of the hyperbola, connect AO, and extend AO to intersect the third-quadrant branch of the hyperbola at point B. Construct an isosceles triangle ABC with AB as the base, where angle C is an obtuse angle and the vertex C is located in the second quadrant.", "ocr_zh": "在平面直角坐标系中，作双曲线y等于4/x，在双曲线第一象限的分支上取一点A，连接AO，并延长AO与双曲线第三象限的分支交于点B，以AB为底边作等腰三角形ABC，角C为钝角且顶点C位于第二象限", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_876.jpg" }, { "index": 877, "ocr_en": "In the plane rectangular coordinate system, draw the hyperbola y equals 4/x. Take a point A on the branch of the hyperbola in the first quadrant, connect AO, and extend AO to intersect the branch of the hyperbola in the third quadrant at point B. Construct an isosceles right triangle ABC with AB as the hypotenuse, where the right angle is at angle C (marked with a gray right-angle symbol) and the vertex C is located in the second quadrant.", "ocr_zh": "在平面直角坐标系中，作双曲线y等于4/x，在双曲线第一象限的分支上取一点A，连接AO，并延长AO与双曲线第三象限的分支交于点B，以AB为斜边作等腰直角三角形ABC，角C为直角（标注灰色直角符号）且顶点C位于第二象限", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_877.jpg" }, { "index": 878, "ocr_en": "In the plane rectangular coordinate system, draw the hyperbola y equals 4/x. Take a point A on the branch of the hyperbola in the first quadrant, connect AO , and extend AO to intersect the branch of the hyperbola in the third quadrant at point B. Construct an equilateral triangle ABC with AB as a side, where the vertex C is located in the fourth quadrant (the angle C is marked with 60 degrees in gray).", "ocr_zh": "在平面直角坐标系中，作双曲线y等于4/x，在双曲线第一象限的分支上取一点A，连接AO，并延长AO与双曲线第三象限的分支交于点B，以AB为边作等边三角形ABC，顶点C位于第四象限（角C灰色标注60度）", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_878.jpg" }, { "index": 879, "ocr_en": "Construct a horizontal line segment AB. Take the midpoint O of line segment AB as the center, and use AB as the diameter to construct an upper semicircle.. On the semicircular arc AB, take a trisection point C such that the arc AC is one - half of the arc BC (mark the arc AC in red). Connect AC. Take a point P on the arc AC. Connect PC and PB. From point B, draw a perpendicular BQ to PC, intersecting the extension of PC at point Q, with the foot of the perpendicular being Q (mark the right angle symbol at angle BQC).", "ocr_zh": "作水平线段AB，以线段AB的中点O为圆心，AB为直径，作上半圆，在半圆弧AB上取三等分点C，有圆弧AC等于圆弧BC的二分之一（标红圆弧AC），连接AC，在圆弧AC上取一点P，连接PC和PB,从B点向PC作垂线BQ交PC的延长线于Q点，垂足为Q（标注直角符号）", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_879.jpg" }, { "index": 880, "ocr_en": "In the plane rectangular coordinate system, draw a circle with the coordinate origin O as the center. Take a point A on the positive x-axis such that OA is twice the radius of the circle. Take a point B on the circle in the second quadrant and connect AB. Construct an equilateral triangle ABC with AB as the side, where the vertex C is in the first quadrant.", "ocr_zh": "在平面直角坐标系中，以坐标原点O为圆心作圆，在x轴正半轴取一点A，有OA等于两倍的圆半径，在第二象限的圆上取一点B，连接AB，以AB为边作等边三角形ABC，顶点C在第一象限", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_880.jpg" }, { "index": 881, "ocr_en": "In the plane rectangular coordinate system, draw a circle with the coordinate origin O as the center. Take a point A on the positive x-axis such that OA is twice the radius of the circle. Take a point B on the circle in the second quadrant and connect AB. Construct an equilateral triangle ABC with AB as the side, where the vertex C is in the first quadrant. Connect OB and OC with dashed lines. Take a point O' on CO such that CO' equals OB (mark \"CO' equals BO\" on the right side of the diagram). Connect O'A with a dashed line. Shade the triangular regions BOA and CO'A in yellow.", "ocr_zh": "在平面直角坐标系中，以坐标原点O为圆心作圆，在x轴正半轴取一点A，有OA等于两倍的圆半径，在第二象限的圆上取一点B，连接AB，以AB为边作等边三角形ABC，顶点C在第一象限，虚线连接OB，OC，在CO上取点O’有CO’等于OB（在图片右侧标注CO’等于BO），虚线连接O’A，将三角形区域BOA和三角形区域CO’A标黄", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_881.jpg" }, { "index": 882, "ocr_en": "Left Diagram:\nConstruct a horizontal line segment AB. Take the midpoint O of line segment AB as the center, and use AB as the diameter to construct an upper semicircle.. On the semicircular arc AB, take a trisection point C such that arc AC is one-half of arc BC (mark arc AC in red). Connect AC, and connect BC with a dashed line (mark the right angle symbol at angle ACB). Take a point P on arc AC, connect PC and PB. From point B, draw a perpendicular BQ to PC, intersecting the extension of PC at point Q, with the foot of the perpendicular being Q (mark the right angle symbol at angle BQC).\n\nRight Diagram:\nConstruct a horizontal line segment AB. Take the midpoint O of line segment AB as the center, and use AB as the diameter to construct an upper semicircle.. On the semicircular arc AB, take a trisection point C such that arc AC is one-half of arc BC (mark arc AC in red). Connect AC, and connect BC with a dashed line (mark the right angle symbol at angle ACB). Take a point P on arc AC, connect PC and PB. From point B, draw a perpendicular BQ to PC, intersecting the extension of PC at point Q, with the foot of the perpendicular being Q (mark the right angle symbol at angle BQC). Take the midpoint O’ of BC, connect O’Q with a dashed line. Shade region BQC in yellow and region APB in purple.", "ocr_zh": "左图：\n作水平线段AB，以线段AB的中点O为圆心，AB为直径，作上半圆，在半圆弧AB上取三等分点C，有圆弧AC等于圆弧BC的二分之一（标红圆弧AC），连接AC，虚线连接BC（角ACB标注直角符号），在圆弧AC上取一点P，连接PC和PB,从B点向PC作垂线BQ交PC的延长线于Q点，垂足为Q（角BQC标注直角符号）\n右图：\n作水平线段AB，以线段AB的中点O为圆心，AB为直径，作上半圆，在半圆弧AB上取三等分点C，有圆弧AC等于圆弧BC的二分之一（标红圆弧AC），连接AC，虚线连接BC（角ACB标注直角符号），在圆弧AC上取一点P，连接PC和PB,从B点向PC作垂线BQ交PC的延长线于Q点，垂足为Q（角BQC标注直角符号），取BC的中点O‘，虚线连接O’Q，将区域BQC标黄，区域APB标紫", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_882.jpg" }, { "index": 883, "ocr_en": "Construct a horizontal line segment AB. Take the midpoint O of line segment AB as the center, and use AB as the diameter to construct an upper semicircle.. On the semicircular arc AB, take a trisection point C such that arc AC is one-half of arc BC (mark arc AC in red). Connect AC, and connect BC with a dashed line (mark the right angle symbol at angle ACB). Take a point P on arc AC, connect PC and PB. From point B, draw a perpendicular BQ to PC, intersecting the extension of PC at point Q, with the foot of the perpendicular being Q (mark the right angle symbol at angle BQC).", "ocr_zh": "作水平线段AB，以线段AB的中点O为圆心，AB为直径，作上半圆，在半圆弧AB上取三等分点C，有圆弧AC等于圆弧BC的二分之一（标红圆弧AC），连接AC，虚线连接BC（角ACB标注直角符号），在圆弧AC上取一点P，连接PC和PB,从B点向PC作垂线BQ交PC的延长线于Q点，垂足为Q（角BQC标注直角符号）", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_883.jpg" }, { "index": 884, "ocr_en": "Construct a horizontal line segment AB. Take the midpoint O of line segment AB as the center, and use AB as the diameter to construct a lower semicircle.. Take a point C on the lower semicircular arc, connect AC. Construct a square ACDE downward with AC as the side, and connect OE.", "ocr_zh": "作水平线段AB，以线段AB的中点O为圆心，AB为直径，作下半圆，在下半圆弧上取一点C，连接AC，以AC为边向下作正方形ACDE，连接OE", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_884.jpg" }, { "index": 885, "ocr_en": "Construct a horizontal line segment AB. Take the midpoint O of line segment AB as the center, and use AB as the diameter to construct a lower semicircle. Take a point C on the lower semicircular arc, connect OC with a blue dashed line (mark \"2\" on the line). Connect AC, construct a square ACDE downward with AC as the side, and connect OE. Rotate triangle AOC counterclockwise about point A until side AC coincides with side AE., resulting in triangle AO’E (both AO’ and O’E are connected with blue dashed lines, mark \"2\" on O’E). Draw a circle with center O’ and radius O’A using a black dashed line. Shade the triangular regions AOC and AO’E in yellow.", "ocr_zh": "作水平线段AB，以线段AB的中点O为圆心，AB为直径，作下半圆，在下半圆弧上取一点C，蓝色虚线连接OC（线上标注2），连接AC，以AC为边向下作正方形ACDE，连接OE，以A点为轴逆时针旋转三角形AOC至边AC和边AE重合，得到三角形AO’E（其中AO’和O’E均以蓝色虚线连接，O‘E上标注2），以O’为圆心O’A为半径以黑色虚线作圆，将三角形区域AOC和三角形区域AO’E标黄", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_885.jpg" }, { "index": 886, "ocr_en": "Construct triangle ABC with AB side horizontal. Take the midpoint O of side AB, and draw a circle with center O and radius one-half of OB. Take a point P on the circle (point P is located inside triangle ABC). Connect PB and PC, where PB is perpendicular to PC (mark the right angle symbol at angle CPB), and PB equals PC.", "ocr_zh": "作三角形ABC（AB边水平），取边AB的中点O，以O点为圆心，OB的二分之一为半径作圆，在圆上取一点P（P点位于三角形ABC内），连接PB、PC，PB垂直于PC（角CPB标注直角符号），且PB等于PC", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_886.jpg" }, { "index": 887, "ocr_en": "Construct triangle ABC with AB side horizontal. Take the midpoint O of side AB, and draw a circle with center O and radius one-half of OB. Take a point P on the circle (point P is located inside triangle ABC). Connect PB and PC, where PB is perpendicular to PC (mark the right angle symbol at angle CPB), and PB equals PC. Rotate triangle BPC counterclockwise about point B until side BP coincides with side BO, resulting in triangle BOO’ (both OO’ and O’B are connected with blue dashed lines, mark the right angle symbol at angle BOO’). Connect O’C with a blue dashed line. Shade the triangular region BOP in yellow and the triangular region BO’C in gray.", "ocr_zh": "作三角形ABC（AB边水平），取边AB的中点O，以O点为圆心，OB的二分之一为半径作圆，在圆上取一点P（P点位于三角形ABC内），连接PB、PC，PB垂直于PC（角CPB标注直角符号），且PB等于PC, 以B点为轴逆时针旋转三角形BPC至边BP和边BO重合，得到三角形BOO’（其中OO’和O’B均以蓝色虚线连接，角BOO’标注直角符号），蓝色虚线连接O’C，将三角形区域BOP标黄，将三角形区域BO’C标灰", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_887.jpg" }, { "index": 888, "ocr_en": "Construct a horizontal line segment AB. Take the midpoint O of line segment AB as the center, and use AB as the diameter to draw a circle. Extend AB to point C such that BC equals OB. Take a point P on the circle (P is located on the upper semicircle), connect PC. Construct an isosceles triangle PCD with PC as the base above PC (angle PDC is an obtuse angle, and the vertex D is outside the circle; PD is equal to DC, mark angle DPC as 30 degrees in gray). Connect OD.", "ocr_zh": "作水平线段AB，以线段AB的中点O为圆心，AB为直径，作圆，延长AB至C使得BC等于OB，在圆上取一点P（P位于上半圆），连接PC，以PC为底边在PC的上方作等腰三角形PCD（角PDC为钝角，且顶点D位于圆外，PD等于DC，灰色标注角DPC为30度），连接OD", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_888.jpg" }, { "index": 889, "ocr_en": "Construct a horizontal line segment AB. Take the midpoint O of line segment AB as the center, and use AB as the diameter to draw a circle. Extend AB to point C such that BC equals OB. Take a point P on the circle (P is located on the upper semicircle), connect PC. Construct an isosceles triangle PCD with PC as the base above PC (angle PDC is an obtuse angle, the vertex D is outside the circle, PD is equal to DC, and angle DPC is marked as 30 degrees in gray). Connect OD. Construct an isosceles triangle OCO’ with OC as the base above OC (angle OO’C is an obtuse angle, the vertex O’ is outside the circle, O’C is equal to O’O, angle O’OC is marked as 30 degrees in gray, and sides OO’ and O’C are connected with dashed lines). Shade the triangular regions DO’C and POC in yellow.", "ocr_zh": "作水平线段AB，以线段AB的中点O为圆心，AB为直径，作圆，延长AB至C使得BC等于OB，在圆上取一点P（P位于上半圆），连接PC，以PC为底边在PC的上方作等腰三角形PCD（角PDC为钝角，且顶点D位于圆外，PD等于DC，灰色标注角DPC为30度），连接OD,以OC为底边在OC的上方作等腰三角形OCO’ （角OO’C为钝角，且顶点O’位于圆外，O’C等于O’O，灰色标注角O’OC为30度,边OO’和边O’C以虚线连接），将三角形区域DO’C和三角形区域POC标黄", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_889.jpg" }, { "index": 890, "ocr_en": "Construct a horizontal line segment AB. Take the midpoint O of line segment AB as the center, and use AB as the diameter to draw a circle. Extend AB to point C such that BC equals AB. Take a point P on the circle (P is located on the upper semicircle), connect PC. Construct a right triangle PCD with PC as the hypotenuse above PC (angle PDC is a right angle, mark the right angle symbol). Connect OD", "ocr_zh": "作水平线段AB，以线段AB的中点O为圆心，AB为直径，作圆，延长AB至C使得BC等于AB，在圆上取一点P（P位于上半圆），连接PC，以PC为斜边在PC的上方作直角三角形PCD（角PDC为直角，标注直角符号），连接OD", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_890.jpg" }, { "index": 891, "ocr_en": "Construct a horizontal line segment AB. Take the midpoint O of line segment AB as the center, and use AB as the diameter to draw a circle. Extend AB to point C such that BC equals AB. Take a point P on the circle (P is located on the upper semicircle), connect PC. Construct a right triangle PCD with PC as the hypotenuse above PC (angle PDC is a right angle, mark the right angle symbol). Connect OD (mark \"12\" above line OD). Construct a right triangle OCO’ with OC as the hypotenuse above OC, similar to right triangle PCD (angle OO’C is a right angle, do not mark the right angle symbol; O’ is an auxiliary point located inside right triangle PDC, and sides OO’ and O’C are connected with blue solid lines). Draw a circle with auxiliary point O’ as the center and radius 3 using a dashed line (mark \"r equals 3\" above the circle). Shade the triangular regions O’DC and OPC in yellow.", "ocr_zh": "作水平线段AB，以线段AB的中点O为圆心，AB为直径，作圆，延长AB至C使得BC等于AB，在圆上取一点P（P位于上半圆），连接PC，以PC为斜边在PC的上方作直角三角形PCD（角PDC为直角，标注直角符号），连接OD（在线OD上方标注12）, 以OC为斜边在OC的上方作与直角三角形PCD相同的直角三角形OCO’（角OO’C为直角，不标注直角符号，且其中O’为辅助点位于直角三角形PDC内，边OO’和边O’C以蓝色实线连接），以辅助点O’为圆心，半径为3以虚线作圆（圆上方标注r等于3），将三角形区域O’DC和三角形区域OPC标黄", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_891.jpg" }, { "index": 892, "ocr_en": "Construct a horizontal line segment AB. Take the trisection point O of line segment AB such that OB is twice OA. Draw a circle centered at O with radius OA. Select a point P on the upper semicircle of this circle. Connect BP. Construct a right triangle PBC above BP with BP as a leg (angle PBC is a right angle, marked with a right angle symbol, and BC is twice the length of BP). Connect AC.", "ocr_zh": "作水平线段AB，取线段AB的三等分点O，有OB等于两倍的OA，以O点为圆心，OA为半径作圆，在圆上取一点P（P位于上半圆），连接BP，以BP为直角边在BP的上方作直角三角形PBC（角PBC为直角，标注直角符号，且BC等于两倍的BP），连接AC", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_892.jpg" }, { "index": 893, "ocr_en": "Construct a right triangle ACB with angle ACB being a right angle (do not mark the right angle symbol). Draw a circle with center A and radius one-half of AC. Take a point D on the circle (D is located on the right semicircle), connect BD. Take the midpoint M of BD and the midpoint O’ of AB. Draw a circle with center O’ and radius O’M.", "ocr_zh": "作直角三角形ACB，角ACB为直角（不标注直角符号），以A为圆心，AC的二分之一为半径作圆，在圆上取一点D（D位于右半圆），连接BD，取BD的中点M，取AB的中点O’，以O’为圆心,O’M为半径作圆", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_893.jpg" }, { "index": 894, "ocr_en": "Construct a right triangle ACB with angle ACB being a right angle (do not mark the right angle symbol). Draw a circle with center A and radius one-half of AC. Take a point D on the circle (D is located on the right semicircle), connect BD. Take the midpoint M of BD", "ocr_zh": "作直角三角形ACB，角ACB为直角（不标注直角符号），以A为圆心，AC的二分之一为半径作圆，在圆上取一点D（D位于右半圆），连接BD，取BD的中点M", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_894.jpg" }, { "index": 895, "ocr_en": "Construct a horizontal line segment AB. Take the trisection point O of line segment AB such that OB equals twice OA. Draw a circle with center O and radius OA. Take a point P on the circle (P is located on the upper semicircle). Connect OP with a blue dashed line (mark \"1\" on line OP). Connect BP. Construct a right triangle PBC above BP with BP as a leg (angle PBC is a right angle, mark the right angle symbol, and BC equals twice BP). Connect AC. Draw O’B perpendicular to OB with B as the foot (O’B equals twice OB, angle O’BO is a right angle but do not mark the symbol; point O’ is located above point B, and the perpendicular O’B is drawn as a blue dashed line). Connect O’O with a blue dashed line (mark \"2√5\" on line O’O) and connect O’C with a blue dashed line (mark \"2\" on line O’C).", "ocr_zh": "作水平线段AB，取线段AB的三等分点O，有OB等于两倍的OA，以O点为圆心，OA为半径作圆，在圆上取一点P（P位于上半圆），蓝色虚线连接OP（在OP线上标注1）连接BP，以BP为直角边在BP的上方作直角三角形PBC（角PBC为直角，标注直角符号，且BC等于两倍的BP），连接AC，以B作为垂足作O’B垂直于OB,且有O’B等于两倍的OB（角O’BO为直角不标注直角符号，O’点位于B点的上方，且垂线O’B采用蓝色虚线），用蓝色虚线连接O’O线上标注2根号五，用蓝色虚线连接O’C线上标注2", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_895.jpg" }, { "index": 896, "ocr_en": "Construct a sector OAB with center O and radius OB, where angle AOB is a right angle. Extend OB to point C such that OB equals twice BC. Take a point D on arc AB, connect CD. Construct a right triangle DCP upward with CD as the hypotenuse, where angle DPC is a right angle (mark the gray right angle symbol), and DC equals three times CP. Connect OP.", "ocr_zh": "以O为圆心，OB长为半径作扇形OAB，角AOB为直角。延长OB至C,使得OB等于两倍的BC，取弧AB上一点D，连接CD，以CD为斜边向上作直角三角形DCP，角DPC为直角，标注灰色直角符号，且DC等于三倍的CP,连接OP", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_896.jpg" }, { "index": 897, "ocr_en": "Construct a sector OAB with center O and radius OB, where angle AOB is a right angle. Extend OB to point C such that OB equals twice BC (mark \"9\" on line OC). Take a point D on arc AB, connect OD with a dashed line (mark \"6\" on line OD). Connect CD and mark the right angle symbol at angle ODC. Construct a right triangle DCP upward with CD as the hypotenuse, where angle DPC is a right angle (mark the gray right angle symbol), and DC equals three times CP. Connect OP. Construct a right triangle OO’C upward with OC as the hypotenuse, where angle OO’C is a right angle (mark the gray right angle symbol; point O’ falls on line segment DP, sides OO’ and O’C are both drawn as dashed lines, and mark \"6√2\" on line OO’). Mark \"2\" on line O’P. Shade the triangular regions ODC and O’PC in yellow.", "ocr_zh": "以O为圆心，OB长为半径作扇形OAB，角AOB为直角。延长OB至C,使得OB等于两倍的BC（OC线下标注9），取弧AB上一点D，虚线连接OD（线上标注6），连接CD，角ODC标注直角符号，以CD为斜边向上作直角三角形DCP，角DPC为直角，标注灰色直角符号，且DC等于三倍的CP,连接OP，以OC为斜边向上作直角三角形OO’C，角OO’C为直角，标注灰色直角符号（O‘点落在线段DP上,边OO’和边O’C均采用虚线，且OO’线上标注6根号2），O’P线上标注2，将三角形区域ODC和三角形区域O’PC标黄", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_897.jpg" }, { "index": 898, "ocr_en": "Construct a right triangle OAB with angle OAB being a right angle (mark the gray right angle symbol), and mark angle BOA as 30 degrees in gray. Draw a circle with center O and radius one-half of OA. Take a point C on the circle (point C is located on the right semicircle), connect AC. Take the midpoint M of line segment AC and connect BM. Take a point O’ on line segment OA, and draw a circle with the length of O’M as the radius using a blue solid line.", "ocr_zh": "作直角三角形OAB，角OAB为直角，标注灰色直角符号，角BOA灰色标注30度，以O点为圆心，OA长的二分之一为半径作圆，在圆上取一点C（C点位于右半圆），连接AC，取线段AC的中点M，连接BM。在线段OA上取一点O’，以O’M的长为半径用蓝色实线作圆", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_898.jpg" }, { "index": 899, "ocr_en": "Construct a right triangle OAB with angle OAB being a right angle (mark the gray right angle symbol), and mark angle BOA as 30 degrees in gray. Draw a circle with center O and radius one-half of OA. Take a point C on the circle (point C is located on the right semicircle), connect AC. Take the midpoint M of line segment AC and connect BM.", "ocr_zh": "作直角三角形OAB，角OAB为直角，标注灰色直角符号，角BOA灰色标注30度，以O点为圆心，OA长的二分之一为半径作圆，在圆上取一点C（C点位于右半圆），连接AC，取线段AC的中点M，连接BM。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_899.jpg" }, { "index": 900, "ocr_en": "Construct horizontal line segment AB, take the midpoint M of line segment AB, use M as the center to draw a circle with a blue dashed line (the radius of the circle is less than the length of MB), take a point P on the circle (point P is located on the upper semicircle), connect MP (mark 1 on the line), connect BP, rotate line segment PB 90 degrees counterclockwise around point P to obtain line segment PC, connect AC and BC, mark angle CBP in gray as 45 degrees, rotate line segment MB 90 degrees counterclockwise around point M to obtain the dashed line segment MM’, and connect M’B with a dashed line.", "ocr_zh": "作水平线段AB，取线段AB的中点M，以M为圆心，以蓝色虚线作圆（圆的半径小于MB的长），在圆上取一点P（P点位于上半圆），连接MP（线上标注1），连接BP，将线段PB绕点P逆时针旋转九十度得到线段PC，连接AC、BC，角CBP灰色标注45度，将线段MB绕点M逆时针旋转九十度得到虚线线段MM’，虚线连接M’B", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_900.jpg" }, { "index": 901, "ocr_en": "Construct a horizontal line segment AB. Take the midpoint M of line segment AB (both AM and BM are marked with \"2\" below the segments). Take a point P above line segment AB, connect MP (mark \"1\" on segment MP), and connect BP. Rotate segment PB 90 degrees counterclockwise around point P to obtain segment PC (mark the right angle symbol at angle BPC), then connect AC.", "ocr_zh": "作水平线段AB，取线段AB的中点M（AM、BM线段下均标注2），在线段AB上方取一点P，连接MP（MP线段上标注1），连接BP，将线段PB绕点P逆时针旋转九十度得到线段PC（角BPC标注直角符号），连接AC", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_901.jpg" }, { "index": 902, "ocr_en": "Construct a square ABCD. Take a point P in the plane (point P is located inside the square) such that BP is perpendicular to PC, with the foot of the perpendicular at P, and angle BPC is a right angle (mark the right angle symbol). Connect PD, and rotate line segment DP 90 degrees clockwise about point D to obtain line segment DQ.", "ocr_zh": "作正方形ABCD，取平面内一点P(P点位于正方形内)，有BP垂直于PC，垂足为P，角BPC为直角（标注直角符号），连接PD，将线段DP绕点D顺时针旋转九十度得到线段DQ", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_902.jpg" }, { "index": 903, "ocr_en": "Construct a square ABCD. Take a point P in the plane (point P is located inside the square) such that BP is perpendicular to PC with the foot at P, and mark the right angle symbol at angle BPC. Connect PD, then rotate segment DP 90 degrees clockwise about point D to obtain segment DQ. Take the midpoint O of BC, draw a circle with center O and diameter BC using blue dashed lines. Connect OD, rotate segment OD 90 degrees clockwise about point D to obtain segment DO’. Draw a circle with center O’ using blue dashed lines, which has the same size as circle O.", "ocr_zh": "作正方形ABCD，取平面内一点P(P点位于正方形内)，有BP垂直于PC，垂足为P，角BPC为直角（标注直角符号），连接PD，将线段DP绕点D顺时针旋转九十度得到线段DQ。取BC的中点O，以O点为圆心BC为直径以蓝色虚线作圆，连接OD，将线段OD绕点D顺时针旋转九十度得到线段DO’，以O’点为圆心以蓝色虚线作与圆O相同大小的圆。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_903.jpg" }, { "index": 904, "ocr_en": "Construct a horizontal line segment AB. Take the midpoint M of line segment AB, draw a circle with center M and radius one-half of MB using a blue dashed line. Take a point P on the circle (P is located on the upper semicircle), connect MP (mark \"1\" on the line) and connect BP. Rotate segment PB 90 degrees counterclockwise about point P to obtain segment PC, then connect AC and BC. Mark angle CBP as 45 degrees in gray. Rotate segment MB 90 degrees counterclockwise about point M to obtain the dashed segment MM’, and connect M’B with a dashed line. Draw a circle with center M’ and radius M’C using a blue dashed line. Shade the triangular regions MPB and M’BC in yellow, and mark \"√2\" above the triangular region M’CB.", "ocr_zh": "作水平线段AB，取线段AB的中点M，以M为圆心,MB的二分之一长为半径以蓝色虚线作圆，在圆上取一点P（P点位于上半圆），连接MP（线上标注1），连接BP，将线段PB绕点P逆时针旋转九十度得到线段PC，连接AC、BC，角CBP灰色标注45度，将线段MB绕点M逆时针旋转九十度得到虚线线段MM’，虚线连接M’B，以M’点为圆心，M’C的长为半径以蓝色虚线作圆，将三角形区域MPB和三角形区域M’BC标黄，在三角形区域M’CB上方标注根号2", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_904.jpg" }, { "index": 905, "ocr_en": "Construct triangle ABC. Construct square BCFE with BC as a side, connect diagonals BF and EC of the square, which intersect at point O (mark \"r = √2\" in red below segment BO), then connect AO. Draw a circle with center A and radius AC using blue dashed lines, and mark \"R = 2\" on segment AC. Construct an isosceles right triangle AHB with AB as the base, where angle AHB is a right angle (mark the red right angle symbol), and mark angle HAB as 45 degrees in gray. Shade the triangular regions ABC and BHO in gray.", "ocr_zh": "作三角形ABC，以BC为边作正方形BCFE，连接正方形对角线BF、EC相交于点O(线段BO下方红色标注r等于根号2)，连接AO。以A点为圆心，AC长为半径以蓝色虚线作圆，在线段AC上标注R等于2，以AB为底边作等腰直角三角形AHB，角AHB为直角，标注红色直角符号，角HAB标注灰色45度，将三角形区域ABC和三角形区域BHO标灰", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_905.jpg" }, { "index": 906, "ocr_en": "Construct triangle ABC. Construct a square BCFE with BC as a side (where F and E are auxiliary points). Connect the diagonals BF and EC of the square, which intersect at point O, then connect AO.", "ocr_zh": "作三角形ABC，以BC为边作正方形BCFE（F、E为辅助点），连接正方形对角线BF、EC相交于点O，连接AO", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_906.jpg" }, { "index": 907, "ocr_en": "Construct triangle ABC. Construct a square BCFE with BC as a side, connect the diagonals BF and EC of the square intersecting at point O, then connect AO. Draw a circle with center A and radius AC using blue dashed lines.", "ocr_zh": "作三角形ABC，以BC为边作正方形BCFE，连接正方形对角线BF、EC相交于点O，连接AO。以A点为圆心，AC长为半径以蓝色虚线作圆", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_907.jpg" }, { "index": 908, "ocr_en": "Construct a square ABCD. Take the midpoint O of BC, and take a point E inside the square. Connect OE, DE, and AE. Rotate segment DE 90 degrees counterclockwise about point D to obtain segment DF, then connect DF, OF, and CF.", "ocr_zh": "作正方形ABCD，取BC的中点O，在正方形内取一点E，连接OE、DE、AE，将线段DE绕点D逆时针旋转九十度得到线段DF，连接OF、CF", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_908.jpg" }, { "index": 909, "ocr_en": "Construct a square ABCD. Take the midpoint O of BC, and take a point E inside the square. Connect OE (mark \"2\" below OE), draw a circle with center O and radius OE using blue dashed lines. Connect AE and DE. Rotate segment DE 90 degrees counterclockwise about point D to obtain segment DF, then connect OF and CF. Connect OD with a dashed line, rotate segment DO 90 degrees counterclockwise about point D to obtain segment DO’, and connect OO’ and O’F with dashed lines (mark \"2\" below O’F). Draw a circle with center O’ and radius O’F using blue dashed lines. Shade the triangular region DEO in green and the triangular region DFO’ in purple, and mark \"5√2 - 2\" on the right side of the diagram.", "ocr_zh": "作正方形ABCD，取BC的中点O，在正方形内取一点E，连接OE（OE下方标注2）,以O为圆心，OE长为半径以蓝色虚线作圆，连接AE、DE，将线段DE绕点D逆时针旋转九十度得到线段DF，连接OF、CF，虚线连接OD，将线段DO绕点D逆时针旋转九十度得到线段DO’，虚线连接OO’,O’F（O’F下方标注2），以O’为圆心，O’F长为半径以蓝色虚线作圆，将三角形区域DEO标绿和三角形区域DFO’标紫，在图片右侧标注5根号2-2", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_909.jpg" }, { "index": 910, "ocr_en": "\tConstruct a square ABCD. Take the midpoint O of BC and connect OD with a dashed line. Take a point E inside the square, connect OE, and draw a circle with center O and radius OE using blue dashed lines. Connect AE and DE. Rotate segment DE 90 degrees counterclockwise about point D to obtain segment DF, then connect OF and CF.", "ocr_zh": "作正方形ABCD，取BC的中点O，虚线连接OD，在正方形内取一点E，连接OE,以O为圆心，OE长为半径以蓝色虚线作圆，连接AE、DE，将线段DE绕点D逆时针旋转九十度得到线段DF，连接OF、CF", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_910.jpg" }, { "index": 911, "ocr_en": "In the plane rectangular coordinate system, construct an inverse proportional function (located in the first and third quadrants). Draw a linear function through the origin O , intersecting the inverse proportional function at points A and B (point A is the intersection in the first quadrant, and point B is the intersection in the second quadrant). Take a point C on the negative x-axis, draw a circle with center C and radius half of OC. Take a point P on the circle (P is located on the upper semicircle), connect AP , and take the midpoint M of AP ", "ocr_zh": "在平面直角坐标系中作反比例函数（位于第一、三象限），过原点O作一次函数交反比例函数于A，B两点（A点为第一象限交点，B点为第二象限交点），在x轴负半轴取一点C，以C为圆心OC的二分之一长为半径作圆，在圆上取一点P（P位于上半圆），连接AP，取AP的中点M。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_911.jpg" }, { "index": 912, "ocr_en": "In the plane rectangular coordinate system, draw an inverse proportional function (lying in the first and third quadrants). Draw a linear function y equals 2x passing through the origin O, which intersects the inverse proportional function at points A and B (point A is the intersection in the first quadrant, labeled as (t, 2t) on the right side of point A, and point B is the intersection in the second quadrant). Take a point C on the negative x-axis, use C as the center and half the length of OC as the radius to make a circle. Take a point P on the circle (where P is located on the upper half of the circle), connect AP, and take the midpoint M of AP. Connect OM and AC with dashed lines. Take a point H on the line segment AC , and use H as the center to draw a circle with a blue solid line.", "ocr_zh": "在平面直角坐标系中作反比例函数（位于第一、三象限），过原点O作一次函数y等于2x交反比例函数于A(A点右侧标注（t，2t）)，B两点（A点为第一象限交点，B点为第二象限交点），在x轴负半轴取一点C，以C为圆心OC的二分之一长为半径作圆，在圆上取一点P（P位于上半圆），连接AP，取AP的中点M。虚线连接OM，AC，在线段AC上取一点H，以H点为圆心蓝色实线作圆", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_912.jpg" }, { "index": 913, "ocr_en": "Construct triangle ABC, take a point P inside the triangle, and connect PA, PB, and PC.", "ocr_zh": "作三角形ABC，取三角形内一点P，连接PA、PB、PC", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_913.jpg" }, { "index": 914, "ocr_en": "Left diagram:\nConstruct an equilateral triangle ABC (where AB = AC = BC). Take a point O inside the equilateral triangle, and draw the circumcircle of equilateral triangle ABC with point O as the center. Take a point P on the arc BC, and connect PA, PB, and PC. \n\nRight diagram: \nConstruct an equilateral triangle ABC (where AB = AC = BC). Take a point P in the plane (point P is below the equilateral triangle), and connect PA, PB, and PC.", "ocr_zh": "左图：\n作等边三角形ABC（AB等于AC等于BC），取等边三角形内一点O，以点O为圆心作等边三角形ABC的外接圆，在圆弧BC上取一点P，连接PA、PB、PC\n右图：\n作等边三角形ABC（AB等于AC等于BC），在平面内取一点P（P点在等边三角形下方），连接连接PA、PB、PC", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_914.jpg" }, { "index": 915, "ocr_en": "Left Figure: Draw a horizontal line l. Take two points A and B above line l (point A is located at the lower left of point B). Connect A and B with a red solid line. From point A, draw a perpendicular AA' to line l with a red solid line, where A' is the foot of the perpendicular. From point B, draw a perpendicular BB' to line l with a dashed line, where B' is the foot of the perpendicular. \n\nMiddle Figure: Draw a horizontal line l. Take two points A and B above line l (point A is located at the lower left of point B). From point A, draw a perpendicular AA' to line l with a dashed line, where A' is the foot of the perpendicular. From point B, draw a perpendicular BB' to line l with a dashed line, where B' is the foot of the perpendicular. Take a point P on segment A'B'. Connect P to A and P to B with a red solid line. \n\nRight Figure: Draw a horizontal line l. Take two points A and B above line l (point A is located at the lower left of point B). Take a point P below and to the right of point A (point P is above line l). Connect P to A and P to B with red solid lines. From point A, draw a perpendicular AA' to line l with a dashed line, where A' is the foot of the perpendicular. From point B, draw a perpendicular BB' to line l with a dashed line, where B' is the foot of the perpendicular. From point P, draw a perpendicular PP' to line l with a red solid line, where P' is the foot of the perpendicular.", "ocr_zh": "左图：作水平直线l，在直线l上方取两点A,B（A点位于B点左下方），红色实线连接AB，从A点以红色直线作AA’垂直于直线l，垂足为A’，从B点以虚线作BB’垂直于直线l，垂足为B’。\n中间图：作水平直线l，在直线l上方取两点A,B（A点位于B点左下方），从A点以虚线作AA’垂直于直线l，垂足为A’，从B点以虚线作BB’垂直于直线l，垂足为B’，在线段A’B’之间取一点P，以红色实线连接PA、PB.\n右图：作水平直线l，在直线l上方取两点A，B（A点位于B点左下方），在A点右下方取一点P（P点位于直线l上方），以红色直线连接PA、PB，从A点以虚线作AA’垂直于直线l，垂足为A’，从B点以虚线作BB’垂直于直线l，垂足为B’，从P点以红色实线作PP’垂直于直线l，垂足为P’", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_915.jpg" }, { "index": 916, "ocr_en": "Draw a horizontal line l. Take two points A and B above line l (point A is located at the lower left of point B). Take a point P below and to the right of point A (point P is above line l). Connect P to A and P to B with red solid lines. From point A, draw a perpendicular AA' to line l with a dashed line, where A' is the foot of the perpendicular. From point B, draw a perpendicular BB' to line l with a dashed line, where B' is the foot of the perpendicular. From point P, draw a perpendicular PP' to line l with a red solid line, where P' is the foot of the perpendicular.", "ocr_zh": "作水平直线l，在直线l上方取两点A，B（A点位于B点左下方），在A点右下方取一点P（P点位于直线l上方），以红色直线连接PA、PB，从A点以虚线作AA’垂直于直线l，垂足为A’，从B点以虚线作BB’垂直于直线l，垂足为B’，从P点以红色实线作PP’垂直于直线l，垂足为P’", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_916.jpg" }, { "index": 917, "ocr_en": "Left Figure:\nConstruct quadrilateral ABPC, connect points BC and AP. Take a point O inside quadrilateral ABPC, and with O as the center, construct the circumcircle of quadrilateral ABPC.\n\nRight Figure:\nConstruct quadrilateral ABPC, and connect points BC and AP.", "ocr_zh": "左图：作四边形ABPC，连接BC、AP，在四边形ABPC内取一点O，以O点为圆心作四边形ABPC的外接圆\n右图：作四边形ABPC，连接BC、AP", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_917.jpg" }, { "index": 918, "ocr_en": "Construct a square ABCD, take a point P inside the square ABCD, and connect points PA, PB, ", "ocr_zh": "作正方形ABCD，在正方形ABCD内取一点P，连接PA、PB、PC", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_918.jpg" }, { "index": 919, "ocr_en": "Left Figure:\nConstruct an equilateral triangle ABC (AB = AC = BC). Construct the circumcircle of equilateral triangle ABC. Take a point P on the minor arc BC. Connect points PA, PB, and PC with lines. Rotate triangle BPC counterclockwise by 60 degrees around point B to triangle BP'A (where P' falls on segment AP). Connect B to P' with a dashed line. Shade the triangular regions ABP' and BPC in pink.\n\nRight Figure:\nConstruct an equilateral triangle ABC (AB = AC = BC). Take a point P in the plane below the equilateral triangle. Connect points PA, PB, and PC with lines. Rotate triangle BPC counterclockwise by 60 degrees around point B to triangle BP'A (connecting P'A and P'B with dashed lines). Connect P' to P with a dashed line. Shade the triangular regions ABP' and BPC in pink.", "ocr_zh": "左图：\n作等边三角形ABC（AB等于AC等于BC），作等边三角形ABC的外接圆，在劣弧BC上取一点P，连接PA、PB、PC，将三角形BPC绕点B逆时针旋转60度至三角形BP’A（P’落在AP上），虚线连接BP’，将三角形区域ABP’和三角形区域BPC标粉红\n右图：\n作等边三角形ABC（AB等于AC等于BC），在平面内取一点P（P点在等边三角形下方），连接PA、PB、PC，将三角形BPC绕点B逆时针旋转60度至三角形BP’A，（其中边P’A、P’B以虚线连接），虚线连接P’P，将三角形区域ABP’和三角形区域BPC标粉红", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_919.jpg" }, { "index": 920, "ocr_en": "Construct an equilateral triangle ABC (AB = AC = BC). Take a point P in the plane below the equilateral triangle. Connect points PA, PB, and PC with lines. Rotate triangle BPC counterclockwise by 60 degrees around point B to triangle BP'A (connecting P'A and P'B with dashed lines). Connect P' to P with a dashed line. Shade the triangular regions ABP' and BPC in pink.", "ocr_zh": "作等边三角形ABC（AB等于AC等于BC），在平面内取一点P（P点在等边三角形下方），连接PA、PB、PC，将三角形BPC绕点B逆时针旋转60度至三角形BP’A，（其中边P’A、P’B以虚线连接），虚线连接P’P，将三角形区域ABP’和三角形区域BPC标粉红", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_920.jpg" }, { "index": 921, "ocr_en": "Construct triangle ABC. With BC as a side, construct equilateral triangle BCD downward (where BC = BD = CD, and sides BD and CD are connected by dashed lines). Construct the circumcircle of the equilateral triangle. Connect AD to intersect the circumcircle of the equilateral triangle at point P (where segment PD is drawn as a red dashed line, and segment AP is drawn as a red solid line). Connect PB and PC with red solid lines. Take a point P’ inside triangle ABC and outside the circle (with point P’ located in the upper left of point P). Connect P’A, P’B, and P’C with blue solid lines, and connect P’D with a blue dashed line.", "ocr_zh": "作三角形ABC，以BC为边，向下作等边三角形BCD（BC等于BD等于CD，其中边BD、CD以虚线连接），作等边三角形外接圆，连接AD交等边三角形外接圆于P点（其中线段PD用红色虚线，线段AP用红色实线），用红色实线连接PB、PC，在三角形ABC内、圆外取一点P’（P’点位于P点左上方），蓝色实线连接P’A、P’B、P’C，蓝色虚线连接P’D", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_921.jpg" }, { "index": 922, "ocr_en": "Construct horizontal line l . Take two points A and B above line l (point A is located lower left of point B). Take a point P lower right of point A (point P is located above line l ). From point A , draw AA' perpendicular to line l with a dashed line, with foot A' From point B draw BB' perpendicular to line l with a dashed line, with foot B' . From point P , draw PP' perpendicular to line l with a blue dashed line, with foot P'. Connect AB with a dashed line. Construct equilateral triangle ABC upward with AB as a side (where BC equals BA equals CA, and sides AC and BC are connected by dashed lines). Connect PA , PB , and PC with blue dashed lines. From point C , draw CP0’ perpendicular to line l with a red solid line, with foot P0’. Draw the circumcircle of the equilateral triangle with a dashed line, intersecting segment CP0’ at point P0 Connect BP0 with a red solid line. From point A , draw AB0 perpendicular to BB’ with a dashed line, with foot B0. Construct equilateral triangle AB0C0 upward with AB0 as a side (where AB0 equals AC0 equals B0C0) with dashed lines. Connect CC0 with a green dashed line. From point C0, draw C0H perpendicular to CP0 with a dashed line, with foot H . Mark segment BB0 as a green dashed line.", "ocr_zh": "作水平直线l，在直线l上方取两点A，B（A点位于B点左下方），在A点右下方取一点P（P点位于直线l上方），从A点以虚线作AA’垂直于直线l，垂足为A’，从B点以虚线作BB’垂直于直线l，垂足为B’，从P点以蓝色虚线作PP’垂直于直线l，垂足为P’，虚线连接AB，以AB为边向上做等边三角形ABC（BC等于BA等于CA，其中边AC、BC以虚线连接），以蓝色虚线连接PA、PB、PC，从C点以红色实线作CP0’垂直于直线l，垂足为P0’，以虚线作等边三角形外接圆交线段CP0’于P0点，红色实线连接BP0，从A以虚线作AB0垂直于BB’，垂足为B0，以AB0为边向上以虚线作等边三角形AB0C0（AB0等于AC0等于B0C0）,绿色虚线连接CC0，从C0点以虚线作C0H垂直于CP0，垂足为H，将线段BB0标为绿色虚线。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_922.jpg" }, { "index": 923, "ocr_en": "Left diagram \nConstruct quadrilateral ABPC, connect AP and BC. Take a point O inside the quadrilateral, and draw the circumcircle of quadrilateral ABPC with center O. Take a point P’ on PA, connect P’B with a dashed line. Color the triangular region ABC pink, and color the triangular region BPP’ light blue. \n\nRight diagram: \nConstruct quadrilateral ABPC, connect AP and BC. Take a point P’ inside triangle ABC (P’ is on the right side of segment AP), connect P’A, P’B, and P’P with dashed lines. Color the triangular region ABC pink, and color the triangular region BPP’ light blue.", "ocr_zh": "左图：\n作四边形ABPC，连接AP、BC，在四边形内取一点O，以O点为圆心作四边形ABPC外接圆，在PA上取一点P’，虚线连接P’B，将三角形区域ABC标粉，将三角形区域BPP’标浅蓝\n右图：\n作四边形ABPC，连接AP、BC，在三角形ABC内取一点P’（P’在线段AP右侧），虚线连接P’A、P’B、P’P，将三角形区域ABC标粉，将三角形区域BPP’标浅蓝", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_923.jpg" }, { "index": 924, "ocr_en": "Construct square ABCD. Take the midpoint O of segment AB, and draw a circle with O as the center and AB as the diameter using dashed lines. Take a point P on the lower right semicircle, connect PA, PB, and PC. Extend CP to intersect the upper left semicircle at point Q (QP is represented by a dashed line). Connect QA and QB with dashed lines. From point Q, draw QH perpendicular to CB, intersecting the extension of CB at point H (the foot of the perpendicular is H), and BH is denoted by a dashed line.", "ocr_zh": "作正方形ABCD，取线段AB的中点O，以O点为圆心，AB为直径以虚线作圆，在右下半圆取一点P，连接PA、PB、PC，延长CP交左半圆于Q点（QP用虚线表示），虚线连接QA、QB，从Q点作QH垂直于CB交CB延长线于H点，垂足为H，BH以虚线表示", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_924.jpg" }, { "index": 925, "ocr_en": "Construct a circle. Take two points A and C on the circle, with point A located to the left of point C. Find the midpoint M of arc AC, and take a point B on arc AM. Connect AB and BC. From point M, draw MD perpendicular to BC, with the foot of the perpendicular at D, and mark a gray right-angle symbol.", "ocr_zh": "作圆，在圆上取两点A、C，点A在点C左侧，取弧AC中点M，在弧AM上取一点B，连接AB、BC，从M点作MD垂直于BC，垂足为D，标记灰色直角符号", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_925.jpg" }, { "index": 926, "ocr_en": "Construct angle MON. Take two points A and B on OM (point A is to the right of point B). Draw a circle through points A and B, tangent to ON at point C. Connect AC and BC. Take a point C’ on OC, and connect C’A and C’B. Take a point C'' on CN, and connect C''A and C''B.", "ocr_zh": "作角MON，在OM上取两点A、B（A点在B点右侧），过A、B两点作圆与ON相切于点C，连接AC、BC，在OC上取一点C’，连接C’A、C’B，在CN上取一点C’’，连接C’’A、C’’B", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_926.jpg" }, { "index": 927, "ocr_en": "\tTake a point O in the plane, draw a circle with O as the center. Take a point P to the right of the circle. From point P, draw PC tangent to the circle at point C. Take a point B on the circle (point B is located lower left of point C). Connect BC, connect PB to intersect the circle at point A, and connect AC.", "ocr_zh": "在平面取一点O，以O为圆心作圆，在圆外右侧取一点P，从P点作PC与圆相切于C点，在圆上取一点B（B点位于C点的左下方），连接BC，连接PB交圆于点A，连接AC", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_927.jpg" }, { "index": 928, "ocr_en": "Construct angle MQN (with M as an auxiliary point). Take two points A and B on ray QM (point A is to the left of point B). Draw a circle through points A and B, intersecting ray QN at points C and D (point C is to the left of point D). Take a point E on segment CD, connect EA and EB. Take a point F on ray DN, connect FA and FB. Take a point P on segment QC (mark an arrow on the left side of point P, pointing from left to right along QC).", "ocr_zh": "作角MQN（M为辅助点），在射线QM上取两点A、B（A点在B点左侧）,过A、B两点作圆与射线QN相交于C、D两点（C点位于D点左侧），在线段CD上取一点E，连接EA、EB，在射线DN上取一点F，连接FA、FB，在线段QC上取一点P（在P点左侧标注一个沿QC从左指向右的箭头）", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_928.jpg" }, { "index": 929, "ocr_en": "Construct a plane rectangular coordinate system. Take two points A and B on the positive x-axis, with point A located to the left of point B, such that OB equals five times OA.", "ocr_zh": "作平面直角坐标系，在x轴正半轴取两点A、B，点A位于点B的左侧，OB等于五倍的OA", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_929.jpg" }, { "index": 930, "ocr_en": "Construct a plane rectangular coordinate system with grids. The coordinate of point C is (3, 0), the coordinate of point B is (0, 3), and the coordinate of point A is (0, 7). Connect AC and BC.", "ocr_zh": "作带网格的平面直角坐标系，C点坐标为（3,0），B点坐标为（0,3），A点坐标为（0,7），连接AC，BC", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_930.jpg" }, { "index": 931, "ocr_en": "Take a point O in the plane, draw a circle with O as the center. Take four points A, B, C, D counterclockwise on the circle. Connect AC, AD, BC, BD. Take a point E to the right of the circle, connect AE and BE.", "ocr_zh": "在平面内取一点O，以O为圆心作圆，在圆上逆时针依次取四点A,B,C,D,连接AC、AD、BC、BD,在圆外右侧取一点E，连接AE、BE", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_931.jpg" }, { "index": 932, "ocr_en": "Construct a plane rectangular coordinate system with coordinate scales. Point A is located at (1, 0), and point B is located at (5, 0).", "ocr_zh": "作带坐标尺度的平面直角坐标系，A点位于（1,0），B点位于（5,0）", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_932.jpg" }, { "index": 933, "ocr_en": "Construct a quadrilateral ABCD where AD is parallel to BC, and CD is perpendicular to BC with the foot of the perpendicular at point C (mark the right-angle symbol). Take a point M on the line segment AD, then connect MB and MC.", "ocr_zh": "作四边形ABCD，AD平行于BC，CD垂直于BC，垂足为C（标注直角符号），在线段AD上取一点M，连接MB、MC", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_933.jpg" }, { "index": 934, "ocr_en": "Construct a quadrilateral ABCD where AD is parallel to BC, and CD is perpendicular to BC with the foot of the perpendicular at point C (mark the right-angle symbol)", "ocr_zh": "作四边形ABCD，AD平行于BC，CD垂直于BC，垂足为C（标注直角符号）", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_934.jpg" }, { "index": 935, "ocr_en": "Construct a quadrilateral ABCD where AD is parallel to BC, and CD is perpendicular to BC with the foot of the perpendicular at point C (mark the right-angle symbol). Take a point N on the line segment AD, then connect NB and NC.", "ocr_zh": "作四边形ABCD，AD平行于BC，CD垂直于BC，垂足为C（标注直角符号），在线段AD上取一点N，连接NB、NC", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_935.jpg" }, { "index": 936, "ocr_en": "Take a point O in the plane, draw a circle with O as the center. Take a point C on the upper semicircle, and take two points A and B counterclockwise in sequence on the lower semicircle. Connect CA and CB. Take a point D above the circle (point D is located at the upper right of point C), and connect DA and DB.", "ocr_zh": "在平面取一点O，以O为圆心作圆，在上半圆取一点C，在下半圆逆时针依次取两点A、B，连接CA、CB，在圆上方取一点D（D点位于C点的右上方），连接DA、DB", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_936.jpg" }, { "index": 937, "ocr_en": "Construct a plane rectangular coordinate system. Take two points A and B on the positive x-axis, with point A located to the left of point B. Draw a straight line through the origin O (the line passes through the first and third quadrants).", "ocr_zh": "作平面直角坐标系，在x轴正半轴取两点A、B，A点在B点的左侧，过原点O作直线（直线经过第一三象限）", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_937.jpg" }, { "index": 938, "ocr_en": "Construct a plane rectangular coordinate system. Take two points A and B on the positive x-axis, with point A located to the left of point B. Draw a straight line l (where l is a linear function passing through the first, second, and third quadrants).", "ocr_zh": "作平面直角坐标系，在x轴正半轴取两点A、B，A点在B点的左侧，作直线l（l为一次函数，且经过第一二三象限）", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_938.jpg" }, { "index": 939, "ocr_en": "Construct a square ABCF. Take a point E on BC and connect AE. Take a point D on CF such that DE is perpendicular to AE with the foot of the perpendicular at E (mark the right-angle symbol), then connect AD. Mark AB as a, BE as 1, EC as a, CD as 1, DF as a-1, and AF as a+1 in red. Mark angle DAE as 45 degrees in red, mark angle DAF as α in gray, and mark angle BAE as β in gray.", "ocr_zh": "作正方形ABCF，在BC上取一点E，连接AE，在CF上取一点D，有DE垂直于AE，垂足为E（标注直角符号），连接AD，红色标注AB为a、BE为1、EC为a、CD为1、DF为a-1、AF为a+1，红色标注角DAE为45度，灰色标注角DAF为α，灰色标注角BAE为β", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_939.jpg" }, { "index": 940, "ocr_en": "Construct triangle ACB with side BC horizontal. Draw AD perpendicular to BC from point A, with the foot of the perpendicular at D, and mark the right-angle symbol.", "ocr_zh": "作三角形ACB,边BC水平，从A点作AD垂直于BC，垂足为D，标注直角符号", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_940.jpg" }, { "index": 941, "ocr_en": "All points below are auxiliary points. Construct a rectangle ABCD. Take a point E on AD and connect BE. Draw a dashed line from E perpendicular to BC, intersecting at H (mark the right angle). Take a point F on CD. Connect EF and BF with solid red lines, where EF is perpendicular to BF at F (mark the right angle). Label BC as 1, CF as tanα, FD as tanβ, DE as tanα·tanβ. Draw a red arrow left from EF labeled √(1+tan²α)·tanβ and from FB labeled √(1+tan²α) (both arrows extend left of the square). Mark angles EFD and FBC as α in red, angle FBE as β in red. Shade triangle EBH light blue.", "ocr_zh": "以下所有点均为辅助点，作长方形ABCD，在AD上取一点E,连接BE，从E点以虚线作EH垂直于BC，垂足为H（标注直角符号），在CD上取一点F，红色实线连接EF、BF，有EF垂直于BF，垂直为F（标注直角符号），在BC边标注1，CF边标注tanα，FD边标注tanβ，DE边标注tanα·tanβ，EF边用红色箭头向左引出标注根号下（1+tan2α）·tanβ，FB边用红色箭头向左引出标注根号下（1+tan2α）(均引出至正方形左侧)，红色标注角EFD和角FBC为α，红色标注角FBE为β，将三角形区域EBH标浅蓝", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_941.jpg" }, { "index": 942, "ocr_en": "Construct a rectangle ABCF (with AF and BC as the lengths, and AB and CF as the widths). Take a point E on BC and connect AE. Take a point D on CF, then connect DA and DE, where DE is perpendicular to AE with the foot of the perpendicular at E (mark the right-angle symbol). Mark AB as 2, BE as 1, EC as 2, CD as 1, DF as 1, and FA as 3 in red. Mark angle EAD as 45 degrees in red, mark angle DAF as α in gray, and mark angle EAB as β in gray. Shade the triangular regions EAB and ECD light blue.", "ocr_zh": "作长方形ABCF（AF、BC为长，AB、CF为宽），在BC上取一点E，连接AE，在CF上取一点D，连接DA、DE，有DE垂直于AE，垂足为E，标记直角符号，红色标注AB为2、BE为1、EC为2、CD为1、DF为1、FA为3，红色标注角EAD为45度，灰色标注角DAF为α，灰色标注角EAB为β，将三角形区域EAB和三角形区域ECD标浅蓝", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_942.jpg" }, { "index": 943, "ocr_en": "Construct a rectangle ABCF (with AF and BC as the lengths, and AB and CF as the widths). Take a point E on BC and connect AE. Take a point D on CF, then connect DA and DE, where DE is perpendicular to AE with the foot of the perpendicular at E (mark the right-angle symbol). Mark AB as 2, BE as 1, EC as 2, CD as 1, DF as 1, and FA as 3 in red. Mark angle EAD as 45 degrees in red, mark angle DAF as α in gray, and mark angle EAB as β in gray. Shade the triangular regions EAB and ECD light blue. On the right side of the rectangle, mark \"1. 2. 3\" in red at the upper part and \"tanα equals 1/3, tanβ equals 1/2\" in blue at the lower part.", "ocr_zh": "作长方形ABCF（AF、BC为长，AB、CF为宽），在BC上取一点E，连接AE，在CF上取一点D，连接DA、DE，有DE垂直于AE，垂足为E，标记直角符号，红色标注AB为2、BE为1、EC为2、CD为1、DF为1、FA为3，红色标注角EAD为45度，灰色标注角DAF为α，灰色标注角EAB为β，将三角形区域EAB和三角形区域ECD标浅蓝，在长方形右侧标注，右侧上方红色标注‘‘1.2.3’’，右侧下方蓝色标注tanα等于1/3、tanβ等于1/2", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_943.jpg" }, { "index": 944, "ocr_en": "Left diagram: \nConstruct an isosceles triangle ABC with AB equal to AC. \n\n\nMiddle diagram: \nConstruct an isosceles triangle ABC with AB equal to AC. Draw a red dashed line AD from point A perpendicular to BC, with the foot of the perpendicular at D (mark the right-angle symbol). Take a point O on AD and connect OC with a red dashed line, where AO equals OC. Label DC as 1, and mark AO as 2−x, OC as 2−x, and OD as x in red. \n\n\nRight of the middle diagram (text): \nFirst line: (2−x)² − x² equals 1 \nSecond line: 4(1−x) equals 1 \nThird line: x equals 3/4 \n\n\nRight diagram: \nConstruct an isosceles triangle ABC with AB equal to AC. Draw a red dashed line CD from point C perpendicular to AB, with the foot of the perpendicular at D (mark the gray right-angle symbol). Label AD as 3m, DB as 2m, CD as 4m, and AC as 5m.", "ocr_zh": "左图：\n作等腰三角形ABC，AB等于AC\n中间图：\n作等腰三角形ABC，AB等于AC，从A点以红色虚线作AD垂直于BC，垂足为D（标注直角符号），在AD上取一点O，红色虚线连接OC，有AO等于OC，标注DC为1，红色标注AO为2-x、OC为2-x、OD为x\n中间图右侧表述：\n第一行：（2-x）2-x2等于1\n第二行：4（1-x）等于1\n第三行：x等于3/4\n右图：\n作等腰三角形ABC，AB等于AC，从C点以红色虚线作CD垂直于AB，垂足为D，标注灰色直角符号，标注AD为3m、DB为2m、CD为4m、AC为5m", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_944.jpg" }, { "index": 945, "ocr_en": "Construct a square ABCD. Take a point F on AD and connect CF, then take a point E on AB and connect CE.", "ocr_zh": "作正方形ABCD，在AD上取一点F，连接CF，在AB上取一点E,连接CE", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_945.jpg" }, { "index": 946, "ocr_en": "All points below are auxiliary points. Construct angle MON (where OM and ON are line segments), mark angle MON as α in gray. Extend MO to H such that OH equals ON, connect dashed line NH. Both ON and OH are represented by red dashed lines. Mark angle OHN as ½α in gray. Above the angle, label \"construct an isosceles triangle outward for the half-angle\".", "ocr_zh": "以下点均为辅助点，作角MON（其中OM和ON为线段），灰色标注角MON为α，延长MO至H，有OH等于ON，连接虚线NH，ON和OH均以红色虚线表示，灰色标注角OHN为\n1/2 α，在角上方标注半角向外构造等腰", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_946.jpg" }, { "index": 947, "ocr_en": "All points below are auxiliary points. Construct angle MON (where OM and ON are line segments), mark angle MON as α in gray. Extend MO to H such that OH equals ON, then connect NH. Represent ON with a red dashed line and OH with a red solid line. Mark angle OHN as ½α in gray. Draw the angle bisector of angle HON with a red dashed line. Above the angle, label \"construct an isosceles triangle inward for the double angle\".", "ocr_zh": "以下点均为辅助点，作角MON（其中OM和ON为线段），灰色标注角MON为α，延长MO至H，有OH等于ON，连接NH，ON以红色虚线表示，OH以红色实线表示，灰色标注角OHN为1/2 α，红色虚线作角HON的角平分线，在角上方标注倍角向内构造等腰", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_947.jpg" }, { "index": 948, "ocr_en": "All points below are auxiliary points. Construct a right triangle ABC with angle ABC as the right angle (mark the gray right-angle symbol), and mark angle BAC as α in red. Take a point D on AB, connect CD with a blue solid line, extend BC to E with a blue solid line, and connect DE. Mark AD as 5, DB as 4, BC as 3, CE as 5, DC as 5 in red; mark angle CDB as 2α in red, angle DCB as 2β in red, and angle CED as β in red. \n\n\nLeft-side labels (all in blue): \nFirst line: Yute Edition \"1 2 3 4 5\" + Double and Half Angles \nSecond line: α + β = 45° \nThird line: tanα = 1/3, tanβ = 1/2 \n\n\nRight-side labels (all in blue unless specified): \nFirst line: \"1\"/2 + \"1/3\" = 45° right arrow \"1.2.3.4.5\" (in red) \nSecond line: \"1\"/2 + \"1/2\" = \"4\"/3, \"1/3\" + \"1/3\" = \"3\"/4", "ocr_zh": "以下点均为辅助点，作直角三角形三角形ABC，角ABC为直角标注灰色直角符号，角BAC红色标注α，在AB上取一点D，蓝色实线连接CD，蓝色实线延长BC至E，连接DE，其中有红色标注AD为5、DB为4、BC为3、CE为5、DC为5，角CDB红色标注2α,角DCB红色标注2β，角CED红色标注β。\n图左侧标注（均为蓝色）：\n第一行：于特版‘‘1 2 3 4 5’’+倍半角\n第二行：α+β=45°\n第三行：tanα=1/3 tanβ=1/2\n图右侧标注（除单独标注外均为蓝色）：\n第一行：‘‘1’’/2+‘‘1/3’’=45°右箭头‘‘1.2.3.4.5’’（红色）\n第二行：‘‘1’’/2+‘‘1/2’’=‘‘4’’/3 ‘‘1/3’’+‘‘1/3’’=’’3’’/4", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_948.jpg" }, { "index": 949, "ocr_en": "Construct a square ABCD. Take the midpoint P of BC and connect PA. Fold triangle PAB along PA to obtain triangle PAE. Connect DE. Draw CF perpendicular to DE from point C, with the foot of the perpendicular at F (intersecting the extension of DE at point F). Mark line segments AB and BP as dashed lines.", "ocr_zh": "作正方形ABCD，取BC中点P，连接PA，将三角形PAB沿着PA翻折得到三角形PAE，连接DE，从C点作CF垂直于DE，垂足为F，交DE延长线于点F，将线段AB和线段BP标为虚线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_949.jpg" }, { "index": 950, "ocr_en": "Take a point O in the plane, draw a circle with center O. Take a point P outside the circle (P is located to the right of the circle). Draw PA from point P tangent to circle O at point A, connect OP. Draw AC perpendicular to OP from point A with the foot of the perpendicular at C, extend AC to intersect the circle at point B, connect BO, extend BO to intersect the extension of PA at point E (point E is outside the circle).", "ocr_zh": "在平面内取一点O，以O点为圆心作圆,取圆外一点P（P位于圆的右侧），从P点作PA与圆O相切于A点，连接OP，过A点作AC垂直于OP，垂足为C，延长AC与圆相交于B点,连接BO，延长BO与PA的延长线相交于E点（E点位于圆外）", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_950.jpg" }, { "index": 951, "ocr_en": "Construct a square ABCD, connect AC and BD intersecting at point O. Take a point E on BC and connect AE. Draw BF perpendicular to AE from point B with the foot of the perpendicular at F, then connect OF. Draw a red dashed line OH from point O perpendicular to AE with the foot of the perpendicular at H (mark the gray right-angle symbol), and shade triangle OHF green. \n\nOn the right side of the square, mark in red: \nFirst line: AO = √5 \nSecond line: AH = 2 \nThird line: HO = HF = BF = 1 \nFourth line: OF = √2", "ocr_zh": "作正方形ABCD，连接AC、BD相交于O点，在BC上取一点E，连接AE，过B点作BF垂直于AE垂直为F，连接OF，过O点以红色虚线作OH垂直于AE垂足为H，标注灰色直角符号，将三角形区域OHF标绿。\n在正方形右侧红色标注：\n第一行：AO=根号5\n第二行：AH=2\n第三行：HO=HF=BF=1\n第四行：OF=根号2", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_951.jpg" }, { "index": 952, "ocr_en": "Construct a square ABCD. Take the midpoint P of BC and connect PA. Fold triangle PAB along PA to obtain triangle PAE. Connect DE. Draw CF perpendicular to DE from point C, with the foot of the perpendicular at F (intersecting the extension of DE at point F). Mark line segments AB and BP as dashed lines. Draw a dashed perpendicular line from point A to DE, with the foot of the perpendicular at point 3. Mark angle PAE as 1 and angle EA3 as 2 in red. Label 45° on line segment AE. Shade triangle DFC light blue and triangle A3D green.\nOn the right side of the square, mark in blue:\nFirst line: ‘‘1’’/2 + ‘‘1’’/3 = 45°\nSecond line: FC/DF = 1/3 right arrow DF = 6", "ocr_zh": "作正方形ABCD，取BC中点P，连接PA，将三角形PAB沿着PA翻折得到三角形PAE，连接DE，从C点作CF垂直于DE，垂足为F，交DE延长线于点F，将线段AB和线段BP标为虚线。过A点以虚线作垂线垂直于DE，垂足为3，红色标注角PAE为1、角EA3为2，在线段AE上标注45度，将三角形区域DFC标浅蓝，将三角形区域A3D标绿\n在正方形右侧蓝色标注：\n第一行：‘‘1’’/2+‘‘1’’/3=45°\n第二行：FC/DF=1/3右箭头DF=6", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_952.jpg" }, { "index": 953, "ocr_en": "Left diagram: \nConstruct a square ABCD. Take a point M on DC and a point N on AB (M is to the left of N), then connect MN. Fold quadrilateral DMNA to the right along MN so that point A falls on BC at point E. Connect AE with a dashed line intersecting MN at point K. Mark line segments DA, DM, and AN as dashed lines. \n\n\nRight diagram: \nConstruct a square ABCD. Take a point M on DC and a point N on AB (M is to the left of N), then connect MN. Fold quadrilateral DMNA to the right along MN so that point A falls on BC at point E. Connect AE with a dashed line intersecting MN at point K. Mark line segments DA, DM, and AN as dashed lines. Mark DA as 9k, AN as 5k, NB as 4k, BE as 3k, NE as 5k, and CE as 6k in red.", "ocr_zh": "左图：\n作正方形ABCD，在DC上取一点M，在AB上取一点N（M在N的左侧），连接MN，将四边形DMNA沿MN向右翻折，使A点落在BC边上得到点E，虚线连接AE交MN于点K，将线段DA、DM、AN标为虚线\n右图：\n作正方形ABCD，在DC上取一点M，在AB上取一点N（M在N的左侧），连接MN，将四边形DMNA沿MN向右翻折，使A点落在BC边上得到点E，虚线连接AE交MN于点K，将线段DA、DM、AN标为虚线，红色标注DA为9k、AN为5K、NB为4k、BE为3k、NE为5k、CE为6k", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_953.jpg" }, { "index": 954, "ocr_en": "Left diagram: \nConstruct a square ABCD, connect AC and BD intersecting at point O. Take a point E on BC and connect AE. Draw BF perpendicular to AE from point B with the foot of the perpendicular at F, then connect OF\nRight diagram: \nConstruct a square ABCD, connect AC and BD intersecting at point O. Take a point E on BC and connect AE. Draw BF perpendicular to AE from point B with the foot of the perpendicular at F, then connect OF. Draw a red dashed line OH from point O perpendicular to AE with the foot of the perpendicular at H (mark the gray right-angle symbol), and shade triangle OHF green. \n\nOn the right side of the square, mark in red: \nFirst line: AO = √5 \nSecond line: AH = 2 \nThird line: HO = HF = BF = 1 \nFourth line: OF = √2", "ocr_zh": "左图：\n作正方形ABCD，连接AC、BD相交于O点，在BC上取一点E，连接AE，过B点作BF垂直于AE垂足为F，连接OF\n右图：\n作正方形ABCD，连接AC、BD相交于O点，在BC上取一点E，连接AE，过B点作BF垂直于AE垂足为F，连接OF，过O点以红色虚线作OH垂直于AE垂足为H，标注灰色直角符号，将三角形区域OHF标绿。\n在正方形右侧红色标注：\n第一行：AO=根号5\n第二行：AH=2\n第三行：HO=HF=BF=1\n第四行：OF=根号2", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_954.jpg" }, { "index": 955, "ocr_en": "Construct a square ABCD. Take the midpoint P of BC and connect PA. Fold triangle PAB along PA to obtain triangle PAE. Connect DE. Draw CF perpendicular to DE from point C, with the foot of the perpendicular at F (intersecting the extension of DE at point F). Mark line segments AB and BP as dashed lines. Draw a dashed perpendicular line from point A to DE, with the foot of the perpendicular at point 3. Mark angle PAE as 1 and angle EA3 as 2 in red. Label 45° on line segment AE. Shade triangle DFC light blue and triangle A3D green.", "ocr_zh": "作正方形ABCD，取BC中点P，连接PA，将三角形PAB沿着PA翻折得到三角形PAE，连接DE，从C点作CF垂直于DE，垂足为F，交DE延长线于点F，将线段AB和线段BP标为虚线。过A点以虚线作垂线垂直于DE，垂足为3，红色标注角PAE为1、角EA3为2，在线段AE上标注45度，将三角形区域DFC标浅蓝，将三角形区域A3D标绿", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_955.jpg" }, { "index": 956, "ocr_en": "Left diagram: \nIn a plane, take a point O, draw a circle with O as the center, take a point P outside the circle (P is on the right side of the circle). From point P, draw PA tangent to circle O at point A. Connect OP, draw AC perpendicular to OP through point A with the foot at C. Extend AC to intersect the circle at point B, connect BO, and extend BO to intersect the extension of PA at point E (point E is outside the circle). \n\nRight diagram: \nIn a plane, take a point O, draw a circle with O as the center, take a point P outside the circle (P is on the right side of the circle). From point P, draw PA tangent to circle O at point A. Connect OP, draw AC perpendicular to OP through point A with the foot at C. Extend AC to intersect the circle at point B, connect BO, and extend BO to intersect the extension of PA at point E (point E is outside the circle). Mark on the right side of the diagram: \n- First line (in red): Directly indicate: tan∠ABE = 1/2 \n- Second line (in blue): Hint: '1'/2 + '1'/2 = '4'/3 \n- Third line (in blue): ∠PBE = 90° \n- Fourth line (in blue): PB = 3, BE = 4, PE = 5, sin∠E = 3/5", "ocr_zh": "左图：\n在平面内取一点O，以O点为圆心作圆,取圆外一点P（P位于圆的右侧），从P点作PA与圆O相切于A点，连接OP，过A点作AC垂直于OP，垂足为C，延长AC与圆相交于B点,连接BO，延长BO与PA的延长线相交于E点（E点位于圆外）\n右图：\n在平面内取一点O，以O点为圆心作圆,取圆外一点P（P位于圆的右侧），从P点作PA与圆O相切于A点，连接OP，过A点作AC垂直于OP，垂足为C，延长AC与圆相交于B点,连接BO，延长BO与PA的延长线相交于E点（E点位于圆外），在图右侧标注：\n第一行：红色标注直接给出：tan角ABE=1/2\n第二行：蓝色标注提示：‘‘1’’/2+‘‘1’’/2=‘‘4’’/3\n第三行：蓝色标注：角PBE=90度\n第四行：蓝色标注：PB=3，BE=4，PE=5，sin角E=3/5", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_956.jpg" }, { "index": 957, "ocr_en": "Construct a square ABCD, connect AC and BD intersecting at point O. Take a point E on BC and connect AE. Draw BF perpendicular to AE from point B with the foot of the perpendicular at F, then connect OF", "ocr_zh": "作正方形ABCD，连接AC、BD相交于O点，在BC上取一点E，连接AE，过B点作BF垂直于AE垂足为F，连接OF", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_957.jpg" }, { "index": 958, "ocr_en": "Construct a square ABCD. Take a point M on DC and a point N on AB (M is to the left of N), then connect MN. Fold quadrilateral DMNA to the right along MN so that point A falls on BC at point E. Connect AE with a dashed line intersecting MN at point K. Mark line segments DA, DM, and AN as dashed lines. ", "ocr_zh": "作正方形ABCD，在DC上取一点M，在AB上取一点N（M在N的左侧），连接MN，将四边形DMNA沿MN向右翻折，使A点落在BC边上得到点E，虚线连接AE交MN于点K，将线段DA、DM、AN标为虚线", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_958.jpg" }, { "index": 959, "ocr_en": "Construct a square ABCD. Take a point M on DC and a point N on AB (M is to the left of N), then connect MN. Fold quadrilateral DMNA to the right along MN so that point A falls on BC at point E. Connect AE with a dashed line intersecting MN at point K. Mark line segments DA, DM, and AN as dashed lines. Mark DA as 9k, AN as 5k, NB as 4k, BE as 3k, NE as 5k, and CE as 6k in red.", "ocr_zh": "作正方形ABCD，在DC上取一点M，在AB上取一点N（M在N的左侧），连接MN，将四边形DMNA沿MN向右翻折，使A点落在BC边上得到点E，虚线连接AE交MN于点K，将线段DA、DM、AN标为虚线，红色标注DA为9k、AN为5K、NB为4k、BE为3k、NE为5k、CE为6k", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_959.jpg" }, { "index": 960, "ocr_en": "Construct a square ABCD. Take the midpoint N of CD, and take a point M on AD, then connect MB and MN.", "ocr_zh": "作正方形ABCD，取CD的中点N，在AD上取一点M，连接MB、MN", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_960.jpg" }, { "index": 961, "ocr_en": "Construct a trapezoid ABCD with AD parallel to BC, and AD perpendicular to CD at point D. Take a point E on CD, then connect AE and BE.", "ocr_zh": "作梯形ABCD，AD平行于BC，AD垂直于CD垂足为D，在CD上取一点E，连接AE、BE", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_961.jpg" }, { "index": 962, "ocr_en": "Construct an isosceles right triangle ABC with AC equals BC and angle C as the right angle (mark the right angle symbol). Take the midpoint D of segment BC, select a point F on AC and a point E on AB, then connect EF. Fold triangle AEF along EF so that point A coincides with point D. Mark AF and AE as dashed lines.", "ocr_zh": "作等腰直角三角形ABC，AC等于BC，角C为直角，标注直角符号，取线段BC的中点D，在AC上取一点F，在AB上取一点E，连接EF，将三角形AEF沿EF折叠使A点与D点重合，将AF和AE标为虚线", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_962.jpg" }, { "index": 963, "ocr_en": "Construct a square ABCD. Take the midpoint N of CD, and select a point M on AD. Connect MN and BM. Take a point E on MN, and connect BE and BN with red dashed lines. Mark angle ABM as angle 1 in red, angle MBE as angle 2 in red, angle EBN as angle 3 in red, and angle NBC as angle 4 in red. Color the triangular area ABM green, and color the triangular area MBE light blue.", "ocr_zh": "作正方形ABCD，取CD的中点N，在AD上取一点M，连接MN、BM，在MN上取一点E，红虚线连接BE、BN，将角ABM红色标注1，将角MBE红色标注2，将角EBN红色标注3，将角NBC红色标注4，将三角形区域ABM标绿，将三角形区域MBE标浅蓝", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_963.jpg" }, { "index": 964, "ocr_en": "Construct trapezoid ABCD with AD parallel to BC and AD perpendicular to CD at point D. Take a point E on CD, connect AE and BE. Take an auxiliary point M on AE, connect BM with a red dashed line. Draw BH (with H as an auxiliary point) through point B perpendicular to AD, intersecting the extension of AD at point H (both BH and HA are marked as red dashed lines). Color the triangular area MBE light blue, and color the triangular area EBC green.", "ocr_zh": "作梯形ABCD，AD平行于BC，AD垂直于CD，垂足为D，在CD上取一点E,连接AE和BE，在AE上取辅助点M，红色虚线连接BM，过B点作BH（H为辅助点）垂直于AD，交AD延长线于H点，垂足为H（BH和HA均为红色虚线标），将三角形区域MBE标浅蓝，将三角形区域EBC标绿", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_964.jpg" }, { "index": 965, "ocr_en": "Construct a right triangle ABC (with C as an auxiliary point), where angle BAC is the right angle and AB equals AC. Take the midpoint D of BC and connect AD. Select a point E on AC, connect BE, take a point G on BE, and connect AG. Connect DG and extend it to intersect AC at F. Connect DE with a red dashed line. Draw GM through point G perpendicular to DE with a red solid line, with the foot at M (M is an auxiliary point). Draw DN through point D perpendicular to BE with a red solid line, with the foot at N (N is an auxiliary point).", "ocr_zh": "作直角三角形ABC（C为辅助点），角BAC为直角，AB等于AC，取BC的中点D，连接AD，在AC上取一点E，连接BE，在BE上取一点G连接AG，连接DG延长DG交AC于F，红色虚线连接DE，过G点红色实线作GM垂直于DE,垂足为M（M为辅助点），过D点红色实线作DN垂直BE，垂足为N（N为辅助点）", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_965.jpg" }, { "index": 966, "ocr_en": "Construct a right triangle ABC (with C as an auxiliary point), where angle BAC is the right angle and AB equals AC. Take the midpoint D of BC and connect AD. Select a point E on AC, connect BE, take a point G on BE, and connect AG. Connect DG and extend it to intersect AC at F.", "ocr_zh": "作直角三角形ABC（C为辅助点），角BAC为直角，AB等于AC，取BC的中点D，连接AD，在AC上取一点E，连接BE，在BE上取一点G连接AG，连接DG延长DG交AC于F", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_966.jpg" }, { "index": 967, "ocr_en": "Construct an isosceles right triangle ABC with AC = BC and ∠C as the right angle (mark the right-angle symbol). Take the midpoint D of segment BC, select a point F on AC and a point E on AB, then connect EF. Fold triangle AEF along EF so that point A coincides with point D. Mark AF and AE as dashed lines, connect AD with a red dashed line, and draw DH through point D perpendicular to AB with a red dashed line, with the foot at H. Mark on the right side of the diagram: \n- First line: '1'/2 + '1'/3 = 45° \n- Second line: '1'/3 + '1'/3 = '3'/4", "ocr_zh": "作等腰直角三角形ABC，AC等于BC，角C为直角，标注直角符号，取线段BC的中点D，在AC上取一点F，在AB上取一点E，连接EF，将三角形AEF沿EF折叠使A点与D点重合，将AF和AE标为虚线，红色虚线连接AD，过D点以红色虚线作DH垂直于AB，垂足为H，在图右侧标注，第一行：‘‘1’’/2+‘‘1’’/3=45度，第二行：‘‘1’’/3+‘‘1’’/3=‘‘3’’/4", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_967.jpg" }, { "index": 968, "ocr_en": "Left diagram: \nConstruct a right triangle ABC with angle.ACB as the right angle. Draw circle A with point A as the center, where the circle intersects side AB at point D and side AC at point E. Connect DE and extend it to intersect the extension of segment BC at point P. \n\nRight diagram: \nConstruct a right triangle ABD with angle.ADB as the right angle. Draw circle A with point A as the center, where the circle intersects side AB at point C and side AD at point E. Connect CE and extend it to intersect the extension of segment BD at point P.", "ocr_zh": "左图：\n作直角三角形ABC，角ACB为直角，以A点为圆心作圆A，圆A与边AB相交于点D，与 边AC相交于点E，连接DE并延长，与线段BC的延长线交于点P\n右图：\n作直角三角形ABD，角ADB为直角，以A点为圆心作圆A，圆A与边AB相交于点C，与 边AD相交于点E，连接CE并延长，与线段BD的延长线交于点P", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_968.jpg" }, { "index": 969, "ocr_en": "Construct quadrilateral ABCD with angle.ABC as the right angle. Connect AC, take a point E on AC, and connect BE. Draw AM perpendicular to BE through point A with a red dashed line, with the foot at M (M is an auxiliary point). Draw DN perpendicular to AC through point D with a red dashed line, with the foot at N (N is an auxiliary point). The following annotations are all in blue: \n- AD = 2x, AN = x, DN = √3 x, NC = 4x, DC = √19 \n- AM = 2y, BM = y, AB = 3x = √5 y, BC = 4x", "ocr_zh": "作四边形ABCD，角ABC为直角，连接AC，在AC上取一点E，连接BE，过A点红色虚线作AM垂直BE，垂足为M（M为辅助点），过D点红色虚线作DN垂直AC，垂足为N（N为辅助点），以下标注均为蓝色，标注AD为2x、AN为x、DN为根号3 x、NC为4x、DC为根号19、AM为2y、BM为y、AB为3x=根号5 y、BC为4x", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_969.jpg" }, { "index": 970, "ocr_en": "Construct quadrilateral ABCD with angle.ABC as the right angle. Connect AC, take a point E on AC, and connect BE", "ocr_zh": "作四边形ABCD，角ABC为直角，连接AC，在AC上取一点E，连接BE", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_970.jpg" }, { "index": 971, "ocr_en": "Left diagram: \nConstruct a plane rectangular coordinate system. Take a point A on the positive x-axis, a point C on the negative x-axis, and a point B on the positive y-axis. Connect AB and BC. \n\nRight diagram: \nConstruct a plane rectangular coordinate system. Take a point A on the positive x-axis, a point C on the negative x-axis, and a point B on the positive y-axis. Connect AB and BC. Take a point A₁ on segment OC, connect BA₁ with a red dashed line. Mark OA₁ = 3, OB = 4, BA₁ = 5, CA₁ = 5. On the left side of the diagram, mark in blue: \"4\"/3 = \"1\"/2 + \"1\"/2.", "ocr_zh": "左图：\n作平面直角坐标系，在x轴正半轴取一点A，在x轴负半轴取一点C，在y轴正半轴取一点B，连接AB，BC\n右图：\n作平面直角坐标系，在x轴正半轴取一点A，在x轴负半轴取一点C，在y轴正半轴取一点B，连接AB，BC，在线段OC上取一点A1 红色虚线连接BA1 标注OA1为3、OB为4,、BA1为5、CA1为5，在图左侧蓝色标注，‘‘4’’/3=‘‘1’’/2+‘‘1’’/2", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_971.jpg" }, { "index": 972, "ocr_en": "Construct rectangle ABCD, where AD is the length and AB is the width. Connect the diagonal AC, take the midpoint G of AC. Draw the perpendicular bisector of AC through point G, which intersects the line BC at point E, the extension of segment AB at point F, and AD at point H (H is an auxiliary point).", "ocr_zh": "作矩形ABCD，AD为长，AB为宽，连接对角线AC，取AC中点G，过点G作AC垂直平分线交直线BC于点E，交线段AB延长线与点F，交AD于H点（H点为辅助点）", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_972.jpg" }, { "index": 973, "ocr_en": "Construct a plane rectangular coordinate system. Take a point A on the positive x-axis, a point C on the negative x-axis, and a point B on the positive y-axis. Connect AB and BC", "ocr_zh": "作平面直角坐标系，在x轴正半轴取一点A，在x轴负半轴取一点C，在y轴正半轴取一点B，连接AB，BC", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_973.jpg" }, { "index": 974, "ocr_en": "Left diagram: \nConstruct rectangle ABCD with AD as the length and AB as the width. Connect diagonal AC, take midpoint G of AC, and draw the perpendicular bisector of AC through point G, which intersects line BC at point E, the extension of segment AB at point F, and AD at point H (H is an auxiliary point). \n\nRight diagram: \nConstruct rectangle ABCD with AD as the length and AB as the width. Connect diagonal AC, take midpoint G of AC, and draw the perpendicular bisector of AC through point G, which intersects line BC at point E, the extension of segment AB at point F, and AD at point H (H is an auxiliary point). Color the triangular area GEC green and the triangular area BEF light blue. Mark on the right side of the diagram: \"4\"/3 = \"1\"/2 + \"1\"/2.", "ocr_zh": "左图：\n作矩形ABCD，AD为长，AB为宽，连接对角线AC，取AC中点G，过点G作AC垂直平分线交直线BC于点E，交线段AB延长线与点F，交AD于H点（H点为辅助点）\n右图：\n作矩形ABCD，AD为长，AB为宽，连接对角线AC，取AC中点G，过点G作AC垂直平分线交直线BC于点E，交线段AB延长线与点F，交AD于H点（H点为辅助点），将三角形区域GEC标绿，将三角形区域BEF标浅蓝，在图右侧标注：‘‘4’’/3=‘‘1’’/2+‘‘1’’/2", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_974.jpg" }, { "index": 975, "ocr_en": "Construct rectangle ABCD with AD as the length and AB as the width. Connect diagonal AC, take midpoint G of AC, and draw the perpendicular bisector of AC through point G, which intersects line BC at point E, the extension of segment AB at point F, and AD at point H (H is an auxiliary point). Color the triangular area GEC green and the triangular area BEF light blue. Mark on the right side of the diagram: \"4\"/3 = \"1\"/2 + \"1\"/2.", "ocr_zh": "作矩形ABCD，AD为长，AB为宽，连接对角线AC，取AC中点G，过点G作AC垂直平分线交直线BC于点E，交线段AB延长线与点F，交AD于H点（H点为辅助点），将三角形区域GEC标绿，将三角形区域BEF标浅蓝，在图右侧标注：‘‘4’’/3=‘‘1’’/2+‘‘1’’/2", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_975.jpg" }, { "index": 976, "ocr_en": "Construct a plane rectangular coordinate system. Take a point A on the positive x-axis, a point C on the negative x-axis, and a point B on the positive y-axis. Connect AB and BC. Take a point A₁ on segment OC, connect BA₁ with a red dashed line. Mark OA₁ = 3, OB = 4, BA₁ = 5, CA₁ = 5. On the left side of the diagram, mark in blue: \"4\"/3 = \"1\"/2 + \"1\"/2.", "ocr_zh": "作平面直角坐标系，在x轴正半轴取一点A，在x轴负半轴取一点C，在y轴正半轴取一点B，连接AB，BC，在线段OC上取一点A1 红色虚线连接BA1 标注OA1为3、OB为4,、BA1为5、CA1为5，在图左侧蓝色标注，‘‘4’’/3=‘‘1’’/2+‘‘1’’/2", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_976.jpg" }, { "index": 977, "ocr_en": "Construct triangle ADB (with B as an auxiliary point). Draw AC perpendicular to BD through point A, with the foot at C, and mark a gray right-angle symbol. Mark angle DAC as 135° in red, and mark the adjacent supplementary angle of angle DAB as 45° in red. Mark angle CDA, angle CAB, and angle ABC with gray angle symbols. On the left side of the diagram, mark in blue: \"2\" + \"3\" = 135°, and on the right side, mark in blue: \"1/3\" + \"1\"/2 = 45°.", "ocr_zh": "作三角形ADB（B为辅助点），过点A作AC垂直BD，垂足为C，标注灰色直角符号，红色标注角DAC为135度，红色标注角DAB的邻补角为45度，角CDA、角CAB、角ABC标灰色角符号，图左侧蓝色标注‘‘2’’+‘‘3’’=135度，图右侧蓝色标注‘‘1/3’’+‘‘1’’/2=45度", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_977.jpg" }, { "index": 978, "ocr_en": "Left diagram: \nConstruct rectangle ABCD with AD as the length and AB as the width. Connect BD, take points M on BC and N on BD respectively, then connect NM and NC. \n\nRight diagram: \nConstruct rectangle ABCD with AD as the length and AB as the width. Connect BD, take points M on BC and N on BD respectively, and connect NM and NC in blue. Fold triangle BDC along BD to obtain triangle BDC₁, with DC₁ and BC₁ as red dashed lines. Connect CC₁ with red dashed lines, connect NC₁ with blue dashed lines, and connect MC₁ with blue solid lines.", "ocr_zh": "左图：\n作矩形ABCD，AD为长，AB为宽，连接BD，在BC、BD上分别取一点M、N，连接NM、NC\n右图：\n作矩形ABCD，AD为长，AB为宽，连接BD，在BC、BD上分别取一点M、N，蓝色连接NM、NC，将三角形BDC沿BD翻折得到三角形BDC1，DC1、BC1为红色虚线，红色虚线连接CC1、蓝色虚线连接NC1、蓝色实线连接MC1", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_978.jpg" }, { "index": 979, "ocr_en": "Construct rectangle ABCD with AD as the length and AB as the width. Take a point E on DC, fold triangle BCE along BE to obtain triangle BEG (point G is located above AD), and BG intersects AD at point F.", "ocr_zh": "作矩形ABCD，AD为长，AB为宽，在DC上取一点E，将三角形BCE沿BE折叠得到三角形BEG（G点位于AD上方），BG与AD相交于F点", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_979.jpg" }, { "index": 980, "ocr_en": "Left diagram: \nConstruct a right triangle ABC with angle ACB as the right angle, mark a gray right-angle symbol, and mark angle A and angle B with gray angle symbols. Mark on the right side of the diagram: \"2\" + \"1\"/2 = 90°. \n\nMiddle diagram: \nConstruct a right triangle ABC with angle ACB as the right angle, mark a gray right-angle symbol, and mark angle A and angle B with gray angle symbols. Mark on the right side of the diagram: \"3\" + \"1\"/3 = 90°. \n\nRight diagram: \nConstruct a right triangle ABC with angle ACB as the right angle, mark a gray right-angle symbol, and mark angle A and angle B with gray angle symbols. Mark on the right side of the diagram: \"n\" + \"1\"/n = 90°.", "ocr_zh": "左图：\n作直角三角形ABC，角ACB为直角，标注灰色直角符号，角A、角B标注灰色角符号，图右侧标注：‘‘2’’+‘‘1’’/2=90度\n中间图：\n作直角三角形ABC，角ACB为直角，标注灰色直角符号，角A、角B标注灰色角符号，图右侧标注：‘‘3’’+‘‘1’’/3=90度\n右图：\n作直角三角形ABC，角ACB为直角，标注灰色直角符号，角A、角B标注灰色角符号，图右侧标注：‘‘n’’+‘‘1’’/n=90度", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_980.jpg" }, { "index": 981, "ocr_en": "Construct rectangle ABCD with AD as the length and AB as the width. Take a point E on DC, fold triangle BCE along BE to obtain triangle BEG (point G is located above AD), and BG intersects AD at point F. Draw FH perpendicular to BC through point F with a red dashed line, with the foot at H. Mark angle AFB and angle FBC with gray angle symbols. Color the triangular area ABF yellow. Mark on the right side of the diagram in blue: \"1\"/3 + \"1\"/3 = \"3\"/4.", "ocr_zh": "作矩形ABCD，AD为长，AB为宽，在DC上取一点E，将三角形BCE沿BE折叠得到三角形BEG（G点位于AD上方），BG与AD相交于F点，过F红色虚线作FH垂直BC，垂足为H，角AFB、角FBC标注灰色角符号，将三角形区域ABF标黄，在图右侧蓝色标注：‘‘1’’/3+‘‘1’’/3=‘‘3’’/4", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_981.jpg" }, { "index": 982, "ocr_en": "Left diagram: \nAll points below are auxiliary points. Construct rectangle ABCD with AD as the length and AB as the width. Take a point E on AB and a point F on BC, then connect DE, DF, and EF. Draw EH through point E perpendicular to CD with a red solid line, with the foot at H. Mark EB = 1, BF = 1, FC = 3, CH = 1, HD = 2, EH = 4 in red. Color the triangular area EFD green. Mark on the right side of the diagram in blue: 2 = \"1\"/3 + 45°. \n\nRight diagram: \nAll points below are auxiliary points. Construct rectangle ABCD with AD as the length and AB as the width. Take a midpoint E on AB and a point F on BC, then connect DE, DF, and EF. Draw EH through point E perpendicular to CD with a red dashed line, with the foot at H. Mark EB = 1, BF = 1, FC = 2, CH = 1, HD = 1, EH = 3 in red. Color the triangular area EFD light blue. Mark on the right side of the diagram in blue: 3 = \"1\"/2 + 45°.", "ocr_zh": "左图：\n以下点均为辅助点，作矩形ABCD，AD为长，AB为宽，AB上取一点E，BC上取一点F，连接DE、DF、EF，过E点红色实线作EH垂直CD，垂足为H，红色标注EB为1、BF为1、FC为3、CH为1、HD为2、EH为4，将三角形区域EFD标绿，图右侧蓝色标注：2=‘‘1’’/3+45度\n右图：\n以下点均为辅助点，作矩形ABCD，AD为长，AB为宽，AB上取一中点E，BC上取一点F，连接DE、DF、EF，过E点红色虚线作EH垂直CD，垂足为H，红色标注EB为1、BF为1、FC为2、CH为1、HD为1、EH为3，将三角形区域EFD标浅蓝，图右侧蓝色标注：3=‘‘1’’/2+45度", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_982.jpg" }, { "index": 983, "ocr_en": "Left diagram: \nConstruct rectangle ABCD with AD as the length and AB as the width. Connect BD, take points M on BC and N on BD respectively, then connect NM and NC. \n\n\nMiddle diagram: \nConstruct rectangle ABCD with AD as the length and AB as the width. Connect BD, take points M on BC and N on BD respectively, and connect NM and NC in blue. Fold triangle BDC along BD to obtain triangle BDC₁, with DC₁ and BC₁ as red dashed lines. Connect CC₁ with red dashed lines, connect NC₁ with blue dashed lines, and connect MC₁ with blue solid lines. \n\n\nRight diagram: \nConstruct rectangle ABCD with AD as the length and AB as the width. Connect BD, take points M on BC and N on BD respectively, then connect NM and NC. Fold triangle BDC along BD to obtain triangle BDC₁, with DC₁ and BC₁ as red dashed lines. Connect CC₁ and NC₁ with red dashed lines, and connect MC₁ with blue solid line. Draw C₁H through point C₁ perpendicular to BC with a blue solid line, with the foot at H, where C₁H intersects BD at point G (G is an auxiliary point). Connect CG with a blue dashed line. Mark on the left side of the diagram: \n- First line (blue): BH:C₁H:BC₁ = 3:4:5 \n- Second line (blue): CN + MN ≤ C₁M ≤ C₁H = 16 \n- Third line (red): Perpendicular < Folded < Oblique \n- Fourth line (blue): \"1\"/2 + \"1\"/2 = \"4\"/3", "ocr_zh": "左图：\n作矩形ABCD，AD为长，AB为宽，连接BD，在BC、BD上分别取一点M、N，连接NM、NC\n中间图：\n作矩形ABCD，AD为长，AB为宽，连接BD，在BC、BD上分别取一点M、N，蓝色连接NM、NC，将三角形BDC沿BD翻折得到三角形BDC1，DC1、BC1为红色虚线，红色虚线连接CC1、蓝色虚线连接NC1、蓝色实线连接MC1\n右图：\n作矩形ABCD，AD为长，AB为宽，连接BD，在BC、BD上分别取一点M、N，连接NM、NC，将三角形BDC沿BD翻折得到三角形BDC1，DC1、BC1为红色虚线，红色虚线连接CC1、红色虚线连接NC1、蓝色实线连接MC1，过C1点蓝色实线作C1H垂直BC，垂足为H，C1H交BD于G点（G为辅助点），蓝色虚线连接CG，在图左侧标注：\n第一行蓝色标注：BH:C1H:BC1=3:4:5\n第二行蓝色标注：CN+MN小于等于C1M小于等于C1H等于16\n第三行红色标注：垂小于折小于斜\n第四行蓝色标注：‘‘1’’/2+‘‘1’’/2等于‘‘4’’/3", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_983.jpg" }, { "index": 984, "ocr_en": "Construct rectangle ABCD with AD as the length and AB as the width. Connect BD, take points M on BC and N on BD respectively, then connect NM and NC. Fold triangle BDC along BD to obtain triangle BDC₁, with DC₁ and BC₁ as red dashed lines. Connect CC₁ and NC₁ with red dashed lines, and connect MC₁ with blue solid line. Draw C₁H through point C₁ perpendicular to BC with a blue solid line, with the foot at H, where C₁H intersects BD at point G (G is an auxiliary point). Connect CG with a blue dashed line. Mark on the left side of the diagram: \n- First line (blue): BH:C₁H:BC₁ = 3:4:5 \n- Second line (blue): CN + MN ≤ C₁M ≤ C₁H = 16 \n- Third line (red): Perpendicular < Folded < Oblique \n- Fourth line (blue): \"1\"/2 + \"1\"/2 = \"4\"/3", "ocr_zh": "作矩形ABCD，AD为长，AB为宽，连接BD，在BC、BD上分别取一点M、N，连接NM、NC，将三角形BDC沿BD翻折得到三角形BDC1，DC1、BC1为红色虚线，红色虚线连接CC1、红色虚线连接NC1、蓝色实线连接MC1，过C1点蓝色实线作C1H垂直BC，垂足为H，C1H交BD于G点（G为辅助点），蓝色虚线连接CG，在图左侧标注：\n第一行蓝色标注：BH:C1H:BC1=3:4:5\n第二行蓝色标注：CN+MN小于等于C1M小于等于C1H等于16\n第三行红色标注：垂小于折小于斜\n第四行蓝色标注：‘‘1’’/2+‘‘1’’/2等于‘‘4’’/3", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_984.jpg" }, { "index": 985, "ocr_en": "All points below are auxiliary points. Construct rectangle ABCD with AD as the length and AB as the width. Take a midpoint E on AB and a point F on BC, then connect DE, DF, and EF. Draw EH through point E perpendicular to CD with a red dashed line, with the foot at H. Mark EB = 1, BF = 1, FC = 2, CH = 1, HD = 1, EH = 3 in red. Color the triangular area EFD light blue", "ocr_zh": "以下点均为辅助点，作矩形ABCD，AD为长，AB为宽，AB上取一中点E，BC上取一点F，连接DE、DF、EF，过E点红色虚线作EH垂直CD，垂足为H，红色标注EB为1、BF为1、FC为2、CH为1、HD为1、EH为3，将三角形区域EFD标浅蓝", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_985.jpg" }, { "index": 986, "ocr_en": "Construct rectangle ABCD with AD as the length and AB as the width. Connect BD, take points M on BC and N on BD respectively, and connect NM and NC in blue. Fold triangle BDC along BD to obtain triangle BDC₁, with DC₁ and BC₁ as red dashed lines. Connect CC₁ with red dashed lines, connect NC₁ with blue dashed lines, and connect MC₁ with blue solid lines.", "ocr_zh": "作矩形ABCD，AD为长，AB为宽，连接BD，在BC、BD上分别取一点M、N，蓝色连接NM、NC，将三角形BDC沿BD翻折得到三角形BDC1，DC1、BC1为红色虚线，红色虚线连接CC1、蓝色虚线连接NC1、蓝色实线连接MC1", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_986.jpg" }, { "index": 987, "ocr_en": "Construct a square ABCD. Take a point E on CD and connect AE. Fold triangle ADE along AE to obtain triangle AEF. Connect BD, which intersects AE and AF at points M and O respectively. Draw the angle bisector AN of angle BAF, which intersects BD at point N, and extend AN to intersect BC. Connect OE, and mark AD and DE as dashed lines.", "ocr_zh": "作正方形ABCD，在CD上取一点E，连接AE，将三角形ADE沿AE翻折得到三角形AEF，连接BD，分别交AE、AF于点M，O，作∠BAF的角平分线AN交BD于点N，并延长至与BC相交，连接OE，将AD、DE标虚线", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_987.jpg" }, { "index": 988, "ocr_en": "Construct a square ABCD. Take a point E on CD and connect AE. Fold triangle ADE along AE to obtain triangle AEF. Connect BD, which intersects AE and AF at points M and O respectively. Draw the angle bisector AN of angle BAF, which intersects BD at point N, and extend AN to intersect BC at point H (H is an auxiliary point). Connect OE, and mark AD and DE as dashed lines. Draw OP through point O perpendicular to AD with a red dashed line, with the foot at P, and draw OQ through point O perpendicular to AE with a red dashed line, with the foot at Q (P and Q are auxiliary points). Mark in red: AP = 3m, PD = 4m, AO = 5m, OP = 4m, AB = 12, BN = 4. Color the triangular area AND light blue and the triangular area BNH green.", "ocr_zh": "作正方形ABCD，在CD上取一点E，连接AE，将三角形ADE沿AE翻折得到三角形AEF，连接BD，分别交AE、AF于点M，O，作∠BAF的角平分线AN交BD于点N，并延长至与BC相交于点H（H为辅助点），连接OE，将AD、DE标虚线，过O点以红色虚线作OP垂直AD，垂足为P，过O点以红色虚线作OQ垂直AE，垂足为Q，（P、Q为辅助点）以红色标注AP为3m、PD为4m、AO为5m、OP为4m，AB为12，BN为4，将三角形区域AND标浅蓝，将三角形区域BNH标绿", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_988.jpg" }, { "index": 989, "ocr_en": "Construct a plane rectangular coordinate system. Draw a quadratic function opening upward with its vertex in the fourth quadrant, intersecting the negative x-axis at point A and the negative y-axis at point C. Take a point D on the positive x-axis, and connect CD and extend it.", "ocr_zh": "作平面直角坐标系，作开口向上的二次函数，顶点位于第四象限，交x轴负半轴于点A，交y轴负半轴于点C，在x轴正半轴取一点D，连接CD并延长", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_989.jpg" }, { "index": 990, "ocr_en": "Construct a plane rectangular coordinate system. Draw a quadratic function opening upward with its vertex in the fourth quadrant, intersecting the negative x-axis at point A and the negative y-axis at point C. Take a point D on the positive x-axis, connect CD and extend it to point E. Take a point P on the parabola in the fourth quadrant, and connect AP and PD with red dashed lines. Take a point F on the extension of DC, connect FP with a red dashed line such that FP is parallel to the x-axis. Denote the intersection of AP and CD as point G (G is an auxiliary point). Mark angle AGD, angle GDP, and angle DPG with gray angle symbols. Color the triangular area AGD light blue, the triangular area DGP light green, and the triangular area FPG green.", "ocr_zh": "作平面直角坐标系，作开口向上的二次函数，顶点位于第四象限，交x轴负半轴于点A，交y轴负半轴于点C，在x轴正半轴取一点D，连接CD并延长至E，在第四象限的抛物线上取一点P，红色虚线连接AP、PD，在DC的延长线上取一点F，红色虚线连接FP有FP平行于x轴，记AP和CD的交点为G（G为辅助点），将角AGD、角GDP、角DPG标灰色角符号，将三角形区域AGD标浅蓝、三角形区域DGP标浅绿、三角形区域FPG标绿", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_990.jpg" }, { "index": 991, "ocr_en": "Left diagram: \nAll points below are auxiliary points. Construct square ABCD. Take a point E on AD and midpoint F on CD, then connect BE, BF, and EF. Draw EH through point E perpendicular to BC with a red dashed line, with the foot at H. Mark in red: ED equals 1, DF equals 2, FC equals 2, CH equals 1, HB equals 3, EH equals 4. Color the triangular area EFB green and the triangular area EBH light blue. Mark on the right side of the diagram in blue: \"1\"/2 + \"1\"/2 = \"4\"/3. \n\nRight diagram: \nAll points below are auxiliary points. Construct rectangle ABCD with AD as the length and AB as the width. Take a point E on AD and midpoint F on CD, then connect BE, BF, and EF. Draw EH through point E perpendicular to BC with a red dashed line, with the foot at H, where EH intersects BF at point G. Mark in red: ED equals 1, DF equals 3, FC equals 3, CH equals 1, HB equals 8, EH equals 6. Color the triangular area EFB green and the triangular area GBH light blue. Mark on the right side of the diagram in blue: \"1/3\" + \"1\"/3 = \"3\"/4.", "ocr_zh": "左图：\n以下点均为辅助点，作正方形ABCD，AD上取一点E，CD上取中点F，连接BE、BF、EF，过E点红色虚线作EH垂直BC，垂足为H，红色标注ED为1、DF为2、FC为2、CH为1、HB为3、EH为4，将三角形区域EFB标绿，将三角形区域EBH标浅蓝，图右侧蓝色标注：‘‘1’’/2+‘‘1’’/2=‘‘4’’/3\n右图：\n以下点均为辅助点，作矩形ABCD，AD为长，AB为宽，AD上取一点E，CD上取中点F，连接BE、BF、EF，过E点红色虚线作EH垂直BC，垂足为H，EH交BF于点G，红色标注ED为1、DF为3、FC为3、CH为1、HB为8、EH为6，将三角形区域EFB标绿，将三角形区域GBH标浅蓝，图右侧蓝色标注：‘‘1/3‘’+‘‘1’’/3=‘‘3’’/4", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_991.jpg" }, { "index": 992, "ocr_en": "All points below are auxiliary points. Construct square ABCD. Take a point E on AD and midpoint F on CD, then connect BE, BF, and EF. Draw EH through point E perpendicular to BC with a red dashed line, with the foot at H. Mark in red: ED equals 1, DF equals 2, FC equals 2, CH equals 1, HB equals 3, EH equals 4. Color the triangular area EFB green and the triangular area EBH light blue.", "ocr_zh": "以下点均为辅助点，作正方形ABCD，AD上取一点E，CD上取中点F，连接BE、BF、EF，过E点红色虚线作EH垂直BC，垂足为H，红色标注ED为1、DF为2、FC为2、CH为1、HB为3、EH为4，将三角形区域EFB标绿，将三角形区域EBH标浅蓝", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_992.jpg" }, { "index": 993, "ocr_en": "All points below are auxiliary points. Construct rectangle ABCD with AD as the length and AB as the width. Take a point E on AD and midpoint F on CD, then connect BE, BF, and EF. Draw EH through point E perpendicular to BC with a red dashed line, with the foot at H, where EH intersects BF at point G. Mark in red: ED equals 1, DF equals 3, FC equals 3, CH equals 1, HB equals 8, EH equals 6. Color the triangular area EFB green and the triangular area GBH light blue.", "ocr_zh": "以下点均为辅助点，作矩形ABCD，AD为长，AB为宽，AD上取一点E，CD上取中点F，连接BE、BF、EF，过E点红色虚线作EH垂直BC，垂足为H，EH交BF于点G，红色标注ED为1、DF为3、FC为3、CH为1、HB为8、EH为6，将三角形区域EFB标绿，将三角形区域GBH标浅蓝", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_993.jpg" }, { "index": 994, "ocr_en": "Construct square ABCD. Take a point E on DC and connect AE. Fold triangle ADE along AE to obtain triangle AFE. Connect BD, take the midpoint O of BD, connect OF and extend OF to intersect CD at point G. Connect BF and BG.", "ocr_zh": "作正方形ABCD，在DC上取一点E，连接AE，将三角形ADE沿AE翻折得到三角形AFE，连接BD，取BD中点O，连接OF并延长OF交CD于点G，连接BF，BG", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_994.jpg" }, { "index": 995, "ocr_en": "Construct square ABCD. Take a point E on DC and connect AE. Fold triangle ADE along AE to obtain triangle AFE. Connect BD, take the midpoint O of BD, connect OF and extend OF to intersect CD at point G. Connect BF and BG. Connect DF with a red dashed line, extend DF to intersect BC at point K. Draw BH through point B perpendicular to DK with a red dashed line, intersecting the extension of DK at H (the foot of the perpendicular).", "ocr_zh": "作正方形ABCD，在DC上取一点E，连接AE，将三角形ADE沿AE翻折得到三角形AFE，连接BD，取BD中点O，连接OF并延长OF交CD于点G，连接BF，BG，红色虚线连接DF，延长DF交BC于K点，过B点以红色虚线作BH垂直DK交DK延长线于H，垂足为H", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_995.jpg" }, { "index": 996, "ocr_en": "Left diagram: \nConstruct a plane rectangular coordinate system. Draw a parabola opening downward with its vertex in the first quadrant, intersecting the negative x-axis at point A, the positive x-axis at point B, and the positive y-axis at point C. Take a point D on OB, connect CD, and rotate segment CD 90° clockwise about point D to obtain segment DE. Mark point E with coordinates (4, 1) in red. Draw a line through point E perpendicular to the x-axis, with the foot at H, and draw CF through point C perpendicular to this line, with the foot at F marked in red as (4, 3). Take two points G₁ and G₂ on this line (G₁ is below point H, G₂ is above point F), marked in red. Connect DG₂ and DF, and connect DG₁ with a dashed line. Color the triangular area DHE pink and the triangular area DHG₁ green. Mark on the right side of the diagram in blue: \n- First line: \"3\" = \"1\"/2 + 45° \n- Second line: G₂(4, 6) \n- Third line: 45° - \"1\"/3 = \"1\"/2 \n- Fourth line: G₁(4, -3/2)", "ocr_zh": "作平面直角坐标系，作开口向下的抛物线，顶点位于第一象限，抛物线交x轴负半轴于点A，交x轴正半轴于点B，交y轴正半轴于点C，取OB上一点D, 连接CD，将线段CD绕点D 顺时针旋转90°得到线段DE，红色标注E点坐标（4,1），过点E作直线垂直x轴，垂足为H，过点C作CF垂直直线，垂足为F红色标注坐标为（4,3），在直线上取两点G1、G2（G1位于H点下方、G2位于F点上方）（红色标出这两点），连接DG2、DF，虚线连接DG1，将三角形区域DHE标粉，将三角形区域DHG1标绿，在图右侧蓝色标注：\n第一行：‘‘3’’=‘‘1’’/2+45度\n第二行：G2(4,6)\n第三行：45度-‘‘1’’/3=‘‘1’’/2\n第四行：G1(4,-3/2)", "class": "Analytic Geometry", "difficulty": 3, "image_path": "fig_996.jpg" }, { "index": 997, "ocr_en": " First Row Left Diagram: \nConstruct a plane rectangular coordinate system. Draw a parabola opening downward with its vertex in the first quadrant, intersecting the negative x-axis at point A, the positive x-axis at point B, and the positive y-axis at point C. Take a point D on OB, connect CD, rotate segment CD 90 degrees clockwise about point D to obtain segment DE, and connect CE. Mark in blue below the diagram: (1). \n\n\nFirst Row Middle Diagram: \nConstruct a plane rectangular coordinate system. Draw a parabola opening downward with its vertex in the first quadrant, intersecting the negative x-axis at point A, the positive x-axis at point B, and the positive y-axis at point C. Take a point D on OB, connect CD, rotate segment CD 90 degrees clockwise about point D to obtain segment DE, and connect CE. Draw a line through point E perpendicular to the x-axis, with the foot at H, and draw CF through point C perpendicular to this line, with the foot at F. Mark in blue below the diagram: (2). \n\n\nFirst Row Right Diagram: \nConstruct a plane rectangular coordinate system. Draw a parabola opening downward with its vertex in the first quadrant, intersecting the negative x-axis at point A, the positive x-axis at point B, and the positive y-axis at point C. Take a point D on OB, connect CD, rotate segment CD 90 degrees clockwise about point D to obtain segment DE, and connect CE. Draw a line through point E perpendicular to the x-axis, with the foot at H, and draw CF through point C perpendicular to this line, with the foot at F. \n\n\nSecond Row Left Diagram: \nConstruct a plane rectangular coordinate system. Draw a parabola opening downward with its vertex in the first quadrant, intersecting the negative x-axis at point A, the positive x-axis at point B, and the positive y-axis at point C. Take a point D on OB, connect CD, rotate segment CD 90 degrees clockwise about point D to obtain segment DE, and connect CE. Draw a line through point E perpendicular to the x-axis, with the foot at H, and draw CF through point C perpendicular to this line, with the foot at F (marked in blue as (4, 3)). Mark in blue: OC = 3, OD = m, EH = m, EF = 3 - m. Color the triangular area OCD green and the triangular area DHE light blue. Mark on the right side of the diagram in blue: \n- First line: (m + 3, 3) \n- Second line: Solution: m = 1 \n- Third line: 45° - \"1\"/3 = \"1\"/2 \n\n\nSecond Row Right Diagram: \nConstruct a plane rectangular coordinate system. Draw a parabola opening downward with its vertex in the first quadrant, intersecting the negative x-axis at point A, the positive x-axis at point B, and the positive y-axis at point C. Take a point D on OB, connect CD, rotate segment CD 90 degrees clockwise about point D to obtain segment DE. Mark point E with coordinates (4, 1) in red. Draw a line through point E perpendicular to the x-axis, with the foot at H, and draw CF through point C perpendicular to this line, with the foot at F marked in red as (4, 3). Take two points G₁ and G₂ on this line (G₁ is below point H, G₂ is above point F), marked in red. Connect DG₂ and DF, and connect DG₁ with a dashed line. Color the triangular area DHE pink and the triangular area DHG₁ green. Mark on the right side of the diagram in blue: \n- First line: \"3\" = \"1\"/2 + 45° \n- Second line: G₂(4, 6) \n- Third line: 45° - \"1\"/3 = \"1\"/2 \n- Fourth line: G₁(4, -3/2)", "ocr_zh": "第一行左图：\n作平面直角坐标系，作开口向下的抛物线，顶点位于第一象限，抛物线交x轴负半轴于点A，交x轴正半轴于点B，交y轴正半轴于点C，取OB上一点D, 连接CD，将线段CD绕点D 顺时针旋转90°得到线段DE，连接CE，图下方蓝色标注（1）\n第一行中间图：\n作平面直角坐标系，作开口向下的抛物线，顶点位于第一象限，抛物线交x轴负半轴于点A，交x轴正半轴于点B，交y轴正半轴于点C，取OB上一点D, 连接CD，将线段CD绕点D 顺时针旋转90°得到线段DE，连接CE，过点E作直线垂直x轴，垂足为H，过点C作CF垂直直线，垂足为F，图下方蓝色标注（2）\n第一行右图：\n作平面直角坐标系，作开口向下的抛物线，顶点位于第一象限，抛物线交x轴负半轴于点A，交x轴正半轴于点B，交y轴正半轴于点C，取OB上一点D, 连接CD，将线段CD绕点D 顺时针旋转90°得到线段DE，连接CE，过点E作直线垂直x轴，垂足为H，过点C作CF垂直直线，垂足为F\n第二行左图：\n作平面直角坐标系，作开口向下的抛物线，顶点位于第一象限，抛物线交x轴负半轴于点A，交x轴正半轴于点B，交y轴正半轴于点C，取OB上一点D, 连接CD，将线段CD绕点D 顺时针旋转90°得到线段DE，连接CE，过点E作直线垂直x轴，垂足为H，过点C作CF垂直直线，垂足为F（蓝色标注（4,3）），蓝色标注OC为3、OD为m、EH为m、EF为3-m，将三角形区域OCD标绿，三角形区域DHE标浅蓝，在图右侧蓝色标注：\n第一行：（m+3,3）\n第二行：解得：m=1\n第三行：45度-‘‘1’’/3=‘‘1’’/2\n第二行右图：\n作平面直角坐标系，作开口向下的抛物线，顶点位于第一象限，抛物线交x轴负半轴于点A，交x轴正半轴于点B，交y轴正半轴于点C，取OB上一点D, 连接CD，将线段CD绕点D 顺时针旋转90°得到线段DE，红色标注E点坐标（4,1），过点E作直线垂直x轴，垂足为H，过点C作CF垂直直线，垂足为F红色标注坐标为（4,3），在直线上取两点G1、G2（G1位于H点下方、G2位于F点上方）（红色标出这两点），连接DG2、DF，虚线连接DG1，将三角形区域DHE标粉，将三角形区域DHG1标绿，在图右侧蓝色标注：\n第一行：‘‘3’’=‘‘1’’/2+45度\n第二行：G2(4,6)\n第三行：45度-‘‘1’’/3=‘‘1’’/2\n第四行：G1(4,-3/2)", "class": "Analytic Geometry", "difficulty": 3, "image_path": "fig_997.jpg" }, { "index": 998, "ocr_en": "Construct a plane rectangular coordinate system. Draw a parabola opening downward with its vertex in the first quadrant, intersecting the negative x-axis at point A, the positive x-axis at point B, and the positive y-axis at point C. Take a point D on OB, connect CD, rotate segment CD 90 degrees clockwise about point D to obtain segment DE, and connect CE. Draw a line through point E perpendicular to the x-axis, with the foot at H, and draw CF through point C perpendicular to this line, with the foot at F. Mark in blue: OC = 3, OD = m, EH = m, EF = 3 - m. Color the triangular area OCD green and the triangular area DHE light blue", "ocr_zh": "作平面直角坐标系，作开口向下的抛物线，顶点位于第一象限，抛物线交x轴负半轴于点A，交x轴正半轴于点B，交y轴正半轴于点C，取OB上一点D, 连接CD，将线段CD绕点D 顺时针旋转90°得到线段DE，连接CE，过点E作直线垂直x轴，垂足为H，过点C作CF垂直直线，垂足为F，蓝色标注OC为3、OD为m、EH为m、EF为3-m，将三角形区域OCD标绿，三角形区域DHE标浅蓝", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_998.jpg" }, { "index": 999, "ocr_en": "Construct a plane rectangular coordinate system. Draw a parabola opening downward with its vertex in the first quadrant, intersecting the negative x-axis at point A, the positive x-axis at point B, and the positive y-axis at point C. Take a point D on OB, connect CD, rotate segment CD 90 degrees clockwise about point D to obtain segment DE, and connect CE. Mark in blue below the diagram: (1). ", "ocr_zh": "作平面直角坐标系，作开口向下的抛物线，顶点位于第一象限，抛物线交x轴负半轴于点A，交x轴正半轴于点B，交y轴正半轴于点C，取OB上一点D, 连接CD，将线段CD绕点D 顺时针旋转90°得到线段DE，连接CE，图下方蓝色标注（1）", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_999.jpg" }, { "index": 1000, "ocr_en": "Construct a plane rectangular coordinate system. Draw a parabola opening downward with its vertex in the first quadrant, intersecting the negative x-axis at point A, the positive x-axis at point B, and the positive y-axis at point C. Take a point D on OB, connect CD, rotate segment CD 90 degrees clockwise about point D to obtain segment DE, and connect CE. Draw a line through point E perpendicular to the x-axis, with the foot at H, and draw CF through point C perpendicular to this line, with the foot at F. Mark in blue below the diagram: (2)", "ocr_zh": "作平面直角坐标系，作开口向下的抛物线，顶点位于第一象限，抛物线交x轴负半轴于点A，交x轴正半轴于点B，交y轴正半轴于点C，取OB上一点D, 连接CD，将线段CD绕点D 顺时针旋转90°得到线段DE，连接CE，过点E作直线垂直x轴，垂足为H，过点C作CF垂直直线，垂足为F，图下方蓝色标注（2）", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1000.jpg" }, { "index": 1001, "ocr_en": "Construct a plane rectangular coordinate system. Draw a parabola opening downward with its vertex in the first quadrant, intersecting the negative x-axis at point A, the positive x-axis at point B, and the positive y-axis at point C. Take a point D on OB, connect CD, rotate segment CD 90 degrees clockwise about point D to obtain segment DE, and connect CE. Draw a line through point E perpendicular to the x-axis, with the foot at H, and draw CF through point C perpendicular to this line, with the foot at F (marked in blue as (4, 3)). Mark in blue: OC = 3, OD = m, EH = m, EF = 3 - m. Color the triangular area OCD green and the triangular area DHE light blue. Mark on the right side of the diagram in blue: \n- First line: (m + 3, 3) \n- Second line: Solution: m = 1 \n- Third line: 45° - \"1\"/3 = \"1\"/2 ", "ocr_zh": "作平面直角坐标系，作开口向下的抛物线，顶点位于第一象限，抛物线交x轴负半轴于点A，交x轴正半轴于点B，交y轴正半轴于点C，取OB上一点D, 连接CD，将线段CD绕点D 顺时针旋转90°得到线段DE，连接CE，过点E作直线垂直x轴，垂足为H，过点C作CF垂直直线，垂足为F（蓝色标注（4,3）），蓝色标注OC为3、OD为m、EH为m、EF为3-m，将三角形区域OCD标绿，三角形区域DHE标浅蓝，在图右侧蓝色标注：\n第一行：（m+3,3）\n第二行：解得：m=1\n第三行：45度-‘‘1’’/3=‘‘1’’/2", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1001.jpg" }, { "index": 1002, "ocr_en": "Construct a plane rectangular coordinate system. Draw a parabola opening downward with its vertex in the first quadrant, intersecting the negative x-axis at point A, the positive x-axis at point B, and the positive y-axis at point C. Take a point D on OB, connect CD, rotate segment CD 90 degrees clockwise about point D to obtain segment DE, and connect CE. Draw a line through point E perpendicular to the x-axis, with the foot at H, and draw CF through point C perpendicular to this line, with the foot at F. ", "ocr_zh": "作平面直角坐标系，作开口向下的抛物线，顶点位于第一象限，抛物线交x轴负半轴于点A，交x轴正半轴于点B，交y轴正半轴于点C，取OB上一点D, 连接CD，将线段CD绕点D 顺时针旋转90°得到线段DE，连接CE，过点E作直线垂直x轴，垂足为H，过点C作CF垂直直线，垂足为F", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1002.jpg" }, { "index": 1003, "ocr_en": "Construct square ABCD. Take a point E on DC and connect AE. Fold triangle ADE along AE to obtain triangle AFE. Connect BD, take the midpoint O of BD, connect OF and extend OF to intersect CD at point G. Connect BF and BG. Mark angle FAE as 1 in red and angle EAD as 2 in red.", "ocr_zh": "作正方形ABCD，在DC上取一点E，连接AE，将三角形ADE沿AE翻折得到三角形AFE，连接BD，取BD中点O，连接OF并延长OF交CD于点G，连接BF，BG，角FAE红色标注1，角EAD红色标注2", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1003.jpg" }, { "index": 1004, "ocr_en": "Construct square ABCD. Take a point E on DC and connect AE. Fold triangle ADE along AE to obtain triangle AFE. Connect BD, take the midpoint O of BD, connect OF and extend OF to intersect CD at point G. Connect BF and BG. Mark angle FAE as 1 in red and angle EAD as 2 in red. Draw FH through point F perpendicular to BC with a red dashed line, with the foot at H (H is an auxiliary point). Color the triangular area FBH light blue.", "ocr_zh": "作正方形ABCD，在DC上取一点E，连接AE，将三角形ADE沿AE翻折得到三角形AFE，连接BD，取BD中点O，连接OF并延长OF交CD于点G，连接BF，BG，角FAE红色标注1，角EAD红色标注2，过F点红色虚线作FH垂直BC，垂足为H（H为辅助点），将三角形区域FBH标浅蓝", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1004.jpg" }, { "index": 1005, "ocr_en": "Construct square ABCD. Take a point E on DC and connect AE. Fold triangle ADE along AE to obtain triangle AFE. Connect BD, take the midpoint O of BD, connect OF and extend OF to intersect CD at point G. Connect BF and BG. Take a point K on BC, connect AK and FK with red dashed lines. Extend BF with a red dashed line to intersect DC at H. Draw FJ through point F perpendicular to DC with a red dashed line, with the foot at J.", "ocr_zh": "作正方形ABCD，在DC上取一点E，连接AE，将三角形ADE沿AE翻折得到三角形AFE，连接BD，取BD中点O，连接OF并延长OF交CD于点G，连接BF，BG，BC上取一点K，红色虚线连接AK、FK，红色虚线延长BF交DC于H，过F红色虚线作FJ垂直DC，垂足为J", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1005.jpg" }, { "index": 1006, "ocr_en": "Construct square ABCD. Take a point E on DC and connect AE. Fold triangle ADE along AE to obtain triangle AFE. Connect BD, take the midpoint O of BD, connect OF and extend OF to intersect CD at point G. Connect BF and BG. Connect DF with a dashed line. Bolden segments OG and BG. Extend OG to intersect the extension of BC at point P, with GP and CP as red dashed lines. Color the triangular area BCG green and the triangular area GCP light blue.", "ocr_zh": "作正方形ABCD，在DC上取一点E，连接AE，将三角形ADE沿AE翻折得到三角形AFE，连接BD，取BD中点O，连接OF并延长OF交CD于点G，连接BF，BG，虚线连接DF，加粗线段OG、BG，延长OG交BC延长线于P点，GP、CP均为红色虚线，将三角形区域BCG标绿，将三角形区域GCP标浅蓝", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1006.jpg" }, { "index": 1007, "ocr_en": "Construct square ABCD. Take a point E on DC, connect AE and BD, which intersect at point M. Take the midpoint O of BD. Mark points O, M, and E in red.", "ocr_zh": "作正方形ABCD，在DC上取一点E，连接AE、BD相交于M点，取BD的中点O，O、M、E点红色标注", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1007.jpg" }, { "index": 1008, "ocr_en": "Construct a square ABCD. Take a point E on DC. Connect AE and BD, which intersect at point M. Take the midpoint O of BD. Fold triangle ADE along AE to obtain triangle AFE. Connect DF with a red dashed line, which intersects AE at point N. Connect OF and extend it to intersect CD at point G. Mark the points M, N, E, O, F, and G in red.", "ocr_zh": "作正方形ABCD，在DC上取一点E，连接AE、BD相交于M点，取BD的中点O，将三角形ADE沿AE翻折得到三角形AFE，红色虚线连接DF交AE于N点，连接OF并延长交CD于G点，将M、N、E、O、F、G点标红", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1008.jpg" }, { "index": 1009, "ocr_en": "Construct square ABCD. Take a point E on DC. Connect AE and BD, intersecting at point M. Take the midpoint O of BD. Fold triangle ADE along AE to obtain triangle AFE. Connect DF with a red dashed line, intersecting AE at point N. Connect OF and extend it to intersect CD at point G. Connect BF and BG. Mark points M, N, E, O, F, G in red, and mark angle DAE in red with the label \"1\".", "ocr_zh": "作正方形ABCD，在DC上取一点E，连接AE、BD相交于M点，取BD的中点O，将三角形ADE沿AE翻折得到三角形AFE，红色虚线连接DF交AE于N点，连接OF并延长交CD于G点，连接BF、BG，将M、N、E、O、F、G点标红，将角DAE红色标注1", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1009.jpg" }, { "index": 1010, "ocr_en": "Construct triangle ABC with side BC horizontal and angle BAC as an obtuse angle. Draw AD perpendicular to BC through point A, with the foot of the perpendicular at D. Mark a gray right-angle symbol there. Label AB as √10, AC as √5, AD as 1, BD as 3, and DC as 2 in red.", "ocr_zh": "作三角形ABC，边BC水平，角BAC为钝角，过A点作AD垂直BC，垂足为D，标注灰色直角符号，红色标注AB为根号10、AC为根号5、AD为1、BD为3、DC为2", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1010.jpg" }, { "index": 1011, "ocr_en": "Construct triangle ACB with side CB horizontal and angle as an acute angle. Draw AD perpendicular to BC through point A, with the foot of the perpendicular at D. Mark a right-angle symbol. Label AC as 2√10, CD as 2, DB as 3, AD as 6, and AB as 3√5 in red. In the upper-left corner of the figure, label \"236''\" in red. (2 is incomplete.)", "ocr_zh": "作三角形ACB，边CB水平，角BAC为锐角，过A点作AD垂直BC，垂足为D，标注直角符号，红色标注AC为2根号10、CD为2、DB为3、AD为6、AB为3根号5，图左上角红色标注236‘’（2不完整）", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1011.jpg" }, { "index": 1012, "ocr_en": "Left figure: \nConstruct a plane rectangular coordinate system. Draw a linear function that passes through the first, second, and third quadrants, intersecting the negative x-axis at point C and the positive y-axis at point D. Take two points A and B on the linear function (point A is in the third quadrant, point B is in the first quadrant), and connect OA and OB. \n\n\nMiddle figure: \nConstruct a plane rectangular coordinate system. Draw a linear function that passes through the first, second, and third quadrants, intersecting the negative x-axis at point C and the positive y-axis at point D. Take two points A and B on the linear function (point A is in the third quadrant, point B is in the first quadrant), and connect OA and OB. Draw a dashed line from point A perpendicular to the x-axis, with the foot of the perpendicular at point M (M is an auxiliary point). Draw a dashed line from point B perpendicular to the y-axis, with the foot of the perpendicular at point N (N is an auxiliary point). Color the triangular region MAO green and the triangular region NBO light blue. In the upper-left corner of the figure, label in blue: ''1/3'' + ''1''/2 = 45°. \n\n\nRight figure: \nConstruct a plane rectangular coordinate system. Draw a linear function that passes through the first, second, and third quadrants, intersecting the negative x-axis at point C and the positive y-axis at point D. Take two points A and B on the linear function (point A is in the third quadrant, point B is in the first quadrant), and connect OA and OB. Draw a line segment BE from point B to the right, parallel to the x-axis. From point E, draw EH perpendicular to the x-axis, with the foot of the perpendicular at H. Take a point F on EH, and label the coordinates of F as (2 + m, 3 - 2m) in red, EF as 2m in red, and BE as m in red. Color the triangular region BEF green and the triangular region OFH light blue. In the upper-right corner of the figure, label in blue: \nFirst line: (3 - 2m)/(2 + m) = 1/2, m = 1 \nSecond line: △OFH is congruent to △FBE.", "ocr_zh": "左图：\n作平面直角坐标系，作一次函数（一次函数经过一二三象限），并且交x轴负半轴于点C，交y轴正半轴于点D，在一次函数上取两点A、B（A点位于第三象限、B点位于第一象限），连接OA、OB\n中间图：\n作平面直角坐标系，作一次函数（一次函数经过一二三象限），并且交x轴负半轴于点C，交y轴正半轴于点D，在一次函数上取两点A、B（A点位于第三象限、B点位于第一象限），连接OA、OB，过A点虚线作AM垂直x轴，垂足为M点（M为辅助点），过B点虚线作BN垂直y轴，垂足为N点（N为辅助点），将三角形区域MAO标绿，将三角形区域NBO标浅蓝，在图左上方蓝色标注：‘‘1/3’’+‘‘1’’/2=45度\n右图：\n作平面直角坐标系，作一次函数（一次函数经过一二三象限），并且交x轴负半轴于点C，交y轴正半轴于点D，在一次函数上取两点A、B（A点位于第三象限、B点位于第一象限），连接OA、OB，过B点向右作线段BE平行于x轴，过E点作EH垂直x轴，垂足为H，在EH上取一点F，红色标注F坐标为（2+m，3-2m），红色标注EF为2m，BE为m，将三角形区域BEF标绿，将三角形区域OFH标浅蓝，在图右上方蓝色标注：第一行：（3-2m）/（2+m）=1/2，m=1 第二行：三角形OFH全等于三角形FBE", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1012.jpg" }, { "index": 1013, "ocr_en": "Left figure: \nConstruct an isosceles right triangle ABC with angle BAC as the right angle, mark the right-angle symbol, where AB equals AC. Take the midpoint D of AC, connect BD, and draw CE perpendicular to BD through point C, intersecting the extension of BD at point E (mark the right-angle symbol at the foot E). Label \"(1)\" below the figure. \n\n\nRight figure: \nConstruct an isosceles right triangle ABC with angle BAC as the right angle, mark the right-angle symbol, where AB equals AC. Take a point D on AC (with AD > DC), connect BD, and draw CE perpendicular to BD through point C, intersecting the extension of BD at point E (mark the right-angle symbol at the foot E). Label \"(2)\" below the figure.", "ocr_zh": "左图：\n作等腰直角三角形ABC，角BAC为直角，标注直角符号，AB等于AC，取AC中点D，连接BD，过点C作CE垂直BD，交BD延长线于点E，垂足为E（标注直角符号），图下方标注（1）\n右图：\n作等腰直角三角形ABC，角BAC为直角，标注直角符号，AB等于AC，取AC上一点D（AD大于DC），连接BD，过点C作CE垂直BD，交BD延长线于点E，垂足为E（标注直角符号），图下方标注（2）", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1013.jpg" }, { "index": 1014, "ocr_en": "Construct a plane rectangular coordinate system. Draw a parabola opening upwards with its vertex P located in the third quadrant. Draw a line through point P that passes through the first, third, and fourth quadrants, intersecting the parabola at point Q (Q is to the right of P). The parabola intersects the x-axis at points A and B (A is to the left of B, both on the negative x-axis) and intersects the positive y-axis at point C. Connect AC, AQ, and AP. Take a point M on the parabola between A and P, and connect MC.", "ocr_zh": "作平面直角坐标系，作开口向上的抛物线，抛物线顶点P位于第三象限，过P点作直线经过第一三四象限，直线与抛物线相交于Q点（Q点位于P点右侧），抛物线与x轴相交于A、B两点（A点在B点左侧，且两点均位于x轴负半轴），抛物线与y轴正半轴相交于点C，连接AC、AQ、AP，在抛物线上AP之间取一点M、连接MC", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1014.jpg" }, { "index": 1015, "ocr_en": "Left figure: \nConstruct a plane rectangular coordinate system. Draw a line passing through the first, third, and fourth quadrants, intersecting the positive x-axis at point B and the negative y-axis at point C. Draw a parabola opening downward through points B and C (with the vertex in the first quadrant), and the parabola intersects the positive x-axis at another point A. \n\n\nMiddle figure: \nConstruct a plane rectangular coordinate system. Draw a line passing through the first, third, and fourth quadrants, intersecting the positive x-axis at point B and the negative y-axis at point C. Draw a parabola opening downward through points B and C (with the vertex in the first quadrant), and the parabola intersects the positive x-axis at another point A. Take a point D on line segment CB and a point P on the parabola to the right of the axis of symmetry, then connect PD. \n\n\nRight figure: \nConstruct a plane rectangular coordinate system. Draw a line passing through the first, third, and fourth quadrants, intersecting the positive x-axis at point B and the negative y-axis at point C. Draw a parabola opening downward through points B and C (with the vertex in the first quadrant), and the parabola intersects the positive x-axis at another point A. Take a point D on line segment CB and a point P on the parabola to the right of the axis of symmetry, then connect PD.", "ocr_zh": "左图：\n作平面直角坐标系，作过第一三四象限的直线，交x轴正半轴于点B，交y轴负半轴于点C，经过点B、C作开口向下的抛物线（顶点位于第一象限），抛物线与x轴的正半轴交于另一点A\n中间图：\n作平面直角坐标系，作过第一三四象限的直线，交x轴正半轴于点B，交y轴负半轴于点C，经过点B、C作开口向下的抛物线（顶点位于第一象限），抛物线与x轴的正半轴交于另一点A，在线段CB上取一点D，在对称轴的右侧抛物线上取一点P，连接PD\n右图：\n作平面直角坐标系，作过第一三四象限的直线，交x轴正半轴于点B，交y轴负半轴于点C，经过点B、C作开口向下的抛物线（顶点位于第一象限），抛物线与x轴的正半轴交于另一点A，在线段CB上取一点D，在对称轴的右侧抛物线上取一点P，连接PD", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1015.jpg" }, { "index": 1016, "ocr_en": "Construct a plane rectangular coordinate system. Draw a parabola opening upwards with vertex E in the fourth quadrant, and draw the axis of symmetry of the parabola through vertex E. The parabola intersects the negative x-axis at point A, the positive x-axis at point B, and the negative y-axis at point C. Connect BC, BE, and CE. Draw a line through point B that passes through the first, second, and fourth quadrants, intersecting the positive y-axis at point D.", "ocr_zh": "作平面直角坐标系，作开口向上的抛物线，顶点E位于第四象限，过顶点E作抛物线对称轴，抛物线与x轴负半轴交于A，与x轴正半轴交于B点，与y轴负半轴交于点C，连接BC、BE、CE，过B点作经过一二四象限的直线，交y轴正半轴于点D", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1016.jpg" }, { "index": 1017, "ocr_en": "Construct a right triangle ABC with angle ACB as the right angle. Take a point O inside the triangle, and draw the incircle of triangle ABC with center O using a dashed line. The circle O is tangent to side AC at point E, to side BC at point F, and to AB at point D. Connect OA and OB. Label AE as 3, EC as 1, CF as 1, FB as 2, BD as 2, DA as 3, OF as 1, and OE as 1.", "ocr_zh": "作直角三角形ABC，角ACB为直角，在三角形内取一点O，以O点为圆心，以虚线作三角形ABC的内切圆，圆O和边AC相切于E点、和边BC相切于F点、和AB相切于D点，连接OA、OB，标注AE为3、EC为1、CF为1、FB为2、BD为2、DA为3、OF为1、OE为1", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1017.jpg" }, { "index": 1018, "ocr_en": "Left figure: \nConstruct an isosceles right triangle with angle CBA as the right angle, where BC equals BA. Take a point D on AC, connect BD, draw CE perpendicular to BD with the foot at E, and draw AF perpendicular to the extension of BD with the foot at F. Label \"(1)\" below the figure. \n\n\nRight figure: \nConstruct an isosceles right triangle with angle CBA as the right angle, where BC equals BA. Take a point D on AC, connect BD, draw CE perpendicular to BD with the foot at E. Take the midpoint O of line segment CE, draw OM perpendicular to AC with the foot at M, and extend MO to intersect BC at point N. Label \"(2)\" below the figure.", "ocr_zh": "左图：\n作等腰直角三角形，角CBA为直角，BC等于BA，AC上取一点D，连接BD，作CE垂直BD，垂足为E，作AF垂直BD交BD延长线于F，垂足为F，图下方标注（1）\n右图：\n作等腰直角三角形，角CBA为直角，BC等于BA，AC上取一点D，连接BD，作CE垂直BD，垂足为E，取线段CE的中点O，过O点作OM垂直AC，垂足为M，延长MO交BC于N点，图下方标注（2）", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1018.jpg" }, { "index": 1019, "ocr_en": "Construct square ABCD. Connect AC and BD, intersecting at point O. Take a point Q on the extension of BC, connect AQ, which intersects BD at point E and CD at point P. Connect OQ, which intersects CD at point F. Connect EF.", "ocr_zh": "作正方形ABCD，连接AC、BD相交于O点，BC延长线上取一点Q，连接AQ，交BD于点E，交CD点P，连接OQ，交CD点F，连接EF", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1019.jpg" }, { "index": 1020, "ocr_en": "Construct a plane rectangular coordinate system. Take a point A on the negative x-axis and a point B on the positive y-axis, then connect AB. Take a point C in the first quadrant and connect BC, AC. Draw an inverse proportional function in the first quadrant passing through point C.", "ocr_zh": "作平面直角坐标系，在x轴负半轴取一点A，在y轴正半轴取一点B，连接AB，在第一象限取一点C，连接BC、AC，过C点在第一象限作反比例函数", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1020.jpg" }, { "index": 1021, "ocr_en": "Construct a plane rectangular coordinate system. Take a point M in the first quadrant, and draw circle M with center M (the circle is located in the first quadrant). Take a point A on the positive y-axis.", "ocr_zh": "作平面直角坐标系，在第一象限取一点M，以M为圆心作圆M（圆位于第一象限），在y轴正半轴取一点A", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1021.jpg" }, { "index": 1022, "ocr_en": "Construct an isosceles triangle ABC with AB equals AC and angle BAC as an acute angle. Take the midpoint M of line segment BC and connect AM. Draw CD perpendicular to BC through point C (with the foot at C), where point D is directly above point C. Connect DA and DB.", "ocr_zh": "作等腰三角形ABC，AB等于AC，角BAC为锐角，取线段BC中点M，连接AM，过C点作CD垂直BC，垂足为C，D点位于C点正上方，连接DA、DB", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1022.jpg" }, { "index": 1023, "ocr_en": "Construct a plane rectangular coordinate system. Draw a parabola opening downward with its vertex in the first quadrant. The parabola intersects the positive x-axis at point B, the negative x-axis at point A, and the positive y-axis at point C. Draw a line through point C, which intersects the parabola at another point D. Take a point P on the parabola to the right of the y-axis (located to the left of the axis of symmetry of the parabola), draw PE perpendicular to the x-axis with the foot at E, and PE intersects CD at point F. Take a point M on the parabola between D and B.", "ocr_zh": "作平面直角坐标系，作开口向下的抛物线，顶点位于第一象限，抛物线与x轴正半轴交于B点、与x轴负半轴交于A点、与y轴正半轴交于点C，过C点作直线，直线与抛物线相交于另一点D，在y轴右侧抛物线上取一点P（位于抛物线对称轴左侧），过点P作PE垂直x轴垂足为E，交CD于点F，在抛物线上DB之间取一点M", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1023.jpg" }, { "index": 1024, "ocr_en": "Construct triangle ABC with side AB horizontal. Take two points D and E inside triangle ABC, and connect DE, CD, CE, AD, and BE. Here, AD bisects angle CAB, BE bisects angle CBA, and DE is parallel to AB.", "ocr_zh": "作三角形ABC，边AB水平，取三角形ABC内两点D、E，连接DE、CD、CE、AD、BE，有AD平分角CAB，BE平分角CBA，DE平行AB", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1024.jpg" }, { "index": 1025, "ocr_en": "Construct a plane rectangular coordinate system. Draw a line passing through the first, third, and fourth quadrants, intersecting the positive x-axis at point A and the negative y-axis at point B. Draw a parabola opening upwards through points A and B, with its vertex located in the fourth quadrant. Label \"F\" above the parabola and \"M\" on the right side of the parabola.", "ocr_zh": "作平面直角坐标系，作直线经过第一三四象限，交x轴正半轴于点A、交y轴负半轴于点B，经过A、B点作开口向上的抛物线，抛物线顶点位于第四象限，抛物线上方标注F，抛物线右侧标注M", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1025.jpg" }, { "index": 1026, "ocr_en": "Construct a right triangle ABC with side AB horizontal and angle ACB as the right angle. Take a point D on BC and connect AD.", "ocr_zh": "作直角三角形ABC，边AB水平，角ACB为直角，，在BC上取一点D，连接AD", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1026.jpg" }, { "index": 1027, "ocr_en": "Construct a plane rectangular coordinate system. Draw a parabola opening downward with its vertex in the first quadrant, intersecting the negative x-axis at point A and the positive x-axis at point B. Draw a line passing through the first, second, and fourth quadrants, intersecting the positive y-axis at point C and the positive x-axis at point D. Take a point P on the parabola in the first quadrant, connect CP, draw PF perpendicular to the x-axis from point P with the foot at F, and PF intersects line CD at point E.", "ocr_zh": "作平面直角坐标系，作开口向下的抛物线，抛物线顶点位于第一象限，抛物线与x轴负半轴交于A点、与x轴正半轴交于B两点，作经过一二四象限的直线，与y轴正半轴交于点C，与x轴正半轴交于点D，在第一象限抛物线上取一点P，连接CP，过点P作PF垂直x轴，垂足为F，交直线CD于点E", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1027.jpg" }, { "index": 1028, "ocr_en": "Construct an isosceles triangle ABC with AB equals AC and angle BAC as an acute angle. Draw a ray AO through point A perpendicular to BC, with the foot at O. Take a point D on AO (point D is inside triangle ABC) and connect CD.", "ocr_zh": "作等腰三角形ABC，AB等于AC，角BAC为锐角，过A点作射线AO垂直于BC，垂足为O，在AO上取一点D（D点在三角形ABC内），连接CD", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1028.jpg" }, { "index": 1029, "ocr_en": "Construct a rhombus ABCD with AD parallel to BC and AB parallel to DC, where angle ABC is an obtuse angle. Connect AC, take a point P on line segment AC, and connect PD and PB.", "ocr_zh": "作菱形ABCD，AD平行于BC、AB平行于DC，角ABC为钝角，连接AC，在线段AC上取一点P，连接PD、PB", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1029.jpg" }, { "index": 1030, "ocr_en": "Construct a plane rectangular coordinate system. Draw a parabola opening upwards with its vertex in the fourth quadrant. The parabola intersects the positive x-axis at point C, the negative x-axis at point A, and the negative y-axis at point B. Draw the axis of symmetry of the parabola, which intersects the positive x-axis at point D.", "ocr_zh": "作平面直角坐标系，作开口向上的抛物线，抛物线顶点位于第四象限，抛物线交x轴正半轴于点C、交x轴负半轴于点A、交y轴负半轴于点B，作抛物线对称轴交x轴正半轴于点D", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1030.jpg" }, { "index": 1031, "ocr_en": "Construct a plane rectangular coordinate system. Draw a parabola opening upwards with its vertex in the fourth quadrant. The parabola intersects the positive x-axis at point B, the negative x-axis at point A, and the negative y-axis at point C. Connect AC and BC. Draw a line through point B that passes through the first, second, and fourth quadrants, intersecting the parabola at another point D.", "ocr_zh": "作平面直角坐标系，作开口向上的抛物线，抛物线顶点位于第四象限，抛物线交x轴正半轴于点B、交x轴负半轴于点A、交y轴负半轴于点C，连接AC、BC，过B点作经过第一二四象限的直线与抛物线交于另一点D", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1031.jpg" }, { "index": 1032, "ocr_en": "Left figure: \nConstruct a plane rectangular coordinate system. Draw a parabola opening upwards with its vertex in the fourth quadrant. The parabola intersects the positive x-axis at point C, the negative x-axis at point A, and the negative y-axis at point B. Draw the axis of symmetry of the parabola, which intersects the positive x-axis at point D. \n\n\nRight figure: \nConstruct a plane rectangular coordinate system. Draw a parabola opening upwards with its vertex in the fourth quadrant. The parabola intersects the positive x-axis at point B, the negative x-axis at point A, and the negative y-axis at point C. Connect AC and BC. Draw a line through point B that passes through the first, second, and fourth quadrants, intersecting the parabola at another point D.", "ocr_zh": "左图：\n作平面直角坐标系，作开口向上的抛物线，抛物线顶点位于第四象限，抛物线交x轴正半轴于点C、交x轴负半轴于点A、交y轴负半轴于点B，作抛物线对称轴交x轴正半轴于点D\n右图：\n作平面直角坐标系，作开口向上的抛物线，抛物线顶点位于第四象限，抛物线交x轴正半轴于点B、交x轴负半轴于点A、交y轴负半轴于点C，连接AC、BC，过B点作经过第一二四象限的直线与抛物线交于另一点D", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1032.jpg" }, { "index": 1033, "ocr_en": "Upper figure: \nDraw line segment MN. Take a point B above MN, and draw BC perpendicular to MN through point B with the foot at C. Mark a gray right-angle symbol. Take a point A on line segment MC, and label AC as m and CB as n. \n\nLower figure: \nDraw line segment MN. Take a point B above MN, and draw BC perpendicular to MN through point B with the foot at C. Mark a gray right-angle symbol. Take a point A on line segment MC, and label AC as m and CB as n. Take two points Q’ and Q from left to right on line segment AC. Connect BQ’ and BQ. Draw a line l through point A as a dashed line. Draw Q’P’ perpendicular to line l through point Q’ with the foot at P’ (mark a gray right-angle symbol), and draw QP perpendicular to line l through point Q as a dashed line with the foot at P (mark a gray right-angle symbol). Both points P and P’ are located below line segment MN. Label angle PAN as α, mark BQ’ in red as y, and mark AQ’ in red as x.", "ocr_zh": "上图：\n作线段MN，在MN上方取一点B，过B点作BC垂直MN，垂足为C，标注灰色直角符号，在线段MC上取一点A，标注AC为m、CB为n\n下图：\n作线段MN，在MN上方取一点B，过B点作BC垂直MN，垂足为C，标注灰色直角符号，在线段MC上取一点A，标注AC为m、CB为n，在线段AC上从左往右取两点Q’、Q，连接BQ’、BQ，过点A以虚线作直线l，过Q’点作Q’P’垂直直线l，垂足为P’（标注灰色直角符号），过Q点虚线作QP垂直直线l，垂足为P（标注灰色直角符号），P、P’点均位于线段MN下方，将角PAN标注α，将BQ’红色标注y、AQ’红色标注x", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1033.jpg" }, { "index": 1034, "ocr_en": "Draw line segment MN. Take a point B above MN, and draw BC perpendicular to MN through point B with the foot at C. Mark a gray right-angle symbol. Take a point A on line segment MC, and label AC as m and CB as n. ", "ocr_zh": "作线段MN，在MN上方取一点B，过B点作BC垂直MN，垂足为C，标注灰色直角符号，在线段MC上取一点A，标注AC为m、CB为n", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1034.jpg" }, { "index": 1035, "ocr_en": "Draw line segment MN. Take a point B above MN, and draw BC perpendicular to MN through point B with the foot at C. Mark a gray right-angle symbol. Take a point A on line segment MC, and label AC as m and CB as n. Take two points Q’ and Q from left to right on line segment AC. Connect BQ’ and BQ. Draw a line l through point A as a dashed line. Draw Q’P’ perpendicular to line l through point Q’ with the foot at P’ (mark a gray right-angle symbol), and draw QP perpendicular to line l through point Q as a dashed line with the foot at P (mark a gray right-angle symbol). Both points P and P’ are located below line segment MN. Label angle PAN as α, mark BQ’ in red as y, and mark AQ’ in red as x.", "ocr_zh": "作线段MN，在MN上方取一点B，过B点作BC垂直MN，垂足为C，标注灰色直角符号，在线段MC上取一点A，标注AC为m、CB为n，在线段AC上从左往右取两点Q’、Q，连接BQ’、BQ，过点A以虚线作直线l，过Q’点作Q’P’垂直直线l，垂足为P’（标注灰色直角符号），过Q点虚线作QP垂直直线l，垂足为P（标注灰色直角符号），P、P’点均位于线段MN下方，将角PAN标注α，将BQ’红色标注y、AQ’红色标注x", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1035.jpg" }, { "index": 1036, "ocr_en": "Draw line segment MN. Take a point A above MN, and draw AC perpendicular to MN through point A with the foot at C. Mark a gray right-angle symbol. Take a point B on line segment MC. Take two points P and Pmin from left to right on line segment BC. Connect AP and APmin. Draw a line l through point B as a dashed line. Draw PQ perpendicular to line l through point P with the foot at Q (mark a gray right-angle symbol), and draw PminQmin perpendicular to line l through point Pmin as a dashed line with the foot at Qmin (mark a gray right-angle symbol). Both points Q and Qmin are located below line segment MN. Label angle QBN as α, and mark \"sinα = n\" in red above point B.", "ocr_zh": "作线段MN，在MN上方取一点A，过A点作AC垂直MN，垂足为C，标注灰色直角符号，在线段MC上取一点B，在线段BC上从左往右取两点P、Pmin，连接AP、APmin，过点B以虚线作直线l，过P点作PQ垂直直线l，垂足为Q（标注灰色直角符号），过Pmin点虚线作PminQmin垂直直线l，垂足为Qmin（标注灰色直角符号），Q、Qmin点均位于线段MN下方，将角QBN标注α，在B点上方红色标注sinα=n", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1036.jpg" }, { "index": 1037, "ocr_en": "Construct a right triangle ABC with angle ACB as the right angle. Label BC as k. Draw a dashed line l through point C (where line l intersects side AB). Draw BD perpendicular to line l as a red dashed line through point B, with the foot at D. Draw AE perpendicular to line l as a red dashed line through point A, with the foot at E (mark the right-angle symbol). Label angle ACD as α in red.", "ocr_zh": "作直角三角形ABC，角ACB为直角，BC标注为k，过C点虚线作直线l（直线l与边AB相交），过B点红色虚线作BD垂直直线l，垂足为D，过A点红色虚线作AE垂直直线l，垂足为E（标注直角符号），红色标注角ACD为α", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1037.jpg" }, { "index": 1038, "ocr_en": "Construct triangle ABC with side BC horizontal. Draw AH perpendicular to BC through point A, with the foot at H.", "ocr_zh": "作三角形ABC，边BC水平，过A点作AH垂直BC，垂足为H", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1038.jpg" }, { "index": 1039, "ocr_en": "Construct triangle ABC with side BC horizontal. Draw AH perpendicular to BC through point A, with the foot at H. Draw a horizontal line l as a dashed line through point A. Construct point C’, the symmetric point of point C with respect to line l. Connect AC’. Draw C’B’ through point C’ parallel to BC (B’ for auxiliary description). Extend HA as a dashed line to intersect B’C’ at point H’ (H’ for auxiliary description). Label BH as x, HC as 2−x, AH as 1, and C’H’ as 2−x in red. Label angle BAH as α and angle C’AH’ as β.", "ocr_zh": "作三角形ABC，边BC水平，过A点作AH垂直BC，垂足为H，过A点虚线作水平直线l，作C点关于直线l的对称点C’，连接AC’，过C’作C’B’平行于BC（B’为辅助描述所加），虚线延长HA交B’C’于点H’（ H’为辅助描述所加），红色标注BH为x、HC为2-x、AH为1、C’H’为2-x，角BAH为α、角C’AH’为β", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1039.jpg" }, { "index": 1040, "ocr_en": "Take a point O in the plane, draw a circle with center O. Take two points A and B outside circle O (point A is on the upper side of the circle, point B is on the left side of the circle). Take a point P on the circle (point P is on the upper-left semicircle). Connect OB, BP, and PA.", "ocr_zh": "平面取一点O，以O点为圆心作圆，在圆O外取两点A、B（A点位于圆上侧，B点位于圆左侧），在圆上取一点P（P点位于左上半圆），连接OB、BP、PA", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1040.jpg" }, { "index": 1041, "ocr_en": "Take a point O in the plane, draw a circle with center O. Take two points A and B outside circle O (point A is on the upper side of the circle, point B is on the left side of the circle). Take a point P on the circle (point P is on the upper-left semicircle). Connect OB, BP, and PA. Take a point C on line segment OB (point C is inside the circle), and connect CP. Shade the triangular region OCP in pink.", "ocr_zh": "平面取一点O，以O点为圆心作圆，在圆O外取两点A、B（A点位于圆上侧，B点位于圆左侧），在圆上取一点P（P点位于左上半圆），连接OB、BP、PA，线段OB上取一点C，连接CP（C点位于圆内），将三角形区域OCP标粉", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1041.jpg" }, { "index": 1042, "ocr_en": "Take a point O in the plane and draw a circle with center O. Take two points A and B outside circle O (point A is on the upper side of the circle, point B is on the left side of the circle). Take a point P on the circle (point P is on the upper-left semicircle). Connect OB and BP. Connect AP with a red solid line and extend it to intersect OB at point C (point C is inside the circle).", "ocr_zh": "平面取一点O，以O点为圆心作圆，在圆O外取两点A、B（A点位于圆上侧，B点位于圆左侧），在圆上取一点P（P点位于左上半圆），连接OB、BP，红色实线连接AP并延长交OB于C点（C点位于圆内）", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1042.jpg" }, { "index": 1043, "ocr_en": "Upper figure: \nConstruct triangle BPO with side BO horizontal and angle BPO as an obtuse angle. Take a point C on line segment BO, connect CP, and shade the triangular region CPO in pink. \n\nLower figure: \nConstruct triangle BPO with side BO horizontal and angle BPO as an obtuse angle. Take a point C on line segment BO, connect CP, and shade the triangular region BPO in green.", "ocr_zh": "上图：\n作三角形BPO，边BO水平，角BPO为钝角，在线段BO上取一点C，连接CP，将三角形区域CPO标粉\n下图：\n作三角形BPO，边BO水平，角BPO为钝角，在线段BO上取一点C，连接CP将三角形区域BPO标绿", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1043.jpg" }, { "index": 1044, "ocr_en": "Construct a right triangle ABC with angle ACB as the right angle. Draw a circle with center at point C (the radius of the circle is half the length of BC). Take a point P on the upper-right semicircle, and connect PA and PB.", "ocr_zh": "作直角三角形ABC，角ACB为直角，以C点为圆心作圆（圆半径为BC边长的一半），在右上半圆上取一点P，连接PA、PB", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1044.jpg" }, { "index": 1045, "ocr_en": "Construct triangle BPO with side BO horizontal and angle BPO as an obtuse angle. Take a point C on line segment BO, and connect CP. Label in red: \"P (right side, moving point)\", \"B (right side, fixed point)\", \"O (right side, center of circle)\". Mark the radius r on line PO in black.", "ocr_zh": "作三角形BPO，边BO水平，角BPO为钝角，在线段BO上取一点C，连接CP，红色标注P点右侧（动点）、B点右侧（定点）、O点右侧（圆心），PO线上半径r（r为黑色）", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1045.jpg" }, { "index": 1046, "ocr_en": "Construct a right triangle ACB with angle ACB as the right angle. Draw a circle with center at point C (radius less than the side lengths of the triangle). Take a point D on the upper-right semicircle, and connect DB, DA, DC. Take a point H on line segment CB (H is inside the circle), connect HD, and shade the triangular region HCD in purple.", "ocr_zh": "作直角三角形ACB，角ACB为直角，以C点为圆心作圆（圆半径小于三角形边长），在右上半圆取一点D，连接DB、DA、DC，在线段CB上取一点H（H位于圆内），连接HD，将三角形区域HCD标紫", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1046.jpg" }, { "index": 1047, "ocr_en": "Construct a right triangle ACB with angle ACB as the right angle. Draw a circle with center at point C (radius less than the side lengths of the triangle). Take a point D on the upper-right semicircle, and connect DB, DA", "ocr_zh": "作直角三角形ACB，角ACB为直角，以C点为圆心作圆（圆半径小于三角形边长），在右上半圆取一点D，连接DB、DA", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1047.jpg" }, { "index": 1048, "ocr_en": "Construct a sector COD with angle COD as a right angle. Take a point A on line segment OC, a point B on line segment OD, and a point P on arc CD. Connect AP, PB, and OP. Extend BP to intersect the extension of OC at point H. Shade the triangular region HOP in pink. Label \"Figure 2\" below the diagram. (Figure 2 is incomplete.)", "ocr_zh": "作扇形COD，角COD为直角，线段OC上取一点A、线段OD上取一点B、弧CD上取一点P，连接AP、PB、OP，延长BP交OC延长线于点H，将三角形区域HOP标粉，在图下方标注图2（图2不完整）", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1048.jpg" }, { "index": 1049, "ocr_en": "Construct a sector COD with angle COD as a right angle. Take a point A on line segment OC, a point B on line segment OD, and a point P on arc CD. Connect AP, PB，Label \"Figure 1\" below the diagram.", "ocr_zh": "作扇形COD，角COD为直角，线段OC上取一点A、线段OD上取一点B、弧CD上取一点P，连接AP、PB，在图下方标注图1", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1049.jpg" }, { "index": 1050, "ocr_en": "Construct a right triangle ABC with angle ACB as the right angle. Take a point D inside the triangle, and connect DB, DA, DC. Draw a circle with center at point C and radius CD as a red solid line. Take a point H on line segment BC, connect DH, and shade the triangular region CHD in green.", "ocr_zh": "作直角三角形ABC，角ACB为直角，在三角形内取一点D，连接DB、DA、DC，以C点为圆心，CD长为半径以红色实线作圆，在线段BC上取一点H，连接DH，将三角形区域CHD标绿", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1050.jpg" }, { "index": 1051, "ocr_en": "Construct a right triangle ABC with angle ACB as the right angle. Take a point D inside the triangle, and connect DB, DA", "ocr_zh": "作直角三角形ABC，角ACB为直角，在三角形内取一点D，连接DB、DA", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1051.jpg" }, { "index": 1052, "ocr_en": "Construct a right triangle ABC with angle ACB as the right angle, marking a gray right-angle symbol. Draw a circle with center at point C (radius less than the side lengths of the triangle). Take a point P on the lower-left semicircle, and connect PA and PB.", "ocr_zh": "作直角三角形ABC，角ACB为直角，标注灰色直角符号，以C点为圆心作圆（圆半径小于三角形边长），在左下半圆取一点P，连接PA、PB", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1052.jpg" }, { "index": 1053, "ocr_en": "Construct a right triangle ABC with angle ACB as the right angle, marking a gray right-angle symbol. Draw a circle with center at point C (radius less than the side lengths of the triangle). Take a point P on the lower-left semicircle, and connect PA, PB. Take a point H on line segment BC (H is inside the circle), connect CP, HP, and shade the triangular region CPH in green.", "ocr_zh": "作直角三角形ABC，角ACB为直角，标注灰色直角符号，以C点为圆心作圆（圆半径小于三角形边长），在左下半圆取一点P，连接PA、PB，在线段BC上取一点H（H位于圆内），连接CP、HP，将三角形区域CPH标绿", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1053.jpg" }, { "index": 1054, "ocr_en": "Construct square ABCD. Draw a circle with center at point B and radius equal to half the side length of the square. Take a point P on the upper-right semicircle, and connect PC and PD.", "ocr_zh": "作正方形ABCD，以B点为圆心，正方形边长的一半为半径作圆，在右上半圆取一点P，连接PC、PD", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1054.jpg" }, { "index": 1055, "ocr_en": "Construct square ABCD. Draw a circle with center at point B and radius equal to half the side length of the square. Take a point P on the upper-right semicircle, and connect PC, PD, BD, BP. Take a point H on BD (point H is inside the circle), connect HP. Draw HM as a red solid line through point H perpendicular to BC, with the foot at M. Shade the triangular region BHP in purple.", "ocr_zh": "作正方形ABCD，以B点为圆心，正方形边长的一半为半径作圆，在右上半圆取一点P，连接PC、PD，连接BD、BP,在BD上取一点H(H点位于圆内)，连接HP，过H点红色实线作HM垂直BC，垂足为M，将三角形区域BHP标紫", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1055.jpg" }, { "index": 1056, "ocr_en": "Construct a rhombus ABCD with all four sides equal and angle ABC as an acute angle, where AD is parallel to BC and AB is parallel to CD. Draw a circle with center at point B and radius equal to half the side length of the rhombus. Take a point P on the upper-right semicircle, and connect PC, PD, BP, BD. Take a point H on line segment BD (H is inside the circle), connect HP. Draw HM as a red solid line through point H perpendicular to BC, with the foot at M. Shade the triangular region BHP in green.", "ocr_zh": "作菱形ABCD，菱形四边相等，角ABC为锐角，AD平行BC、AB平行CD，以B点为圆心，菱形边长的一半为半径作圆，在右上半圆取一点P，连接PC、PD、BP、BD，在线段BD上取一点H（H位于圆内），连接HP，过H点红色实线作HM垂直BC，垂足为M，将三角形区域BHP标绿", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1056.jpg" }, { "index": 1057, "ocr_en": "Construct a rhombus ABCD with all four sides equal and angle ABC as an acute angle, where AD is parallel to BC and AB is parallel to CD. Draw a circle with center at point B and radius equal to half the side length of the rhombus. Take a point P on the upper-right semicircle, and connect PC, PD", "ocr_zh": "作菱形ABCD，菱形四边相等，角ABC为锐角，AD平行BC、AB平行CD，以B点为圆心，菱形边长的一半为半径作圆，在右上半圆取一点P，连接PC、PD", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1057.jpg" }, { "index": 1058, "ocr_en": "\tConstruct a plane rectangular coordinate system. Take a point A on the positive x-axis and a point B on the positive y-axis (with OA equals OB). Connect AB with a red solid line. Draw a circle with center at point O and radius OA. Take a point P on the upper-right semicircle, and connect PA, PB. Take a point H on line segment OA, connect OP, PH. Take a point C on the positive x-axis to the right of A, and take a point D outside the circle in the first quadrant, connect PD, PC. Shade the triangular region OPH in green.", "ocr_zh": "作平面直角坐标系，x轴正半轴取一点A、y轴正半轴取一点B（OA等于OB），红色实线连接AB，以O点为圆心、OA长为半径作圆，在右上半圆取一点P，连接PA、PB，线段OA上取一点H，连接OP、PH，在A点右侧x轴正半轴取一点C，在第一象限圆外取一点D，连接PD、PC，将三角形区域OPH标绿", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1058.jpg" }, { "index": 1059, "ocr_en": "Construct triangle ABC. Draw a circle with center at point A, where the radius is less than the side lengths of the triangle. Take a point P on the upper-right semicircle (point P is inside the triangle), and connect PB, PC.", "ocr_zh": "作三角形ABC，以A点为圆心作圆，圆半径小于三角形边长，在右上半圆取一点P（P点位于圆内），连接PB、PC", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1059.jpg" }, { "index": 1060, "ocr_en": "Construct triangle ABC. Draw a circle with center at point A, where the radius is less than the side lengths of the triangle. Take a point P on the upper-right semicircle (point P is inside the circle), and connect PB, PC. Take two points H and M in sequence from left to right on line segment AB (both points are inside the circle), connect PA, PH. Connect CH and CM with red solid lines. Shade the triangular region AHP in pink.", "ocr_zh": "作三角形ABC，以A点为圆心作圆，圆半径小于三角形边长，在右上半圆取一点P（P点位于圆内），连接PB、PC，在线段AB上从左到右依次取两点H、M（两点均位于圆内），连接PA、PH，红色实线连接CH、CM，将三角形区域AHP标粉", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1060.jpg" }, { "index": 1061, "ocr_en": "Construct a plane rectangular coordinate system. Take a point A on the positive x-axis and a point B on the positive y-axis. Take a point C on the positive x-axis to the right of A. Take two points P and D in the first quadrant (point P is to the left of A, and point D is between A and C). Connect PB, PA, PC, PD.", "ocr_zh": "作平面直角坐标系，x轴正半轴取一点A、y轴正半轴取一点B, 在A点右侧x轴正半轴取一点C,在第一象限取两点P、D（P点在A左侧、D点在AC之间），连接PB、PA、PC、PD", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1061.jpg" }, { "index": 1062, "ocr_en": "Construct an equilateral triangle ABC with AB equals AC equals BC. Take a point O inside the triangle and draw the incircle of triangle ABC with center at point O. Take a point P on the right semicircle of circle O, and connect PB, PC.", "ocr_zh": "作等边三角形ABC，AB等于AC等于BC，在三角形内取一点O，以O点为圆心作三角形ABC内切圆，在圆O右半圆取一点P，连接PB、PC", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1062.jpg" }, { "index": 1063, "ocr_en": "Construct a square CABD. Take a point O inside the square and draw the incircle of the square with center at O. Take a point P on the lower-left semicircle of circle O, and connect PA, PB.", "ocr_zh": "作正方形CABD，在正方形CABD内取一点O，以O为圆心作正方形内切圆，在圆O左下半圆上取一点P，连接PA、PB", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1063.jpg" }, { "index": 1064, "ocr_en": "Construct a scaled plane rectangular coordinate system. Take point A at coordinates (-4, -4), point B at (0, 4), and point C at (0, -6). Draw line AB through points A and B, draw line AC through points A and C. Draw a parabola opening downward through points A and B, with its vertex in the second quadrant. Find the intersection point E of line AB and the x-axis. Draw line EG perpendicular to the x-axis through point E (with the foot at E), where point G is the intersection of line EG and the parabola.", "ocr_zh": "作带刻度的平面直角坐标系，取坐标（-4，-4）为点A、坐标（0,4）为点B、坐标（0，-6）为点C，经过A、B两点作直线AB，经过A、C两点作直线AC，经过A、B两点作开口向下的抛物线，抛物线顶点在第二象限，取直线AB与x轴交点E，过E点作直线EG垂直x轴，垂足为E，G点为直线EG和抛物线交点", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1064.jpg" }, { "index": 1065, "ocr_en": "Construct a rectangle ABCD, where AD is the length and AB is the width.", "ocr_zh": "作矩形ABCD，AD为长，AB为宽", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1065.jpg" }, { "index": 1066, "ocr_en": "Construct triangle ABC with AB equals AC. Take the midpoint D of AC and connect BD.", "ocr_zh": "作三角形ABC，AB等于AC，取AC的中点D连接BD", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1066.jpg" }, { "index": 1067, "ocr_en": "Construct a square ABCD. Take a point P inside the square and connect PC, PD.", "ocr_zh": "作正方形ABCD，在正方形ABCD内取一点P，连接PC、PD", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1067.jpg" }, { "index": 1068, "ocr_en": "Construct a scaled plane rectangular coordinate system. Take point A at (4, 0) and point B at (0, 3). Draw a circle with center at point O and radius 2 as a green solid line. Take a point E on the upper-right semicircle, connect EB and EA, and connect OE with a dashed line.", "ocr_zh": "作带刻度的平面直角坐标系，取（4,0）为点A、（0,3）为点B，以O点为圆心，长度2为半径以绿色实线作圆，在右上半圆取一点E，连接EB、EA，虚线连接OE", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1068.jpg" }, { "index": 1069, "ocr_en": "Construct a graduated plane rectangular coordinate system, take (4, 0) as point A and (0, 3) as point B. With point O as the center, draw a circle with radius 2 as a green solid line, which intersects the positive y-axis at point C and the negative y-axis at point D. Take a point E on the upper right semicircle, connect EB and EA, and connect OE with a dashed line. Take a point M between line segment CD, connect ME with a dashed line. With point C as the center, draw circle C with radius CM as a dashed line (point B is outside circle C). With point D as the center, draw a semi-arc with radius DM as a dashed line. Color the region of OEB orange.", "ocr_zh": "作带刻度的平面直角坐标系，取（4,0）为点A、（0,3）为点B，以O点为圆心，长度2为半径以绿色实线作圆，圆与y轴正半轴相交于C点、与y轴负半轴相交于D点，在右上半圆取一点E，连接EB、EA，虚线连接OE，在线段CD之间取一点M，虚线连接ME，以C点为圆心，CM为半径以虚线作圆C（B点在圆C外），以D点为圆心，DM为半径以虚线作半圆弧，将区域OEB部分标橙", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1069.jpg" }, { "index": 1070, "ocr_en": "Left diagram: \nConstruct right triangle CPD with angle CPD as the right angle. Take the midpoint O of line segment CD, and draw the circumcircle of triangle CPD with O as the center and CD as the diameter. Take a point B on line segment OC, connect PB, and draw a tangent from point P to circle O, intersecting the extension of DC at point A. \n\nRight diagram: \nConstruct right triangle CPD with angle CPD as the right angle. Take the midpoint O of line segment CD, and draw the circumcircle of triangle CPD with O as the center and CD as the diameter. Take a point A on line segment OC, connect PA, extend DC to B, and connect BP and extend it.", "ocr_zh": "左图：\n作直角三角形CPD，角CPD为直角，取线段CD的中点O，以O点为圆心,CD为直径作三角形CPD的外接圆，在线段OC上取一点B，连接PB，过P点作圆O的切线与DC的延长线相交于A点\n右图：\n作直角三角形CPD，角CPD为直角，取线段CD的中点O，以O点为圆心,CD为直径作三角形CPD的外接圆，在线段OC上取一点A，连接PA，延长DC至B，连接BP并延长", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1070.jpg" }, { "index": 1071, "ocr_en": "Construct right triangle CPD with angle CPD as the right angle. Take the midpoint O of line segment CD, and draw the circumcircle of triangle CPD with O as the center and CD as the diameter. Take a point B on line segment OC, connect PB, and draw a tangent from point P to circle O, intersecting the extension of DC at point A. ", "ocr_zh": "作直角三角形CPD，角CPD为直角，取线段CD的中点O，以O点为圆心,CD为直径作三角形CPD的外接圆，在线段OC上取一点B，连接PB，过P点作圆O的切线与DC的延长线相交于A点", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1071.jpg" }, { "index": 1072, "ocr_en": "Construct right triangle CPD with angle CPD as the right angle. Take the midpoint O of line segment CD, and draw the circumcircle of triangle CPD with O as the center and CD as the diameter. Take a point A on line segment OC, connect PA, extend DC to B, and connect BP and extend it.", "ocr_zh": "作直角三角形CPD，角CPD为直角，取线段CD的中点O，以O点为圆心,CD为直径作三角形CPD的外接圆，在线段OC上取一点A，连接PA，延长DC至B，连接BP并延长", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1072.jpg" }, { "index": 1073, "ocr_en": "Construct an equilateral triangle ABC with AB equals AC equals BC. Take a point inside the equilateral triangle and draw the incircle of the triangle. Take a point P on the lower right semicircle, and connect PC and PB.", "ocr_zh": "作等边三角形ABC，AB等于AC等于BC，在等边三角形ABC内取一点，作三角形的内切圆，在右下半圆上取一点P，连接PC、PB", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1073.jpg" }, { "index": 1074, "ocr_en": "Construct a rhombus ABCD with all four sides equal and angle ABC as an acute angle. With point B as the center, draw a circle in blue solid line with a radius of half the side length of the rhombus. Take a point P on the upper right semicircle (point P is inside the rhombus), and connect PC and PD.", "ocr_zh": "作菱形ABCD，四边相等，角ABC为锐角，以B点为圆心，菱形边长的一半为半径以蓝色实线作圆，在右上半圆上取一点P（P点在菱形内），连接PC、PD", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1074.jpg" }, { "index": 1075, "ocr_en": "Construct a horizontal line segment AB, take the midpoint O of AB, and draw a circle with O as the center and AB as the diameter. Take the midpoint C of line segment OA, draw CD perpendicular to AB at point C, intersecting the circle at point D. Connect DO and extend it to intersect the circle at another point E. Take a point P on the lower left semicircle, and connect PC and PE.", "ocr_zh": "作水平线段AB，取AB的中点O，以O点为圆心，线段AB为直径作圆，取线段OA的中点C，过C点作CD垂直AB，垂足为C，交圆于D点，连接DO并延长交圆于另一点E，在左下半圆上取一点P，连接PC、PE", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1075.jpg" }, { "index": 1076, "ocr_en": "Construct a square ABCD, take a point inside the square, and draw the incircle of the square with this point as the center. Take a point P on the lower left semicircle, and connect PA and PB.", "ocr_zh": "作正方形ABCD，在正方形内取一点，以该点为圆心作正方形的内切圆，在左下半圆上取一点P，连接PA、PB", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1076.jpg" }, { "index": 1077, "ocr_en": "Construct square ABCD. With point B as the center, draw a circle with a radius equal to half the side length of the square. Take a point P on the upper right semicircle (point P is inside the square and closer to side BC), and connect PC and PD.", "ocr_zh": "作正方形ABCD，以B点为圆心，正方形边长的一半为半径作圆，在右上半圆上取一点P（P点位于正方形内，且更靠近边BC），连接PC、PD", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1077.jpg" }, { "index": 1078, "ocr_en": "Construct square ABCD. With point B as the center, draw a circle with a radius equal to half the side length of the square. Take a point P on the upper right semicircle (point P is inside the square and closer to side BC), and connect PC and PD. Take points N and M on BD and BC respectively (both points N and M are inside the circle), and connect PM, PN, MN, and BP with dashed lines. Let MN intersect BP at point K.", "ocr_zh": "作正方形ABCD，以B点为圆心，正方形边长的一半为半径作圆，在右上半圆上取一点P（P点位于正方形内，且更靠近边BC），连接PC、PD，在BD、BC上分别取点N、M（N、M点均位于圆内），虚线连接PM、PN、MN、BP，MN和BP相交于点K", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1078.jpg" }, { "index": 1079, "ocr_en": "Construct a plane rectangular coordinate system. Draw a parabola opening downward with its vertex in the first quadrant, intersecting the positive x-axis at point A and the positive y-axis at point B. Draw line AB through points A and B. Take a point E on line segment OA, draw a perpendicular to the x-axis through point E (mark the right-angle symbol) with the foot at E, intersecting line AB at point N and the parabola at point P. Draw PM perpendicular to AB through point P (mark the right-angle symbol) with the foot at M. Rotate segment OE counterclockwise about point O to obtain OE' (point E' lies inside triangle BOA), and connect E'A and E'B.", "ocr_zh": "作平面直角坐标系，作开口向下的抛物线，抛物线顶点位于第一象限，抛物线与x轴正半轴交于点A、与y轴正半轴交于点B，过A、B点作直线AB，在线段OA上取一点E，过点E作x轴的垂线，垂足为E（标注直角符号），交直线AB于点N，交抛物线于点P，过点P作PM垂直AB，垂足为M（标注直角符号），将线段OE绕点O逆时针旋转得到OE’（E’点位于三角形BOA内），连接E’A、E’B", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1079.jpg" }, { "index": 1080, "ocr_en": "Construct a plane rectangular coordinate system. Draw a parabola opening downward with its vertex in the first quadrant, intersecting the positive x-axis at point A and the positive y-axis at point B. Draw line AB through points A and B. Take a point E on line segment OA, draw a perpendicular line to the x-axis through point E with the foot at E (mark the right-angle symbol), intersecting line AB at point N and the parabola at point P. Draw PM perpendicular to AB through point P with the foot at M (mark the right-angle symbol).", "ocr_zh": "作平面直角坐标系，作开口向下的抛物线，抛物线顶点位于第一象限，抛物线与x轴正半轴交于点A、与y轴正半轴交于点B，过A、B点作直线AB，在线段OA上取一点E，过点E作x轴的垂线，垂足为E（标注直角符号），交直线AB于点N，交抛物线于点P，过点P作PM垂直AB，垂足为M（标注直角符号）", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1080.jpg" }, { "index": 1081, "ocr_en": "Construct angle MON, draw the angle bisector of angle MON through point O. Take a point P on the angle bisector, draw AP perpendicular to OP through point P (mark the right-angle symbol), and extend AP with a dashed line to intersect ON at point B.", "ocr_zh": "作角MON，过O点作角MON的角平分线，在角平分线上取一点P，过P点作AP垂直OP，垂足为P（标注直角符号），虚线延长AP交ON于点B", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1081.jpg" }, { "index": 1082, "ocr_en": "Construct an isosceles right triangle ABC with AB equals AC and angle BAC as the right angle (mark the gray right-angle symbol). Draw the angle bisector of angle ABC, which intersects AC at point D. Through point C, draw CE perpendicular to BD, intersecting the extension of BD at point E (mark the gray right-angle symbol as the foot). Extend BA and CE with dashed lines, and let them intersect at point F.", "ocr_zh": "作等腰直角三角形ABC，AB等于AC，角BAC为直角（标注灰色直角符号），作角ABC的角平分线交AC于D点，过C点作CE垂直BD交BD的延长线于点E，垂足为E（标注灰色直角符号），虚线延长BA、CE相交于点F", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1082.jpg" }, { "index": 1083, "ocr_en": "Construct an isosceles right triangle ABC where AB equals AC, and angle BAC is a right angle (mark the right-angle symbol). Draw the angle bisector of angle ABC, intersecting AC at point D. Through point C, draw CE perpendicular to BD, intersecting the extension of BD at point E (mark the right-angle symbol at E as the foot).", "ocr_zh": "作等腰直角三角形ABC，AB等于AC，角BAC为直角（标注直角符号），作角ABC的角平分线交AC于D点，过C点作CE垂直BD交BD的延长线于点E，垂足为E（标注直角符号）", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1083.jpg" }, { "index": 1084, "ocr_en": "Construct triangle ABC with side CB horizontal. Draw the angle bisector of angle ABC, intersecting AC at point E. Through point A, draw AD perpendicular to BE with the foot at D, and extend AD with a dashed line to intersect BC at point F. Mark the right-angle symbol at D, and label angle EAD as 1 and angle DAB as 2.", "ocr_zh": "作三角形ABC，边CB水平，作角ABC的角平分线交AC于点E，过A点作AD垂直BE，垂足为D，虚线延长AD交BC于点F，标注角EAD为1、角DAB为2", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1084.jpg" }, { "index": 1085, "ocr_en": "Construct triangle ABC with side CB horizontal. Draw the angle bisector of angle ABC, intersecting AC at point E. Through point A, draw AD perpendicular to BE with the foot at D, marking the right-angle symbol. Label angle EAD as 1 and angle DAB as 2.", "ocr_zh": "作三角形ABC，边CB水平，作角ABC的角平分线交AC于点E，过A点作AD垂直BE，垂足为D，标注直角符号，标注角EAD为1、角DAB为2", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1085.jpg" }, { "index": 1086, "ocr_en": "Construct triangle ABC with side BC horizontal and angle ABC as an obtuse angle. Draw the angle bisector of angle BAC, intersecting BC at point D. Through point B, draw BE perpendicular to AD with the foot at E, marking the right-angle symbol.", "ocr_zh": "作三角形ABC，边BC水平，角ABC为钝角，作角BAC的角平分线交BC于点D，过B点作BE垂直AD，垂足为E，标注直角符号", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1086.jpg" }, { "index": 1087, "ocr_en": "Construct triangle ABC with side BC horizontal. Draw the angle bisector of angle BAC, intersecting BC at point D. Through point B, draw BE perpendicular to AD with the foot at E, and extend BE with a dashed line to intersect AC at point F.", "ocr_zh": "作三角形ABC，边BC水平，作角BAC的角平分线交BC于点D，过B点作BE垂直AD，垂足为E，虚线延长BE交AC于F", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1087.jpg" }, { "index": 1088, "ocr_en": "Construct angle MON and its angle bisector. Take a point P on the angle bisector, then draw PQ parallel to ON, intersecting OM at point Q.", "ocr_zh": "作角MON，作角MON的角平分线，在角平分线上取一点P，过P点作PQ平行ON交OM于点Q", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1088.jpg" }, { "index": 1089, "ocr_en": "Construct angle MAN. Take a point B on AM and a point C on AN, then connect BC. Construct the angle bisectors of the external angles angle CBM and angle BCN, and let these two bisectors intersect at point D. Through point D, draw line EF parallel to BC, intersecting AM and AN at points E and F respectively (in the figure, point M is below point E, point N is below point F, and the letters M and N are incomplete).", "ocr_zh": "作角MAN，在AM上取一点B、在AN上取一点C，连接BC，作外角CBM、外角BCN 的角平分线，两角平分线相交于点D，过D点作EF平行BC，分别交AM、AN于点E、F（图中M点在E点下方、N点在F点下方，M、N点字母不完整）", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1089.jpg" }, { "index": 1090, "ocr_en": "Construct triangle ABC, extend BC to G. Construct the angle bisectors of angle ABC and angle ACG, and let the two bisectors intersect at point D. Through point D, draw DE parallel to BC, intersecting AB at point E and AC at point F. Label the figure as Figure ② below (in the figure, the letter C is incomplete, and Figure ② is incomplete).", "ocr_zh": "作三角形ABC，延长BC至G，作角ABC、角ACG的角平分线，两角平分线相交于点D,过D点作DE平行BC，交 AB 于点 E，交 AC 于点 F，在图下方标注图②（图中C点字母不完整、图②不完整）", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1090.jpg" }, { "index": 1091, "ocr_en": "Construct triangle ABC with side BC horizontal. Construct the angle bisectors of angle ABC and angle ACB, and let the two bisectors intersect at point D. Through point D, draw EF parallel to BC, intersecting AB at point E and AC at point F (in the figure, the letter B is incomplete).", "ocr_zh": "作三角形ABC，边BC水平，作角ABC、角ACB的角平分线，两角平分线相交于点D，过D点作EF平行BC，交AB于点E、交AC于点F（图中B点字母不完整）", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1091.jpg" }, { "index": 1092, "ocr_en": "Construct triangle ABC with side BC horizontal. Draw the angle bisector of angle BAC, intersecting BC at point D. Take points E on BD and F on AD respectively, then connect EF, with EF parallel to AB.", "ocr_zh": "作三角形ABC，边BC水平，作角BAC角平分线交BC于点D，分别在 BD、AD 上取点 E、F，连接EF，有EF平行AB", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1092.jpg" }, { "index": 1093, "ocr_en": "Construct triangle ABC. Draw the angle bisectors of angle ABC and angle ACB, which intersect at point E. Through point E, draw MN parallel to BC, intersecting AB at point M and AC at point N.", "ocr_zh": "作三角形ABC，作角ABC、角ACB的角平分线，两角平分线相交于点E，过E点作MN平行BC，交AB于点M、交AC于点N", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1093.jpg" }, { "index": 1094, "ocr_en": "Construct triangle ABC with side BC horizontal. Draw the angle bisector of angle BAC, intersecting BC at point D. Take points E on BD and F on AD respectively, then connect EF such that EF is parallel to AB. Through point C, draw a dashed line CM parallel to AB, intersecting the extension of AD at point M (with MD drawn as a dashed line).", "ocr_zh": "作三角形ABC，边BC水平，作角BAC角平分线交BC于点D，分别在 BD、AD 上取点 E、F，连接EF，有EF平行AB，过C点虚线作CM平行AB交AD延长线于点M（MD为虚线）", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1094.jpg" }, { "index": 1095, "ocr_en": "Construct trapezoid ABCD with AD parallel to BC. Take a point E on CD, and connect AE and BE, such that AE bisects angle BAD and BE bisects angle ABC.", "ocr_zh": "作梯形ABCD，AD平行BC，在CD上取一点E，连接AE、BE，有AE平分角BAD，BE平分角ABC", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1095.jpg" }, { "index": 1096, "ocr_en": "Construct trapezoid ABCD with AD parallel to BC. Take a point E on CD, and connect AE and BE such that AE bisects angle BAD and BE bisects angle ABC. Take a point F on AB and connect EF with a dashed line.", "ocr_zh": "作梯形ABCD，AD平行BC，在CD上取一点E，连接AE、BE，有AE平分角BAD，BE平分角ABC，在AB上取一点F，虚线连接EF", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1096.jpg" }, { "index": 1097, "ocr_en": "In triangle ABC, angle A is greater than angle C is greater than angle B; Take a point D on BC and connect AD, and AD bisects the angle BAC; Take a little E on AB and make the auxiliary line DE", "ocr_zh": "在三角形ABC中，角A大于角C大于角B;在BC上取一点D，连接AD，AD平分角BAC;在AB上取一点E，做辅助线DE", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1097.jpg" }, { "index": 1098, "ocr_en": "In triangle ABC, angle A is greater than angle C is greater than angle B; Take a point D on BC and connect AD, AD bisects the angle BAC", "ocr_zh": "在三角形ABC中，角A大于角C大于角B;在BC上取一点D，连接AD，AD平分角BAC", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1098.jpg" }, { "index": 1099, "ocr_en": "In the angular AOG, O is the endpoint; There is a point D and OD in the angle AOG, and the DC is perpendicular to the point C, which connects AD; There are some B and E on the OG, and the OB is smaller than the OE; Connect the BD, pass the D point as the auxiliary line DE and the vertical BG intersect at the point E", "ocr_zh": "在角AOG中，O为端点；角AOG内有一点D,OD 平分角AOB，DC垂直OA 于点 C，连接AD;OG上有点B、E，OB小于OE；连接BD，过D点作辅助线DE垂直BG交于点E", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1099.jpg" }, { "index": 1100, "ocr_en": "In the angular AOG, O is the endpoint; There is a little D in the angle AOG, OD bisects the angle AOB, and the DC is perpendicular to OA through D, and the vertical foot is C, which is connected to AD; There is a point B on the OG, connected to BD, connected to 0D", "ocr_zh": "在角AOG中，O为端点；角AOG内有一点D,OD平分角AOB，过D做DC垂直于OA，垂足为C，连接AD;OG上有一点B，连接BD，连接0D", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1100.jpg" }, { "index": 1101, "ocr_en": "In angular NAM, A is the endpoint; There is a bit of D on AN, a bit of E and B on AM, and the AE is less than AB; C is a point on the bisector of the angular MAN, connecting CD, CE, CB; The angular ADC is angle 1, and the angular EBC is angle 2", "ocr_zh": "在角NAM中，A为端点；AN上有点D，AM上有点E、B，且AE小于AB；C是角MAN平分线上一点，连接CD、CE、CB；角ADC为角1,角EBC为角2", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1101.jpg" }, { "index": 1102, "ocr_en": "In angular NAM, A is the endpoint; A bit of D on AN, a bit of B on AM; C is a point on the angular MAN bisector, connecting CD and CB, making the auxiliary line CF perpendicular to AN, perpendicular to F, AD is less than AF; CE over C is perpendicular to AM, the vertical foot is E, and the AE is less than AB; The angular ADC is angular 1, the angular EBC is angular 2, the angular DAC is angular 3, the angular CAE is angular 4, and the angular CDF is angular 5. Angle 1 is an obtuse angle, and angle 2, angle 3, angle 4, and angle 5 are acute angles", "ocr_zh": "在角NAM中，A为端点；AN上有点D，AM上有点B；C是角MAN平分线上一点，连接CD、CB，做辅助线CF垂直于AN，垂足为F，AD小于AF；过C做CE垂直于AM，垂足为E，且AE小于AB；角ADC为角1,角EBC为角2，角DAC为角3，角CAE为角4，角CDF为角5；角1为钝角，角2、角3、角4、角5为锐角", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1102.jpg" }, { "index": 1103, "ocr_en": "There is a triangle ABC, AB perpendicular to AC at point A; There is a point E outside the triangle ABC, which connects BE and CE, and BE is perpendicular to CE at the point E; BE intersects AC at point D; BE bisector ABC", "ocr_zh": "有三角形ABC，AB垂直AC于点A；三角形ABC外有一点E，连接BE、CE，此时BE垂直CE于点E；BE与AC相交于点D；BE平分角ABC", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1103.jpg" }, { "index": 1104, "ocr_en": "There is a triangle ABC, AB perpendicular to AC at point A; There is a point E outside the triangle ABC, which connects BE and CE, and BE is perpendicular to CE at the point E; BE intersects AC at point D; BE bisector ABC; Do auxiliary lines: extend CE and BA respectively and intersect at point F; Angle DBC is angle 1, angle ABD is angle 2, angle DCE is angle 3", "ocr_zh": "有三角形ABC，AB垂直AC于点A；三角形ABC外有一点E，连接BE、CE，此时BE垂直CE于点E；BE与AC相交于点D；BE平分角ABC；做辅助线：分别延长CE、BA，相交于点F；角DBC为角1，角ABD为角2，角DCE为角3", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1104.jpg" }, { "index": 1105, "ocr_en": "There is a square ABCD, and there is a little E on the BC, which connects the AE; There is a little F on the CD, which connects the AF, and the AF bisects the angle DAE", "ocr_zh": "有正方形ABCD，BC上有一点E，连接AE；CD上有一点F，连接AF，AF平分角DAE", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1105.jpg" }, { "index": 1106, "ocr_en": "There is a square ABCD, and there is a little E on the BC, which connects the AE; There is a little F on the CD, which connects to the AF, AF bisecting angle DAE; Do auxiliary line: extend CB to G and connect AG; Angular AFD is Angle 1, Angular FAD is Angle 2, Angular BAG is Angle 3, Angular EAF is Angle 4, and Angular BAE is Angle 5", "ocr_zh": "有正方形ABCD，BC上有一点E，连接AE；CD上有一点F，连接AF，AF平分角DAE；做辅助线：延长CB至G，连接AG；角AFD为角1，角FAD为角2，角BAG为角3，角EAF为角4，角BAE为角5", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1106.jpg" }, { "index": 1107, "ocr_en": "In the quadrilateral ABCD, the angle ABC and the angle ADC are acute angles, and the angle BCD and the angle BAD are obtuse angles. The bisector AE of the angle DAB intersects CD with E and connects BE, and BE exactly bisects the angle ABC", "ocr_zh": "在四边形ABCD中，角ABC与角ADC为锐角，角BCD与角BAD为钝角；角DAB的平分线AE交CD于E，连接BE，且BE恰好平分角ABC", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1107.jpg" }, { "index": 1108, "ocr_en": "In the quadrilateral ABCD, the angle ABC and the angle ADC are acute angles, and the angle BCD and the angle BAD are obtuse angles. There is a little E on the CD, which connects AE and BE; Take the point F on AB and make the auxiliary line EF; The angular AED is angular 1, the angular AEF is angular 2, the angular FEB is angular 3, the angular BEC is angular 4, the angular EAD is angular 5, the angular EAF is angular 6, the angular CBE is angular 7, and the angular EBA is angular 8", "ocr_zh": "在四边形ABCD中，角ABC与角ADC为锐角，角BCD与角BAD为钝角；角DAB的平分线AE交CD于E，连接BE，且BE恰好平分角ABC；AB上取点F，做辅助线EF；角AED为角1，角AEF为角2，角FEB为角3，角BEC为角4，角EAD为角5，角EAF为角6，角CBE为角7，角EBA为角8", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1108.jpg" }, { "index": 1109, "ocr_en": "The isosceles triangle ABC and the isosceles triangle ADE share a vertex A, and there is no other intersecting part except the point A; AB=AC, AD=AE, do the auxiliary line BD, CE, CE and DE have and only one intersection E, CE and AD have no intersection", "ocr_zh": "等腰三角形ABC与等腰三角形ADE有一共用顶点A，除点A外无其他相交部分；AB=AC，AD=AE，做辅助线BD、CE，CE与DE有且仅有一交点E，CE与AD无交点", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1109.jpg" }, { "index": 1110, "ocr_en": "The isosceles triangle ABC and the isosceles triangle ADE have a common vertex A, and there is no other intersecting part except the point A, AB=AC, AD=AE,; Do auxiliary line BD, CE, CE and AD have an intersection", "ocr_zh": "等腰三角形ABC与等腰三角形ADE有一共用顶点A，除点A外无其他相交部分，AB=AC，AD=AE，；做辅助线BD、CE，CE与AD有一交点", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1110.jpg" }, { "index": 1111, "ocr_en": "Left: The isosceles right triangle AOB and the isosceles right triangle COD have a common vertex O, and there is no other intersecting part except the point O, and the angle COB is an acute angle; The angular AOB and the angular COD are at right angles; Connect BD and AC, intersect at point E, connect OE Right: The isosceles right triangle AOB and the isosceles right triangle COD have a common vertex O, and the angle AOB and the angle COD are right angles; CD and OB have only one intersection point, and the angle AOC is an acute angle; Connect BD and AC; Extend the intersection of AC and BD at point E and connect to OE", "ocr_zh": "左图：等腰直角三角形AOB与等腰直角三角形COD有一共用顶点O，除点O外无其他相交部分，角COB为锐角；角AOB与角COD为直角；连接BD、AC,交于点E,连接OE 右图：等腰直角三角形AOB与等腰直角三角形COD有一共用顶点O，角AOB与角COD为直角；CD与OB有且仅有一个交点，角AOC为锐角；连接BD、AC；延长AC与BD交于点E，连接OE", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1111.jpg" }, { "index": 1112, "ocr_en": "The isosceles triangle AOB and the isosceles triangle COD have a common vertex O, and there is no other intersecting part except the point O, OA=OB, OC=OD, and the angle BOC is an acute angle; Connect BD and AC, intersect at point E, and connect to OE", "ocr_zh": "等腰三角形AOB与等腰三角形COD有一共用顶点O，除点O外无其他相交部分，OA=OB，OC=OD，角BOC为锐角；连接BD、AC,交于点E,连接OE", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1112.jpg" }, { "index": 1113, "ocr_en": "If the angular AOD is greater than 90 degrees, the line segment OB and OC are made with O as the endpoint, OA=OB, OC=OD, and the angular AOB is equal to the angular COD", "ocr_zh": "角AOD大于90度，以O为端点做线段OB、OC，OA=OB,OC=OD，角AOB等于角COD", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1113.jpg" }, { "index": 1114, "ocr_en": "The equilateral triangle ABC and the equilateral triangle CED share a vertex C, and there is no other intersecting part except the point C; Connect BE and AD", "ocr_zh": "等边三角形ABC与等边三角形CED有一共用顶点C，除点C外无其他相交部分；连接BE、AD", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1114.jpg" }, { "index": 1115, "ocr_en": "The equilateral triangle ABC and the equilateral triangle CED share a vertex C, and there is no other intersecting part except the point C; Connect BE and AD at point P; The angle CAP is angle 1, and the angle CBE is angle 2; The angular APB is 60 degrees, and the angular ACB is 60 degrees", "ocr_zh": "等边三角形ABC与等边三角形CED有一共用顶点C，除点C外无其他相交部分；连接BE、AD交于点P；角CAP为角1，角CBE为角2；角APB为60度，角ACB为60度", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1115.jpg" }, { "index": 1116, "ocr_en": "The equilateral triangle ABC and the equilateral triangle CED share a vertex C, and there is no other intersecting part except the point C; Connect BE and AD at point P; CP bisected angle BPD", "ocr_zh": "等边三角形ABC与等边三角形CED有一共用顶点C，除点C外无其他相交部分；连接BE、AD交于点P；CP平分角BPD", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1116.jpg" }, { "index": 1117, "ocr_en": "The equilateral triangle ABC and the equilateral triangle CED share a vertex C, and there is no other intersecting part except the point C; Connect BE, AD; The corner BCE is marked yellow, and the corner ACD is marked green", "ocr_zh": "等边三角形ABC与等边三角形CED有一共用顶点C，除点C外无其他相交部分；连接BE、AD；角BCE标黄，角ACD标绿", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1117.jpg" }, { "index": 1118, "ocr_en": "The equilateral triangle ABC and the equilateral triangle CED share a vertex C, and there is no other intersecting part except the point C; Connect BE and AD at point P; CP bisector angle BPD; Do auxiliary lines: pass C to do CH perpendicular to H, cross C to do CK perpendicular to AD to K", "ocr_zh": "等边三角形ABC与等边三角形CED有一共用顶点C，除点C外无其他相交部分；连接BE、AD交于点P；CP平分角BPD；做辅助线：过C做CH垂直BE于H，过C做CK垂直于AD于K", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1118.jpg" }, { "index": 1119, "ocr_en": "The equilateral triangle ABC and the equilateral triangle CED share a vertex C, and there is no other intersecting part except the point C; Connect BE and AD at point P; CP bisector angle BPD; Angle CAP is angle 1, angle CBP is angle 2; Take the point F on the BP and make the auxiliary line CF, BF is equal to AP; The corner BCF is marked yellow, and the corner CAP is marked green", "ocr_zh": "等边三角形ABC与等边三角形CED有一共用顶点C，除点C外无其他相交部分；连接BE、AD交于点P；CP平分角BPD；角CAP为角1，角CBP为角2；在BP上取点F，做辅助线CF，BF等于AP；角BCF标黄，角CAP标绿", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1119.jpg" }, { "index": 1120, "ocr_en": "Left: In the quadrilateral ABP, AC and BP are connected, and the triangle ABC is an equilateral triangle with an angle of 120 degrees APC Right: In the quadrilateral DECP, CD and PE are connected, and the triangle DEC is an equilateral triangle with an angle CPD of 120 degrees", "ocr_zh": "左图：在四边形ABCP中，连接AC、BP，三角形ABC为等边三角形，角APC为120度 右图：在四边形DECP中，连接CD、PE，三角形DEC为等边三角形，角CPD为120度", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1120.jpg" }, { "index": 1121, "ocr_en": "The equilateral triangle ABC and the equilateral triangle CED share a vertex C, and there is no other intersecting part except the point C; Connect BE and AD at point P; Connect CP, CP is marked blue", "ocr_zh": "等边三角形ABC与等边三角形CED有一共用顶点C，除点C外无其他相交部分；连接BE、AD交于点P；连接CP，CP标蓝", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1121.jpg" }, { "index": 1122, "ocr_en": "The equilateral triangle ABC and the equilateral triangle ADE have a common vertex A, AC and DE have one and only one intersection point, AB and DE have no intersection point, and the triangle ABC is greater than the triangle ADE; Connect BD and CE, and extend the intersection of BD and CE at point P; Connect to an AP and the AP is marked blue", "ocr_zh": "等边三角形ABC与等边三角形ADE有一共用顶点A，AC与DE有且仅有一个交点，AB与DE无交点，三角形ABC大于三角形ADE；连接BD、CE，延长BD与CE交于点P；连接AP，AP标蓝", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1122.jpg" }, { "index": 1123, "ocr_en": "Make two equilateral triangles ABD and BCE on the same side of the line ABC, connecting AE and CD at point H, AE and DB at point G, DC and BE at point F, and connecting GF", "ocr_zh": "在直线 ABC 的同一侧作两个等边三角形ABD 和BCE，连接 AE 与 CD交于点H，AE与DB交于点G，DC与BE交于点F，连接GF", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1123.jpg" }, { "index": 1124, "ocr_en": "Above: The equilateral triangle ABC and the equilateral triangle BDE share a common vertex B, and there is no other intersection except for the point B, and the angle ABE is less than 90 degrees; There is only one intersection between AD, CE, AD, and DE, and there is no intersection between AD and BE; Extend the intersection of CE and AD at point P; Connect BP and mark blue Below: The equilateral triangle ABC and the equilateral triangle CDE have a common vertex C, BC and DE have and only one intersection point, and the angle ACD is less than 90 degrees; Connect AD and BE, extend the intersection of AD and BE at point P, connect CP and mark blue", "ocr_zh": "上图：等边三角形ABC与等边三角形BDE有一共用顶点B，除点B外无其他相交部分,角ABE小于90度；连接AD、CE，AD、DE仅有一交点D，AD、BE无交点；延长CE与AD交于点P；连接BP并标蓝 下图：等边三角形ABC与等边三角形CDE有一共用顶点C，BC与DE有且仅有一个交点，角ACD小于90度；连接AD、BE，延长AD与BE交于点P，连接CP并标蓝", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1124.jpg" }, { "index": 1125, "ocr_en": "The equilateral triangle ABC and the equilateral triangle CDE have a common vertex C, BC and DE have and only one intersection point, and the angle ACD is less than 90 degrees; Connect AD and BE, extend the intersection of AD and BE at point P, connect CP and mark blue", "ocr_zh": "等边三角形ABC与等边三角形CDE有一共用顶点C，BC与DE有且仅有一个交点，角ACD小于90度；连接AD、BE，延长AD与BE交于点P，连接CP并标蓝", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1125.jpg" }, { "index": 1126, "ocr_en": "The equilateral triangle ABC and the equilateral triangle BDE share a common vertex B, and there is no other intersection except for the point B, and the angle ABE is less than 90 degrees; There is only one intersection between AD, CE, AD, and DE, and there is no intersection between AD and BE; Extend the intersection of CE and AD at point P; Connect BP and mark blue", "ocr_zh": "等边三角形ABC与等边三角形BDE有一共用顶点B，除点B外无其他相交部分,角ABE小于90度；连接AD、CE，AD、DE仅有一交点D，AD、BE无交点；延长CE与AD交于点P；连接BP并标蓝", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1126.jpg" }, { "index": 1127, "ocr_en": "Make two equilateral triangles ABD and BCE on the same side of the line ABC, connecting AE and CD at point H, AE and DB at point G, DC and BE at point F, and connecting GF", "ocr_zh": "在直线 ABC 的同一侧作两个等边三角形ABD 和BCE，连接 AE 与 CD交于点H，AE与DB交于点G，DC与BE交于点F，连接GF", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1127.jpg" }, { "index": 1128, "ocr_en": "Make two equilateral triangles ABD and BCE on the same side of the line ABC, connecting AE and CD at point H, AE and DB at point G, DC and BE at point F, and connecting GF", "ocr_zh": "在直线 ABC 的同一侧作两个等边三角形ABD 和BCE，连接 AE 与 CD交于点H，AE与DB交于点G，DC与BE交于点F，连接GF", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1128.jpg" }, { "index": 1129, "ocr_en": "The isosceles right triangle ABC and the isosceles right triangle ADE share a vertex A, there is no other intersecting part except the point A, the angle BAC is equal to the angle EAD is equal to 90 degrees, and the angle CAE is less than 90 degrees; Connect CD and BE to point G and AG; AG and BG are marked in red", "ocr_zh": "等腰直角三角形ABC与等腰直角三角形ADE有一共用顶点A，除点A外无其他相交部分,角BAC等于角EAD等于90度，角CAE小于90度；连接CD、BE交于点G，连接AG；AG、BG标红", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1129.jpg" }, { "index": 1130, "ocr_en": "Left: The isosceles right triangle ABC and the isosceles right triangle ADE have a common vertex A, AC and DE have and only one intersection point, the angle BAC is equal to the angle EAD is equal to 90 degrees, and the angle BAE is less than 90 degrees; Connecting CD, BE Right: The isosceles right triangle ABC and the isosceles right triangle ADE have a common vertex A, AC and DE have and only one intersection point, the angle BAC is equal to the angle EAD is equal to 90 degrees, and the angle BAE is less than 90 degrees; Connect CD, BE; Extend the intersection of BE and CD at point G, and BG is marked in red", "ocr_zh": "左图：等腰直角三角形ABC与等腰直角三角形ADE有一共用顶点A，AC与DE有且仅有一个交点，角BAC等于角EAD等于90度，角BAE小于90度；连接CD、BE 右图：等腰直角三角形ABC与等腰直角三角形ADE有一共用顶点A，AC与DE有且仅有一个交点，角BAC等于角EAD等于90度，角BAE小于90度；连接CD、BE；延长BE与CD交于点G，BG标红", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1130.jpg" }, { "index": 1131, "ocr_en": "The equilateral triangle ABD and BCE have a common vertex B, and there is no other intersecting part except point B, connecting AE and CD to intersect at point H; AE, CD are marked blue", "ocr_zh": "等边三角形ABD 和BCE有一共用顶点B，除点B外无其他相交部分，连接 AE 与 CD交于点H；AE、CD标蓝", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1131.jpg" }, { "index": 1132, "ocr_en": "The equilateral triangle ABD and BCE have a common vertex B, EC and BD have and only one intersection point, and the angle ABE is less than 90 degrees; Connect AE, CD and mark blue", "ocr_zh": "等边三角形ABD 和BCE有一共用顶点B，EC与BD有且仅有一个交点，角ABE小于90度；连接AE、CD并标蓝", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1132.jpg" }, { "index": 1133, "ocr_en": "The square ABCD and EDGF have a common vertex D, and there is no other intersecting part except D, and the angle CDG is less than 90 degrees; Connect AG and CE to point H, AG and CE are marked in blue", "ocr_zh": "正方形ABCD与EDGF有一共用顶点D，除D外无其他相交部分，角CDG小于90度；连接AG、CE交于点H，AG、CE标蓝", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1133.jpg" }, { "index": 1134, "ocr_en": "The isosceles right triangle ADC and the isosceles right triangle GDE have a common vertex D, and there is no other intersection except the point D, the angle ADC is equal to the angle EDG is equal to 90 degrees, and the angle CDG is less than 90 degrees; Connect AG and CE to point H", "ocr_zh": "等腰直角三角形ADC与等腰直角三角形GDE有一共用顶点D，除点D外无其他相交部分,角ADC等于角EDG等于90度，角CDG小于90度；连接AG、CE交于点H", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1134.jpg" }, { "index": 1135, "ocr_en": "The two isosceles triangles ABD and BCE have a common vertex B, which has no intersecting part except for point B, and the angle DBE is less than 90 degrees; Connect the AE with the CD to H", "ocr_zh": "两个等腰三角形 ABD 与 BCE，有一共用顶点B，除点B外无其他相交部分,角DBE小于90度； AB等于BD,CB等于EB；连接 AE 与 CD交于H", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1135.jpg" }, { "index": 1136, "ocr_en": "In the isosceles right triangle ABC, the angle BAC is equal to 90 degrees, the BC is dotted with D and F, and DF is equal to CF, connecting AD and AF; Outside the triangle ABC, there is a point E near AC, connecting AE, CE, BE, and the EA is perpendicular to A", "ocr_zh": "在等腰直角三角形ABC中，角BAC等于90度，BC上有点D、F，DF等于CF，连接AD、AF；三角形ABC外，靠近AC处有点E，连接AE、CE、BE,EA垂直AD于A", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1136.jpg" }, { "index": 1137, "ocr_en": "In the planar Cartesian coordinate system xoy, the negative half axis of the x-axis is a little D, and the negative half axis of the y-axis is a little C, which is connected to DC and extends to the fourth quadrant, and E is extended on the extension line of the fourth quadrant. The positive half axis of the x-axis is a little A, B, F from left to right, and connects AE, BE, FE; EF is perpendicular to the x-axis of F", "ocr_zh": "在平面直角坐标系xoy中，x轴负半轴有点D，y轴负半轴有点C连接DC并向第四象限延长，在其第四象限延长线上有点E；x轴正半轴从左到右有点A、B、F，连接AE、BE、FE；EF垂直x轴于F", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1137.jpg" }, { "index": 1138, "ocr_en": "In the planar Cartesian coordinate system xoy, the negative half axis of the x-axis is a little D, and the negative half axis of the y-axis is a little C, which is connected to DC and extends to the fourth quadrant, and E is extended on the extension line of the fourth quadrant. The positive half axis of the x-axis is a little A, G, B, F from left to right, connecting AE, GE, BE, FE; EF perpendicular x-axis to F; M is DE CLP, connected to MG; There are some Q and H on the EF, which are connected to MQ and MH, and the GM is vertical to M, and the MQ is vertical to GE", "ocr_zh": "在平面直角坐标系xoy中，x轴负半轴有点D，y轴负半轴有点C连接DC并向第四象限延长，在其第四象限延长线上有点E；x轴正半轴从左到右有点A、G、B、F，连接AE、GE、BE、FE；EF垂直x轴于F；M为DE中电，连接MG；EF上有点Q、H，连接MQ、MH，GM垂直MH于M，MQ与GE垂直", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1138.jpg" }, { "index": 1139, "ocr_en": "The isosceles right triangle ABC and AED have a common vertex A, AC and ED have only one intersection point, and the angle BAE is less than 90 degrees; The angular BAC is equal to the EAD is equal to 90 degrees; Connect BD, CE, AF to BD to F, EC to G", "ocr_zh": "等腰直角三角形ABC 和AED有一共用顶点A，AC与ED有且仅有一个交点，角BAE小于90度；角BAC等于EAD等于90度；连接 BD、CE,AF 交 BD 于 F,交 EC 于G", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1139.jpg" }, { "index": 1140, "ocr_en": "The isosceles right triangle ABC and AED have a common vertex A, AC and ED have only one intersection point, and the angle BAD is less than 90 degrees; The angular BAC is equal to the EAD is equal to 90 degrees; Connect BE, CD, and mark in red; Extend CD to F, make auxiliary line DF, connect AF and BE to G", "ocr_zh": "等腰直角三角形ABC 和AED有一共用顶点A，AC与ED有且仅有一个交点，角BAD小于90度；角BAC等于EAD等于90度；连接BE、CD，并标红；延长CD至F，做辅助线DF，连接AF与BE交于G", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1140.jpg" }, { "index": 1141, "ocr_en": "The isosceles right triangle ABC and AED have a common vertex A, and there is no other intersecting part except point A; The angular BAC is equal to EAD is equal to 90 degrees, and the angular CAE is less than 90 degrees; Connect CE and BD; G is the CE point, connects GA and extends, and crosses BD to F", "ocr_zh": "等腰直角三角形ABC 和AED有一共用顶点A，除点A外无其他相交部分,；角BAC等于EAD等于90度，角CAE小于90度；连接CE、BD； G 是 CE 上一点，连接GA并延长，交BD于F", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1141.jpg" }, { "index": 1142, "ocr_en": "Take A as the vertex to make the equilateral triangle ABC, the isosceles triangle ADE, where AD=AE, and the angle DAE is equal to 120 degrees; AC and BE intersect at point H; Connect BE and DC, Q is the midpoint of BE, connect AQ and extend the intersection DC at point R", "ocr_zh": "以A为顶点做等边三角形ABC，等腰三角形ADE，其中AD=AE，且角DAE等于120度；AC、BE交于点H；连接BE、DC，Q为BE中点，连接AQ并延长交DC于点R", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1142.jpg" }, { "index": 1143, "ocr_en": "Take A as the vertex to make the equilateral triangle ABC, the isosceles triangle ADE, where AD=AE, and the angle DAE is equal to 120 degrees; Connect BE, DC, BE and AC at point H; G is the midpoint of DC, which connects AG and extends the intersection of BC at point P and BE at point F", "ocr_zh": "以A为顶点做等边三角形ABC，等腰三角形ADE，其中AD=AE，且角DAE等于120度；连接BE、DC，BE与AC交于点H；G为DC中点，连接AG并延长交BC于点P，交BE于点F", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1143.jpg" }, { "index": 1144, "ocr_en": "In the square ABCD, E is the midpoint of AD and connects to CE, and P is the point above CE and connects AP and BP", "ocr_zh": "在正方形ABCD中，E为AD中点，连接CE，P为CE上一点，连接AP、BP", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1144.jpg" }, { "index": 1145, "ocr_en": "In the rectangular ABCD, connect AC, E is the midpoint of BC, P is the point on the diagonal AC, and connect EP and BP", "ocr_zh": "在矩形ABCD中，连接AC，E为BC中点，P为对角线AC上一点，连接EP、BP", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1145.jpg" }, { "index": 1146, "ocr_en": "In the right triangle ABC, the angle C is the right angle, and the angle ABC is greater than the angle BCA; E is one point on AC and F is one point on AB, connecting EF and BE", "ocr_zh": "在直角三角形ABC中，角C为直角，角ABC大于角BCA；E为AC上一点，F为AB上一点，连接EF、BE", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1146.jpg" }, { "index": 1147, "ocr_en": "The diagonal of the square ABCD intersects at the point O, and E is the point on BC, connecting AE and OE", "ocr_zh": "正方形ABCD的对角线交于点O，E是BC上一点，连接AE、OE", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1147.jpg" }, { "index": 1148, "ocr_en": "In the square ABCD, P is the inner point of the square, connecting AP and BP, G is the first point on CD, M is the last point of BC, and connecting PG and MG", "ocr_zh": "在正方形ABCD中，P为正方形内一点，连接AP、BP，G为CD上一点，M为BC上一点，连接PG、MG", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1148.jpg" }, { "index": 1149, "ocr_en": "In the planar Cartesian coordinate system xoy, the y-axis positive semi-axis is a little A, B, and OA is less than OB; The positive and semi-axial axes of the x-axis are a little C, D, and OC is less than OD, and AC and BD are connected", "ocr_zh": "在平面直角坐标系xoy中，y轴正半轴有点A、B，OA小于OB；x轴正半轴有点C、D，OC小于OD，连接AC、BD", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1149.jpg" }, { "index": 1150, "ocr_en": "In the square ABCD, P is the upper point of the CD, E is the midpoint of AD, connecting EP, and M is the inner point of the square, connecting AM, BM, and AM perpendicular to BM to M, connecting MP", "ocr_zh": "在正方形ABCD中，P为CD上一点，E为AD中点，连接EP，M为正方形内一点，连接AM、BM，AM垂直于BM于M，连接MP", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1150.jpg" }, { "index": 1151, "ocr_en": "In the planar Cartesian coordinate system xoy, the first quadrant has a right-angled triangle AOB, point B is on the X-axis, and the angular OBA is 90 degrees; P is the point on AO, C is the point on AB, and D is the midpoint of BO; Connect to PD and PC", "ocr_zh": "在平面直角坐标系xoy中，第一象限有直角三角形AOB，点B在X轴上，角OBA为90度；P为AO上一点，C为AB上一点，D为BO中点；连接PD、PC", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1151.jpg" }, { "index": 1152, "ocr_en": "In the diamond-shaped ABCD, AC and BD are connected; E is the midpoint of BC; P and M are the points on AC and AB, respectively, and are connected to MP and PE", "ocr_zh": "在菱形ABCD中，连接AC、BD；E为BC中点；P、M分别为AC、AB上的点，连接MP、PE", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1152.jpg" }, { "index": 1153, "ocr_en": "In the acute triangle ABC, BD bisects the angle ABC and crosses AC to D. There is a bit of M on the BD and a little N on the BC, connecting MN and MC", "ocr_zh": "在锐角三角形ABC中，BD平分角ABC并交AC于D；BD上有点M，BC上有点N，连接MN、MC", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1153.jpg" }, { "index": 1154, "ocr_en": "There is a square ABCD, connected to the diagonal AC, there are a little E and F on the AC, and the AE is less than AF", "ocr_zh": "有正方形ABCD，连接对角线AC，AC上有点E、F，AE小于AF", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1154.jpg" }, { "index": 1155, "ocr_en": "In the planar Cartesian coordinate system xoy, the fourth quadrant has a square ABOC, O is at the coordinate origin, B is on the negative half axis of the x-axis, and C is on the positive half axis of the y-axis. D is the midpoint of BO, E is the point above OC; Connect AD, DE, AE", "ocr_zh": "在平面直角坐标系xoy中，第四象限有正方形ABOC，O在坐标原点，B在x轴负半轴上，C在y轴正半轴上；D为BO中点，E为OC上一点；连接AD、DE、AE", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1155.jpg" }, { "index": 1156, "ocr_en": "The diagonal of the diamond-shaped ABCD intersects the point O, the diamond is connected to the AH bisecting angle BAC, the line segment is AH, and the line AB is connected to the CE, PE, OP", "ocr_zh": "菱形ABCD的对角线交于点O，菱形外有点H，连接AH平分角BAC，线段AH上有点E，直线AB上有点P，连接CE、PE、OP", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1156.jpg" }, { "index": 1157, "ocr_en": "The angle AOB is an acute angle, with a little M on the OA, a little N on the OB, and a little P in the corner BOA, connecting NM, MP, NP, OP", "ocr_zh": "角AOB为锐角，OA上有点M，OB上有点N，角BOA内有一点P，连接NM、MP、NP、OP", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1157.jpg" }, { "index": 1158, "ocr_en": "There is a square ABCD, connected diagonal AC, and there is a bit of P on AC; AB is a little E, F, AE is equal to EF is equal to FB; Connect PE and PF", "ocr_zh": "有正方形ABCD，连接对角线AC，AC上有点P；AB上有点E、F，AE等于EF等于FB；连接PE、PF", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1158.jpg" }, { "index": 1159, "ocr_en": "In the planar Cartesian coordinate system xoy, the second quadrant has a rectangular ABCO, O is at the coordinate origin, A is on the negative half axis of the x-axis, and C is on the positive half axis of the y-axis. A little D on CO, a little E and F on AO; Connect BF, BD, ED", "ocr_zh": "在平面直角坐标系xoy中，第二象限有矩形ABCO，O在坐标原点，A在x轴负半轴上，C在y轴正半轴上；CO上有点D，AO上有点E、F；连接BF、BD、ED", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1159.jpg" }, { "index": 1160, "ocr_en": "In the diamond-shaped ABCD, Q is the midpoint of AB, connecting the diagonal BD, and there is a point P on the BD; Connect to the AP and QP", "ocr_zh": "在菱形ABCD中，Q为AB中点，连接对角线BD，BD上有点P；连接AP、QP", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1160.jpg" }, { "index": 1161, "ocr_en": "In the equilateral triangle ABC, N is the midpoint of AB and AD is the midline on the edge of BC; M is the last point on AD, connecting MN and BM", "ocr_zh": "在等边三角形ABC中，N为AB中点，AD是BC边上的中线；M为AD上一点，连接MN、BM", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1161.jpg" }, { "index": 1162, "ocr_en": "There is a square ABCD that connects the diagonal AC; N and M are the points on AC and CD, respectively, and connect ND and NM", "ocr_zh": "有一正方形ABCD，连接对角线AC；N、M分别为AC、CD上的点，连接ND、NM", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1162.jpg" }, { "index": 1163, "ocr_en": "There is a diamond-shaped ABCD, which connects the diagonal lines AC and BD at the point O; E is the midpoint of AB, and P is the upper point of AC; Connect DP and EP", "ocr_zh": "有菱形ABCD，连接对角线AC、BD交于点O；E为AB中点，P为AC上一点；连接DP、EP", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1163.jpg" }, { "index": 1164, "ocr_en": "The base of the isosceles triangle ABC is BC, with a little E and D on BC and F on AC, which connects EF to form a vertical bisector of waist AC; There is a bit of M on the EF, and it is connected to DM and MC", "ocr_zh": "等腰三角形ABC的底为BC，BC上有点E、D，AC上有点F，连接EF构成腰AC的垂直平分线；EF上有点M，连接DM、MC", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1164.jpg" }, { "index": 1165, "ocr_en": "In a planar Cartesian coordinate system, there are a few Cs and D on the x-axis, and C is to the left of D; The first quadrant is a bit A, and the second quadrant is a bit B; Connect AD and BC", "ocr_zh": "在平面直角坐标系中，x轴上有点C、D，且C在D左侧；第一象限有点A，第二象限有点B；连接AD、BC", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1165.jpg" }, { "index": 1166, "ocr_en": "In the square ABCD, connect the diagonal AC; A bit of E on AD and a bit of F on AB; Connect DF to AC to H, connect CE to DF to G; There is a little P on the CG, which is connected to HP and PQ", "ocr_zh": "在正方形ABCD中，连接对角线AC；AD上有点E，AB上有点F；连接DF交AC于H，连接CE交DF于G；CG上有一点P，连接HP、PQ", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1166.jpg" }, { "index": 1167, "ocr_en": "In the square ABCD, there is a bit E on AB, which connects DE; Rotate ED 90 degrees clockwise around point E to EF; Connect DF and CF", "ocr_zh": "在正方形ABCD中，AB上有点E，连接DE；将ED绕点E顺时针旋转90度到EF；连接DF、CF", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1167.jpg" }, { "index": 1168, "ocr_en": "In the planar Cartesian coordinate system xoy, the first quadrant has an angular AOB, O is at the coordinate origin, B is on the x-axis positive semi-axis, and A is in the first quadrant. M and N are the points on OA and OB, respectively. P is one point within the angular AOB; Connect to MN, MP, NP", "ocr_zh": "在平面直角坐标系xoy中，第一象限有角AOB，O在坐标原点，B在x轴正半轴上，A在第一象限内；M、N分别为OA、OB上的点；P为角AOB内一点；连接MN、MP、NP", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1168.jpg" }, { "index": 1169, "ocr_en": "In the sector BOC, the OD bisects the angle BOC at the BC arc at the point D. E is a point on the radius OB, connecting CE and ED, and the inner mark of CED is the shaded part", "ocr_zh": "在扇形BOC中，OD平分角BOC交BC弧于点D；E为半径OB上一点，连接CE、ED，CED内标为阴影部分", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1169.jpg" }, { "index": 1170, "ocr_en": "In the equilateral triangle ABC, AD is perpendicular to BC and D; E and F are the points on AD and AB, respectively; Connect EF and CE", "ocr_zh": "等边三角形ABC中，AD垂直于BC于D；E、F分别为AD、AB上的点；连接EF、CE", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1170.jpg" }, { "index": 1171, "ocr_en": "In the diamond-shaped ABCD, connect the diagonal AC and BD; The triangle EFG is obtained by translating the triangle ABD along the ray BD direction, and G is on the edge BD. Connect to the EC and GC", "ocr_zh": "在菱形ABCD中，连接对角线AC、BD；将三角形ABD沿射线BD方向平移得到三角形EFG，G在边BD上；连接EC、GC", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1171.jpg" }, { "index": 1172, "ocr_en": "In the rectangular ABCD, connect the diagonal BD; M and N are the points on BD and AB, respectively, and connect AM and MN", "ocr_zh": "在矩形ABCD中，连接对角线BD；M、N分别为BD、AB上的点，连接AM、MN", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1172.jpg" }, { "index": 1173, "ocr_en": "In the planar Cartesian coordinate system xoy, A and B are the two points on the bisector of the first quadrant angle, and OA is less than OB. C is the middle point of the first quadrant, AC is parallel to the x-axis, BC is parallel to the y-axis, and AC is perpendicular to BC to C. D is a point on the y-axis, connecting AD and BD", "ocr_zh": "在平面直角坐标系xoy中，A、B为第一象限角平分线上的两点，OA小于OB；C为第一象限中一点，AC平行于x轴，BC平行于y轴，AC垂直于BC于C；D为y轴上一点，连接AD、BD", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1173.jpg" }, { "index": 1174, "ocr_en": "The triangle ADE and the triangle BCE are two equilateral triangles located on the same side of the straight line AB and sharing the vertex E, connecting the CD; P and F are the midpoints of CD and AB, respectively, and connect AP, EP, FP, and BP", "ocr_zh": "三角形ADE和三角形BCE是位于直线AB同侧且共用顶点E的两个等边三角形，连接CD；P、F分别是CD、AB的中点，连接AP、EP、FP、BP", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1174.jpg" }, { "index": 1175, "ocr_en": "In the right-angled triangle ABC, the angle BAC is a right angle, and E and D are the points on AC and BC, respectively, connecting AD and DE", "ocr_zh": "在直角三角形ABC中，角BAC为直角，E、D分别为AC、BC上的点，连接AD、DE", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1175.jpg" }, { "index": 1176, "ocr_en": "In the square ABCD, there is a point P and E on AB to connect PD and CE, and B as BG perpendicular to CE and PG to point G", "ocr_zh": "在正方形ABCD中,AB上有点P、E,连接PD、CE,过B作BG垂直于CE于点G连接PG", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1176.jpg" }, { "index": 1177, "ocr_en": "The equilateral triangle ABC has a little D on the side AC, and the connection BD becomes the height of the triangle; There is a bit of E on the BD, connected to CE; Rotate CE clockwise to obtain CF, and point F is outside the triangle ABC and close to AC, connecting DF 、 EF、CF、AF", "ocr_zh": "等边三角形ABC的边AC上有点D，连接BD成为三角形的高；BD上有点E，连接CE；将CE顺时针旋转得到CF，F点在三角形ABC外且靠近AC处，连接DF、EF、AF、CF", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1177.jpg" }, { "index": 1178, "ocr_en": "In the right triangle ABC, the angle C is the right angle, with AB as the side, the square ABDE is made on AB, the point D is made DF perpendicular to CB, and the extension line of CB is crossed to F; Connect BE, there is a little N on the BE, a little P on the AC, and connect AN and PN", "ocr_zh": "在直角三角形ABC中，角C为直角，以AB为边，在AB上做正方形ABDE过点D做DF垂直于CB,交CB的延长线于F；连接BE,BE上有一点N,AC上有一点P,连接AN、PN", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1178.jpg" }, { "index": 1179, "ocr_en": "There is a point A on the side of the cylindrical shape, and a point B on the bottom to make the auxiliary line AB", "ocr_zh": "在圆柱形的侧面上有一点A，上底上有一点B，做辅助线AB", "class": "Solid Geometry", "difficulty": 1, "image_path": "fig_1179.jpg" }, { "index": 1180, "ocr_en": "In the planar Cartesian coordinate system xoy, there is a parabola; The vertex M of the parabola is located in the first quadrant, the positive half axis of the parabolic intersection y axis is C, the negative half axis of the intersection x-axis is A, and the positive half axis of the intersection is B. The line AM intersects the y-axis at point D", "ocr_zh": "在平面直角坐标系xoy中，有一抛物线；抛物线的顶点M位于第一象限，抛物线交y轴正半轴为C，交x轴负半轴于A，交x轴正半轴于B；直线AM与y轴交于点D", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1180.jpg" }, { "index": 1181, "ocr_en": "Left: In the planar Cartesian coordinate system xoy, there is a parabola; The vertex of the parabola is located in the first quadrant, the positive half axis of the parabola is C, the negative half axis of the x-axis is A, and the positive half axis of the x-axis is B. Do auxiliary line: the axis of symmetry of the parabola Right: In the plane Cartesian coordinate system xoy, there is a parabola; The vertex of the parabola is located in the first quadrant, the positive half axis of the parabola is C, the negative half axis of the x-axis is A, and the positive half axis of the x-axis is B. Do Auxiliary Line: The axis of symmetry of the parabola", "ocr_zh": "左图：在平面直角坐标系xoy中，有一抛物线；抛物线的顶点位于第一象限，抛物线交y轴正半轴为C，交x轴负半轴于A，交x轴正半轴于B；做辅助线：抛物线的对称轴 右图：在平面直角坐标系xoy中，有一抛物线；抛物线的顶点位于第一象限，抛物线交y轴正半轴为C，交x轴负半轴于A，交x轴正半轴于B；做辅助线：抛物线的对称轴", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1181.jpg" }, { "index": 1182, "ocr_en": "In the first quadrant of the planar Cartesian coordinate system xoy, there is a diamond-shaped ABCO, O coincides with the origin, and C is on the positive semi-axis of the x-axis; Connect the diagonal OB, a little P on the OB; Connect to an AP", "ocr_zh": "在平面直角坐标系xoy第一象限中，有一菱形ABCO，O与原点重合，C在x轴正半轴上；连接对角线OB，OB上有点P；连接AP", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1182.jpg" }, { "index": 1183, "ocr_en": "In the triangle ABC, AB is equal to AC, and B is done as BE perpendicular AC at the point E; D is a point on BE, connect to the CD", "ocr_zh": "在三角形ABC中，AB等于AC,过B做BE垂直AC于点E;D为BE上一点,连接CD", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1183.jpg" }, { "index": 1184, "ocr_en": "In the equilateral triangle ABC, there is a point D on AC that connects BD, BD bisects the angle ABC, and E is a point on BD that connects AE", "ocr_zh": "在等边三角形ABC中，AC上有一点D,连接BD,BD平分角ABC,E为BD上一点连接AE", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1184.jpg" }, { "index": 1185, "ocr_en": "In the triangle ABC, the angle ABC is an obtuse angle, and the angle A is less than the angle C is smaller than the angle ABC, and P is the point on the AC, connecting BP", "ocr_zh": "在三角形ABC中,角ABC为钝角，且角A小于角C小于角ABC,P为AC上一点,连接BP", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1185.jpg" }, { "index": 1186, "ocr_en": "In the right-angled triangle ABC, the angle ACB is equal to 90 degrees, AD and AE are intercepted on AC and AB respectively, D and E are the center of the circle, a certain length is the radius as the arc, and the two arcs intersect at the point M in the angle BAC. Do the ray AM cross BC to F, P is a point on AF, connect to PC", "ocr_zh": "在直角三角形ABC中，角ACB等于90度，在AC和AB上分别截取AD、AE,以D和E分别为圆心,一定长度为半径作弧，两弧在角BAC内交于点M;做射线AM交BC于F,P为AF上一点,连接PC", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1186.jpg" }, { "index": 1187, "ocr_en": "In the planar Cartesian coordinate system xoy, there is a quadratic function image; The positive half axis of the X axis of the image is C, the negative half axis of the X axis is A, and the negative half axis of the y axis is B, and its vertex is located in the fourth quadrant. D is on the y-axis positive semi-axis, P is on the x-axis, and DP is connected", "ocr_zh": "在平面直角坐标系xoy中，有一二次函数图像；图像交x轴正半轴于C，交x轴负半轴于A，交y轴负半轴于B，其顶点位于第四象限；D在y轴正半轴上，P在x轴上，连接DP", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1187.jpg" }, { "index": 1188, "ocr_en": "In the planar Cartesian coordinate system xoy, there is a point of C on the positive half axis of the x-axis, a point of A and B on the positive half axis of the y-axis, and OA is greater than OB, connecting CA and CB", "ocr_zh": "在平面直角坐标系xoy中，x轴正半轴上有点C，y轴正半轴上有点A、B，OA大于OB，连接CA、CB", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1188.jpg" }, { "index": 1189, "ocr_en": "In the triangle ABC, B is passed to make BD perpendicular to AC to AC to D, P is a point on BD, and PC is connected", "ocr_zh": "在三角形ABC中,过B做BD垂直于AC交AC于D,P为BD上一点,连接PC", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1189.jpg" }, { "index": 1190, "ocr_en": "In the parallelogram ABCD, P is a point on the CD, connecting BP", "ocr_zh": "在平行四边形ABCD中,P为CD上一点,连接BP", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1190.jpg" }, { "index": 1191, "ocr_en": "In the triangle ABC, the angle A is a right angle", "ocr_zh": "在三角形ABC中,角A为直角", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1191.jpg" }, { "index": 1192, "ocr_en": "In the right triangle ABC, the angle ACB is equal to 90 degrees; D is the midpoint of AB, connected to CD; There is a little P on the CD, connect to the AP", "ocr_zh": "在直角三角形ABC中，角ACB等于90度;D为AB的中点,连接CD;CD上有一点P,连接AP", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1192.jpg" }, { "index": 1193, "ocr_en": "In the planar Cartesian coordinate system, there is a straight line with one, two, and four quadrants, which intersects the y-axis and A, and the x-axis and B; There are some P, Q on the y-axis, and P is above A, and Q is below A; Connect the BQ", "ocr_zh": "在平面直角坐标系中，有一过一、二、四象限的直线，交y轴与A,交x轴与B；y轴上有点P、Q，且P在A的上方,Q在A的下方;连接BQ", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1193.jpg" }, { "index": 1194, "ocr_en": "In the diamond-shaped ABCD, connect the diagonal AC; E is a point on the AC and is connected to EB", "ocr_zh": "在菱形ABCD中，连接对角线AC;E为AC上一点,连接EB", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1194.jpg" }, { "index": 1195, "ocr_en": "In the diamond-shaped ABCD, connect the diagonal BD; P is a little bit on the BD, connected to the PC", "ocr_zh": "在菱形ABCD中，连接对角线BD;P为BD上一点,连接PC", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1195.jpg" }, { "index": 1196, "ocr_en": "In the quadrilateral ABCD, AC and BD are connected, and the two line segments intersect at O; AC bisects BD perpendicularly", "ocr_zh": "在四边形ABCD中,连接AC、BD，两线段相交于O；AC垂直平分BD", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1196.jpg" }, { "index": 1197, "ocr_en": "In the planar Cartesian coordinate system, there is a quadratic function image, the image intersects the negative half axis of the x-axis in A, the positive half-axis of the x-axis in B, and the positive half-axis of the y-axis in C. Auxiliary line: make its axis of symmetry cross the x-axis positive half axis to D; P is a point on the y-axis, connected to PD", "ocr_zh": "在平面直角坐标系中，有一二次函数图像，图像交x轴负半轴于A，交x轴正半轴于B，交y轴正半轴于C；辅助线：做其对称轴交x轴正半轴于D；P为y轴上一点，连接PD", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1197.jpg" }, { "index": 1198, "ocr_en": "In a triangular OPM, the angular OPM is equal to 90 degrees, and G is one point on the OM that connects the PG", "ocr_zh": "在三角形OPM中,角OPM等于90度，G为OM上一点,连接PG", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1198.jpg" }, { "index": 1199, "ocr_en": "In the triangle ABC, the angle ACB is equal to 90 degrees; Point C folds along BE and coincides with point D on AB; I do auxiliary line DE, BE", "ocr_zh": "在三角形ABC中,角ACB等于90度;点C沿BE折叠与AB上的点D重合;做辅助线DE、BE", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1199.jpg" }, { "index": 1200, "ocr_en": "In triangle ABC, D is a point on BC, connecting AD, and E is a point on AD", "ocr_zh": "在三角形ABC中，D为BC上一点，连接AD，E为AD上一点", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1200.jpg" }, { "index": 1201, "ocr_en": "There is a line segment AC, and there is a little B above AC and AB is connected", "ocr_zh": "有线段AC，AC上方有点B，连接AB", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1201.jpg" }, { "index": 1202, "ocr_en": "In triangle ABC, D is a point on BC and connects AD; M is the previous point on AD, which is connected to CM", "ocr_zh": "在三角形ABC中，D为BC上一点，连接AD；M为AD上一点，连接CM", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1202.jpg" }, { "index": 1203, "ocr_en": "In the planar Cartesian coordinate system, there is a parabola, the parabola intersects the negative half axis of the x-axis at A, the positive half-axis of the x-axis at B, and the negative half-axis of the y-axis at C. Connect AC and CB; Auxiliary line: Do its axis of symmetry cross the x-axis positive half axis to D", "ocr_zh": "在平面直角坐标系中，有一抛物线，抛物线交x轴负半轴于A，交x轴正半轴于B，交y轴负半轴于C；连接AC、CB；辅助线：做其对称轴交x轴正半轴于D", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1203.jpg" }, { "index": 1204, "ocr_en": "In the planar Cartesian coordinate system, there is a quadratic function image, the image intersects the negative half axis of the x-axis in A, the positive half-axis of the x-axis in C, and the negative half-axis of the y-axis in B. D is on the y-axis positive half-axis, P is on the x-axis positive half-axis, connected to DP", "ocr_zh": "在平面直角坐标系中，有一二次函数图像，图像交x轴负半轴于A，交x轴正半轴于C，交y轴负半轴于B；D在y轴正半轴上，P在x轴正半轴上，连接DP", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1204.jpg" }, { "index": 1205, "ocr_en": "In the right triangle ABC, the angle ACB is equal to 90 degrees, E is a point on the edge of BC, C is the center of the circle, and CE is the radius to make a circle, and AC and BC are respectively intersected at two points D and E; P is a point in the triangle ABC, which connects AP and BP", "ocr_zh": "在直角三角形ABC中，角ACB等于90度，E为BC边上一点，以C为圆心,CE为半径做圆，分别交AC，BC于D、E两点；P为三角形ABC内一点，连接AP、BP", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1205.jpg" }, { "index": 1206, "ocr_en": "In the isosceles right triangle ABC, the angle A is equal to 90 degrees; E and F are the midpoints of AB and AC, with A as the center of the circle and AE as the radius to make the sector AEF; P is the point on the EF arc, connecting BP and CP", "ocr_zh": "在等腰直角三角形ABC中，角A等于90度；E、F是AB、AC的中点，以A为圆心，AE为半径做扇形AEF；P为EF弧上的点,连接BP、CP", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1206.jpg" }, { "index": 1207, "ocr_en": "In the triangle ABC, the angle ACB is equal to 90 degrees; Take C as the center of the circle and CD as the radius to make a circle, connecting AD and BD; CD is smaller than AC and CB", "ocr_zh": "在三角形ABC中,角ACB等于90度;以C为圆心，CD为半径做圆,连接AD、BD;CD小于AC和CB", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1207.jpg" }, { "index": 1208, "ocr_en": "In the circle O, the points A and B are on the circle, and the angle AOB is equal to 90 degrees; There is a point C on the OA, D on the OB, and M on the arc AB, connecting CD, DM, MC", "ocr_zh": "在圆O中,点A、B在圆上,角AOB等于90度；OA上有点C，OB上中点为D，弧AB上有点M，连接CD、DM、MC", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1208.jpg" }, { "index": 1209, "ocr_en": "In the planar Cartesian coordinate system, the origin O is the center of the circle, and OA is the radius to make the circle. A is on the positive semi-axis of the x-axis; The x-axis positive semi-axis is a little N, and the OA is less than ON, the first quadrant is a little M, and the circle O is a little P, connecting PM and PM", "ocr_zh": "在平面直角坐标系中，以原点O为圆心，OA为半径做圆；A在x轴正半轴上;x轴正半轴有点N，且OA小于ON，第一象限中有点M，圆O上有点P,连接PM和PM", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1209.jpg" }, { "index": 1210, "ocr_en": "In the diamond ABCD, the circle is made with the point B as the center of the circle, and the radius of the circle is less than BC, and in the diamond ABCD, there is a point P on the arc of the circle B, which connects PD and PC", "ocr_zh": "在菱形ABCD中，以点B为圆心做圆,圆的半径小于BC,在菱形ABCD中、圆B的弧上有一点P，连接PD、PC", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1210.jpg" }, { "index": 1211, "ocr_en": "In a half-circle, O is the center of the circle and AB is the diameter; There is a little P on the arc AB, and AC and BD are circular tangents that connect CP and PD", "ocr_zh": "在一半圆中，O为圆心，AB为直径；弧AB上有一点P,AC、BD为圆的切线,连接CP,PD", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1211.jpg" }, { "index": 1212, "ocr_en": "In an isosceles right-angled triangle ACB, the angle ACB is equal to 90 degrees; O is the point on AB, and the semicircle with O as the center of the circle is tangent to both AC and BC; P is a point on the semicircle, connecting CP and BP", "ocr_zh": "在等腰直角三角形ACB中,角ACB等于90度;O为AB上一点,以O为圆心的半圆与AC和BC均相切;P为半圆上一点,连接CP、BP", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1212.jpg" }, { "index": 1213, "ocr_en": "In the planar Cartesian coordinate system, there is a triangular AOB, where O coincides with the origin, B is on the y-axis positive semi-axis, and A is on the x-axis positive semi-axis; P is on the first quadrant outside the triangle AOB, connecting BP and PA; There is a point C on the positive half axis of the x-axis, and the OA is less than the OC, and D is the point in the first quadrant, which is connected to the PD", "ocr_zh": "在平面直角坐标系中有三角形AOB，其中O与原点重合，B在y轴正半轴上，A在x轴正半轴上；P在三角形AOB外部的第一象限上，连接BP、PA；x轴正半轴上有一点C，且OA小于OC，D为第一象限中一点，连接PD", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1213.jpg" }, { "index": 1214, "ocr_en": "There is a square ABCD, with the point B as the center of the circle, and the radius less than BC is the circle B; P is the point on the circle B and in the quadrilateral ABCD, connecting PD and P", "ocr_zh": "有正方形ABCD，以点B为圆心,小于BC的半径做圆B；P为圆B上、四边形ABCD内的点，连接PD、PC", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1214.jpg" }, { "index": 1215, "ocr_en": "In a triangular PCB, the angle C is smaller than the angle B is smaller than the angle CPB, and N is the point on the BC, which is connected to the PN", "ocr_zh": "在三角形PCB中，角C小于角B小于角CPB，N为BC上一点，连接PN", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1215.jpg" }, { "index": 1216, "ocr_en": "In the plane Cartesian coordinate system, there are parabolas with one, two, and four quadrants, the parabolic vertex is in the fourth quadrant, the parabola intersects the y-axis positive semi-axis at the point C, and the positive half-axis of the intersecting x-axis is at the two points A and B, and the OA is less than OB", "ocr_zh": "在平面直角坐标系中，有一过一、二、四象限的抛物线，抛物线顶点在第四象限中，抛物线交y轴正半轴于点C，交x轴正半轴于A、B两点，且OA小于OB", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1216.jpg" }, { "index": 1217, "ocr_en": "In the planar Cartesian coordinate system, there is a parabola with one, two, and four quadrants, the parabolic vertex is in the fourth quadrant, the parabola intersects the y-axis positive half axis at the point C, and the positive half-axis of the x-axis is at the two points A and B, and the OA is less than OB; Take point B as the center of the circle, make a circle with a radius smaller than AB, and a point P on the circle to connect AP and CP", "ocr_zh": "在平面直角坐标系中，有一过一、二、四象限的抛物线，抛物线顶点在第四象限中，抛物线交y轴正半轴于点C，交x轴正半轴于A、B两点，且OA小于OB；以点B为圆心，小于AB的半径做圆，圆上有一点P，连接AP、CP", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1217.jpg" }, { "index": 1218, "ocr_en": "In the plane Cartesian coordinate system, there is a quadratic function image, the image and the coordinate axis intersect at the origin O and the point B located on the positive half axis of the x-axis, with O as the center of the circle, and the radius less than OB is the circle O; P is on the circle O, one point in the fourth quadrant, and A is the vertex of the function, connecting AP, AB, BP; C is not in the circle O on OA", "ocr_zh": "在平面直角坐标系中有一二次函数图像，图像与坐标轴交于原点O和位于x轴正半轴的B点两点，以O为圆心，小于OB的半径做圆O；P为圆O上,第四象限中一点,A为函数顶点，连接AP、AB、BP；C在OA上不在圆O内", "class": "Analytic Geometry", "difficulty": 3, "image_path": "fig_1218.jpg" }, { "index": 1219, "ocr_en": "In the isosceles right triangle ABC, the angle B is equal to 90 degrees, with B as the center of the circle to make the circle B tangent to AC, P is on the circle B, a point in the triangle ABC, connecting AP and CP", "ocr_zh": "在等腰直角三角形ABC中，角B等于90度，以B为圆心做圆B与AC相切，P为圆B上、三角形ABC内一点，连接AP、CP", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1219.jpg" }, { "index": 1220, "ocr_en": "Left: In the planar Cartesian coordinate system, there is a parabola with the positive half axis of the y-axis at the point C, and the positive half-axis of the x-axis at the points A and B, and the OA is less than OB. The axis of symmetry of the parabola of the auxiliary line, the positive half axis of the intersection x-axis is at the point E; A straight line with a passing point A, in the first quadrant, the parabolic symmetry axis is at the point D right: in the plane Cartesian coordinate system, there is a parabola with the positive half axis of the y-axis at the point C, and the positive half axis of the x-axis at the points A and B, and OA is less than OB; Take point B as the center of the circle, the radius less than AB is the circle B, and the point P is on the circle B, a point in the parabola of the first quadrant, connecting AP and CP", "ocr_zh": "左图：在平面直角坐标系中，有一抛物线交y轴正半轴于点C，交x轴正半轴于点A、B，且OA小于OB；做辅助线抛物线的对称轴，交x轴正半轴于点E；有一过点A的直线，在第一象限中交抛物线对称轴于点D 右图：在平面直角坐标系中，有一抛物线交y轴正半轴于点C，交x轴正半轴于点A、B，且OA小于OB；以点B为圆心，小于AB的半径做圆B， P点在圆B上、第一象限抛物线内一点，连接AP、CP", "class": "Analytic Geometry", "difficulty": 3, "image_path": "fig_1220.jpg" }, { "index": 1221, "ocr_en": "In the planar Cartesian coordinate system, the origin O is the center of the circle, and OP is the radius to make the circle (OP is the auxiliary line). There are some points C AND D ON THE X AND Y AXES, AND C AND D ARE OUTSIDE THE CIRCLE O; There is a little M in the circle O and on the y axis, and PD, PC, MP, MC are connected", "ocr_zh": "在平面直角坐标系中，以原点O为圆心，OP为半径做圆（OP为辅助线）;x、y轴上分别有点C、D，C、D两点均在圆O外；圆O内、y轴上有点M，连接PD、PC、MP、MC", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1221.jpg" }, { "index": 1222, "ocr_en": "In the right triangle ABC, the angle ACB is equal to 90 degrees; There is a dot D inside the triangle, which connects AD, CD, BD", "ocr_zh": "在直角三角形ABC中，角ACB等于90度；三角形内有一点D，连接AD、CD、BD", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1222.jpg" }, { "index": 1223, "ocr_en": "Points A and B are on the circle O, OA and OB are the radius of the circle, and OA is perpendicular to O; C is the midpoint of AO, D is a point on the OB, and P is on the arc AB, which is connected to PC and PD", "ocr_zh": "点A、B在圆O上,OA、OB为圆的半径，且OA垂直OB于O；C为AO的中点，D为OB上一点，P在弧AB上，连接PC、PD", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1223.jpg" }, { "index": 1224, "ocr_en": "In the sector COD, the angular COD is equal to 90 degrees, P is a point on the CD arc, and A and B are the points on OC and OD, respectively, connecting AP and BP", "ocr_zh": "在扇形COD中,角COD等于90度，P为CD弧上一点，A、B分别为OC、OD上的点,连接AP、BP", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1224.jpg" }, { "index": 1225, "ocr_en": "In the right triangle ABC, the angle C is equal to 90 degrees, with C as the center of the circle, the radius less than CB is the circle C, the point P is on the circle C, and in the triangle ABC, AP and BP are connected", "ocr_zh": "在直角三角形ABC中，角C等于90度，以C为圆心，小于CB的半径做圆C，点P在圆C上，且在三角形ABC内，连接AP、BP", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1225.jpg" }, { "index": 1226, "ocr_en": "In the right triangle ABC, the angle C is equal to 90 degrees; Take C as the center of the circle, the radius less than CB is the circle C, the point P is on the circle C, and in the triangle ABC, connect AP and BP; There is a bit of D on the CB, and D is in the circle C at the same time, doing the auxiliary line CP and DP", "ocr_zh": "在直角三角形ABC中，角C等于90度；以C为圆心，小于CB的半径做圆C，点P在圆C上，且在三角形ABC内，连接AP、BP；CB上有点D，D同时在圆C内，做辅助线CP、DP", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1226.jpg" }, { "index": 1227, "ocr_en": "In the plane Cartesian coordinate system, there is a parabola, which intersects with the negative semi-axis of the x-axis at point A, with the positive semi-axis of the x-axis at point B, and with the positive semi-axis of the y-axis at point C. D is the vertex of the parabola, on the first quadrant; M is in the parabola in the first quadrant, connecting DM and OM; Take B as the center of the circle, and make a circle with a radius less than OB that passes the point M", "ocr_zh": "在平面直角坐标系中有一抛物线，抛物线与x轴负半轴交于点A，与x轴正半轴交于点B，与y轴正半轴交于点C；D为抛物线的顶点，在第一象限上；M在第一象限中的抛物线内，连接DM、OM；以B为圆心，小于OB的半径做一过点M的圆", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1227.jpg" }, { "index": 1228, "ocr_en": "In a planar Cartesian coordinate system, A is in the third quadrant, B is on the positive semi-axis of the y-axis, and C is on the negative semi-axis of the y-axis. AC and AB are connected, and AB intersects with the x-axis at point E; The circle is made with E as the center and ED length as the radius, D on the y-axis negative half axis and OD less than OC. M is in the third quadrant, a point above the circle, connecting EM, AM, CM", "ocr_zh": "在平面直角坐标系中，A在第三象限，B在y轴正半轴上，C在y轴负半轴上；连接AC、AB，AB与x轴相交于点E；以E为圆心，ED长为半径做圆，D在y轴负半轴上且OD小于OC；M为第三象限中，圆上一点，连接EM、AM、CM", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1228.jpg" }, { "index": 1229, "ocr_en": "There is a rectangular ABCD, and there is a little P on the BC, which is connected to the AP; Fold the triangle ABP in half along AP to get the triangle AB'P, which connects B'C and B' inside the rectangle", "ocr_zh": "有矩形ABCD，BC上有一点P，连接AP；将三角形ABP沿AP对折，得到三角形AB'P，连接B'C，B'在矩形内部", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1229.jpg" }, { "index": 1230, "ocr_en": "In the square ABCD, E and F are the dots on CD and AD, respectively; Connect BF and AE at point P, and BF perpendicular to AE, connect to PC", "ocr_zh": "在正方形ABCD中，E、F分别是CD、AD上的点；连接BF、AE相交于点P，且BF垂直AE，连接PC", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1230.jpg" }, { "index": 1231, "ocr_en": "In the rectangular ABCD, there is a little P; There is a bit of E on AB, connecting EP, BP, DP", "ocr_zh": "在矩形ABCD中，有一点P；AB上有点E，连接EP、BP、DP", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1231.jpg" }, { "index": 1232, "ocr_en": "In the planar Cartesian coordinate system, there is a square OABC, where O coincides with the origin, C is on the y-axis positive semi-axis, and A is on the x-axis positive semi-axis; There is a little P on the OA, which is connected to the PC; There is a little D on the negative half axis of the x-axis, which connects the CD; Do a DE vertical PC over D to point E, connect AE", "ocr_zh": "在平面直角坐标系中有一正方形OABC，其中O与原点重合，C在y轴正半轴上，A在x轴正半轴上；OA上有一点P，连接PC；在x轴负半轴上有一点D，连接CD；过D做DE垂直PC于点E，连接AE", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1232.jpg" }, { "index": 1233, "ocr_en": "In the isosceles right triangle ABC, ACB is called 90 degrees; M is a point on the AC, connected to the BM; With CM as the diameter, make a circle O cross BM at the point N, and connect AN", "ocr_zh": "在等腰直角三角形ABC中，叫ACB等于90度；M为AC上一点，连接BM；以CM为直径做圆O交BM于点N，连接AN", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1233.jpg" }, { "index": 1234, "ocr_en": "In the rectangular ABCD, there is a point P on the side BC, which connects to the AP; M is a point inside the quadrilateral APCD, connected to CM", "ocr_zh": "在矩形ABCD中，边BC上有一点P，连接AP；M为四边形APCD内一点，连接CM", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1234.jpg" }, { "index": 1235, "ocr_en": "There is an obtuse triangle ABC, and angle C is an obtuse angle", "ocr_zh": "有一钝角三角形ABC，角C为钝角", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1235.jpg" }, { "index": 1236, "ocr_en": "In the right triangle ABC, the angle ACB is equal to 90 degrees; D and E are the points on the edges AB and AC, respectively, and connect CD and BE to intersect O", "ocr_zh": "在直角三角形ABC中，角ACB等于90度；D、E分别为边AB、AC上的点，连接CD、BE相交于O", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1236.jpg" }, { "index": 1237, "ocr_en": "There is a trapezoidal ABCD, which connects its diagonal AC; There is a little M inside the triangular ACD, which connects AM and DM", "ocr_zh": "有一梯形ABCD，连接其对角线AC；在三角形ACD内有一点M，连接AM、DM", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1237.jpg" }, { "index": 1238, "ocr_en": "In the right-angled triangle AOB, the angle O is a right angle; Take O as the center of the circle, and the radius less than the length of OB is made as a circle O; P is a point on AB, and P is a tangent PQ of round O, where Q is the tangent point", "ocr_zh": "在直角三角形AOB中，角O为直角；以O为圆心，小于OB长度的半径做圆O；P为AB上一点，过P做圆O的切线PQ，其中Q为切点", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1238.jpg" }, { "index": 1239, "ocr_en": "In the rectangular ABCD, E is the midpoint of AB, P is the point above BC, and the connecting EP (EB and BP are dashed lines); The triangle PBE is folded along PE to obtain a triangle EFP, and F is in the rectangular ABCD. Connect the CF", "ocr_zh": "在矩形ABCD中，E为AB的中点，P为BC上一点，连接EP（EB、BP为虚线）；将三角形PBE沿PE折叠得到三角形EFP，F在矩形ABCD内；连接CF", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1239.jpg" }, { "index": 1240, "ocr_en": "In the quadrilateral ABCD, AC and BD are connected to the point O; In triangle ABC, AB=AC; In a triangular ACD, AC=AD; In the triangle ABO, AB=BO", "ocr_zh": "在四边形ABCD中，连接AC、BD交于点O；三角形ABC中，AB=AC；三角形ACD中，AC=AD；三角形ABO中，AB=BO", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1240.jpg" }, { "index": 1241, "ocr_en": "In the rectangular ABCD, P is the point on AD, connecting BP; The line segment BA and the line segment BQ are symmetrical with respect to the line where the BP is located, and the PQ is connected", "ocr_zh": "在矩形ABCD中，P为AD上一点，连接BP；线段BA与线段BQ关于BP所在的直线对称，连接PQ", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1241.jpg" }, { "index": 1242, "ocr_en": "In the quadrilateral ABCD, the angle DAB=ABC=90 degrees, the angle ADC is an obtuse angle, the angle BCD is an acute angle, and the AD is less than BC; There is a little E on the BC, which connects to the AE; There is a bit of F on the AE, and BF and DF are connected", "ocr_zh": "在四边形ABCD中，角DAB=角ABC=90度，角ADC为钝角，角BCD为锐角，AD小于BC；BC上有一点E，连接AE；AE上有点F，连接BF、DF", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1242.jpg" }, { "index": 1243, "ocr_en": "In the equilateral triangle ABC, D and E are the points on the edges BC and CA, respectively, and connect AD and BE to F", "ocr_zh": "在等边三角形ABC中，D、E分别为边BC、CA上的点，连接AD、BE交于F", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1243.jpg" }, { "index": 1244, "ocr_en": "A little F inside the square ABCD and an E on the side AD; Connect BE, EF, DF, CF", "ocr_zh": "在正方形ABCD内有点F，边AD上有点E；连接BE、EF、DF、CF", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1244.jpg" }, { "index": 1245, "ocr_en": "A little M inside the rectangular ABCD and a little E on the side BC; Connect AM, CM, EM", "ocr_zh": "在矩形ABCD内有点M，边BC上有点E；连接AM、CM、EM", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1245.jpg" }, { "index": 1246, "ocr_en": "In the quadrilateral ABCD, the angle BAD=angle BCD=90 degrees, the angle CBA is an obtuse angle, and the angle CDA is an acute angle; Connect to the AC and BD", "ocr_zh": "在四边形ABCD中，角BAD=角BCD=90度，角CBA为钝角，角CDA为锐角；连接AC、BD", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1246.jpg" }, { "index": 1247, "ocr_en": "E and F are two points on the edge AD of the square ABCD, and AE=DF; Connect CF to BD at point G, connect AG to BE at point H, and connect DH", "ocr_zh": "E、F分别为正方形ABCD的边AD上两点，且AE=DF；连接CF交BD于点G，连接AG交BE于点H，连接DH", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1247.jpg" }, { "index": 1248, "ocr_en": "There is an isosceles right-angled triangle ABC, and the angle A is equal to 90 degrees; Let D be a point outside the triangle ABC, close to the side of the side AC, and connect BD and CD; Angle D is an acute angle, and angle BCD is an obtuse angle", "ocr_zh": "有等腰直角三角形ABC，角A等于90度；设D为三角形ABC外、靠近边AC一侧的一点，连接BD、CD；角D为锐角，角BCD为钝角", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1248.jpg" }, { "index": 1249, "ocr_en": "In the planar Cartesian coordinate system, O is the origin, B is a point on the y-axis positive semi-axis, and A is a point on the x-axis positive semi-axis, connecting AB; C is the first quadrant, a point outside the triangle BOA, connecting BC and AC; M is a point on the AC and connects to the OM", "ocr_zh": "在平面直角坐标系中，O为原点，B为y轴正半轴上一点，A为x轴正半轴上一点，连接AB；C为第一象限中、三角形BOA外一点，连接BC、AC；M为AC上一点，连接OM", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1249.jpg" }, { "index": 1250, "ocr_en": "In the planar Cartesian coordinate system, O is the origin, B is a point on the y-axis, A is a point on the x-axis, and C is a point in the first quadrant. Connect AC and BC, and AC is greater than BC; M is the midpoint of the AC, which is connected to the OM", "ocr_zh": "在平面直角坐标系中，O为原点，B为y轴正半轴上一点，A为x轴正半轴上一点，C为第一象限内一点；连接AC、BC，AC大于BC；M为AC中点，连接OM", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1250.jpg" }, { "index": 1251, "ocr_en": "In the planar Cartesian coordinate system, there is a straight line passing through the one, two, and four quadrants that intersects the positive half axis of the y-axis at the point A, and the positive half-axis of the intersection of the x-axis at the point B. There is a point Q on the line of the first quadrant, and a point P on the OB, which connects PQ; Rotate the Q point clockwise around the P point to obtain the point Q' in the fourth quadrant, and connect the PQ'", "ocr_zh": "在平面直角坐标系中，有一条过一、二、四象限的直线交y轴正半轴于点A，交x轴正半轴于点B；第一象限的直线上有一点Q，OB上有一点P，连接PQ；将Q点绕P点顺时针旋转一定角度，得到第四象限中的点Q'，连接PQ'", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1251.jpg" }, { "index": 1252, "ocr_en": "In the arc AB, connect the two points A and B, and P is the midpoint of the arc AB; Rotate the arc AB around A counterclockwise at a certain angle to obtain the arc AB', and P' is the midpoint of AB';", "ocr_zh": "在弧 AB中，连接A、B两点，P为弧AB的中点；将弧AB绕A逆时针旋转一定角度得到弧AB'，P'为AB'中点；做辅助线弧BB'", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1252.jpg" }, { "index": 1253, "ocr_en": "In the rectangular ABCD, P is one point above BC and BP is less than CP, and the AP is connected. Rotate the AP counterclockwise with A as the center to obtain AQ (Q is close to AD in the rectangle) and connect QD", "ocr_zh": "在矩形ABCD中，P为BC上一点，BP小于CP，连接AP；以A为中心将AP逆时针旋转一定角度得到AQ（Q在矩形中靠近AD处），连接QD", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1253.jpg" }, { "index": 1254, "ocr_en": "In the isosceles right triangle ABC, the angle ACB is equal to 90 degrees; Take the hypotenuse AB as the diameter, and make a half circle outside the triangle ABC; P is connected to the PC on this semicircle, and M is the PC midpoint", "ocr_zh": "在等腰直角三角形ABC中，角ACB等于90度；以斜边AB为直径，在三角形ABC外做一半圆；P在此半圆上，连接PC，M为PC中点", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1254.jpg" }, { "index": 1255, "ocr_en": "M is the midpoint of the ABCD side CD of the square, and P is the point in the quadrilateral, close to the point CD, connecting MP and BP; The line segment BP is rotated counterclockwise at a certain angle with B as the center to obtain the line segment BQ outside the rectangular ABCD, and the angle CBQ is an acute angle at this time, which is connected to MQ", "ocr_zh": "M是正方形ABCD边CD的中点，P为四边形内、靠近点CD处的一点，连接MP、BP；线段BP以B为中心逆时针旋转一定角度，在矩形ABCD外得到线段BQ，此时角CBQ为锐角，连接MQ", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1255.jpg" }, { "index": 1256, "ocr_en": "Take O as the center of the circle and AB as the diameter to make a circle O; C is a point on the circle O, connected to BC; Rotate BC around point C at a certain angle counterclockwise to obtain CD, connect OD, and the angle DOB is an acute angle", "ocr_zh": "以O为圆心，AB为直径，做圆O； C为圆O上一点,连接BC;将BC绕C点逆时针旋转一定角度得到CD,连接OD，角DOB为锐角", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1256.jpg" }, { "index": 1257, "ocr_en": "In the right triangle ABC, the angle ACB is equal to 90 degrees; P is a point outside the triangle near the edge AB, connecting AP and BP, and AP is less than BP at this time; Q is the midpoint of BP and is connected to QC", "ocr_zh": "在直角三角形ABC中，角ACB等于90度；P为三角形外靠近边AB处的一点，连接AP，BP，此时AP小于BP；Q为BP的中点，连接QC", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1257.jpg" }, { "index": 1258, "ocr_en": "In the square ABCD, E is the point above BC, and BE is less than CE; F is a point on AB, connected to EF; Make an equilateral triangle with EFG as the side and connect the CG within the square", "ocr_zh": "正方形ABCD中，E为BC上一点，BE小于CE；F为AB上一点，连接EF；以EF为边向右侧做等边三角形EFG在正方形内，连接CG", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1258.jpg" }, { "index": 1259, "ocr_en": "In the planar Cartesian coordinate system, the origin O is the center of the circle, and the positive half axis of the x-axis of the circle O is made at the point A. B is the point of the circle O in the second quadrant, connecting AB, and C is the midpoint of the chord AB; There is a straight line with one, three, and four quadrants outside the circle O, and the positive half axis of the x-axis is at point D, and the negative half-axis of the y-axis is at point E", "ocr_zh": "在平面直角坐标系中，以原点O为圆心，做圆O交x轴正半轴于点A；B为圆O在第二象限中的一点，连接AB，C为弦AB的中点；在圆O外有一过一、三、四象限的直线，交x轴正半轴于D点，交y轴负半轴于E点", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1259.jpg" }, { "index": 1260, "ocr_en": "In triangle ABC, the angle C is equal to 90 degrees; The midpoint of AC is P; Rotate the AP clockwise around point A to obtain AQ, and the angle QAB is an obtuse angle and connect BQ", "ocr_zh": "在三角形ABC中角，角C等于90度；AC中点为P;将AP绕A点顺时针旋转一定角度得到AQ，此时角QAB为钝角，连接BQ", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1260.jpg" }, { "index": 1261, "ocr_en": "In the planar Cartesian coordinate system, A is a point on the negative half axis of the x-axis, and B is a point on the positive half axis of the y-axis. Make an equilateral triangle ABP below AB with AB as the side, and P is in the fourth quadrant, connecting AP, BP, OP", "ocr_zh": "在平面直角坐标系中，A为x轴负半轴上一点，B为y轴正半轴上一点；以AB为边在AB下方做等边三角形ABP，此时P在第四象限中，连接AP，BP，OP", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1261.jpg" }, { "index": 1262, "ocr_en": "There is a triangle ABC, the angle A is less than the angle ACB is smaller than the angle B; There is a point P on the side AB, which connects CP; Point A: The point of symmetry with respect to CP is A', connecting CA', PA'", "ocr_zh": "有三角形ABC，角A小于角ACB小于角B；边AB上有一点P，连接CP；点A关于CP的对称点为A'，连接CA'，PA'", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1262.jpg" }, { "index": 1263, "ocr_en": "The square ABCD and the square AEFG share vertex A, and there is no overlap except A, and the angle DAG is less than 90 degrees, connecting CF and BG", "ocr_zh": "正方形ABCD与正方形AEFG共用顶点A，除A外无其他重叠部分，且角DAG小于90度，连接CF、BG", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1263.jpg" }, { "index": 1264, "ocr_en": "The square ABCD and the square AEFG share vertex A, there is no overlap except A, and the angular DAG is less than 90 degrees", "ocr_zh": "正方形ABCD与正方形AEFG共用顶点A，除A外无其他重叠部分，且角DAG小于90度", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1264.jpg" }, { "index": 1265, "ocr_en": "The square ABCD and the square AEFG share vertex A, there is no overlap except A, and the angle DAG is less than 90 degrees, connecting CF, BG, BE; Take the midpoints M and N of CF and BE respectively and connect MN", "ocr_zh": "正方形ABCD与正方形AEFG共用顶点A，除A外无其他重叠部分，且角DAG小于90度，连接CF、BG、BE；分别取CF、BE的中点M、N，连接MN", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1265.jpg" }, { "index": 1266, "ocr_en": "Left: In the square ABCD, O is the midpoint of BC, and E is a point in the square near AB and BC, connecting AE, EO, DE; Rotate the line segment DE around point D counterclockwise to obtain DF, F is outside the square ABCD and the angle FDC is less than 90 degrees. Connecting CF Right: There is a square ABCD", "ocr_zh": "左图：在正方形ABCD中，O为BC中点，E为正方形内靠近AB、BC处一点，连接AE、EO、DE；将线段DE绕D点逆时针旋转一定角度得到DF，F在正方形ABCD外部且角FDC小于90度；连接CF 右图：有正方形ABCD", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1266.jpg" }, { "index": 1267, "ocr_en": "There is an equilateral triangle ABC, E is a point on AC, and an equilateral triangle EFB is made with BE as the side, and F is outside the triangle ABC, close to BC; Connect the CF", "ocr_zh": "有等边三角形ABC，E为AC上一点，以BE为边做等边三角形EFB，F在三角形ABC外，靠近BC处；连接CF", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1267.jpg" }, { "index": 1268, "ocr_en": "There is an equilateral triangle ABC, D is a point on AB, and CD is connected to CD, and CD is the height of the triangle; M is the point on the CD, connecting BMM, using BM as the edge to make an equilateral triangle BMN, and point N is outside the triangle ABC, near CB", "ocr_zh": "有等边三角形ABC，D为AB上一点，连接CD，CD为三角形的高；M为CD上的点，连接BM，以BM为边做等边三角形BMN，N点在三角形ABC外，靠近CB处", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1268.jpg" }, { "index": 1269, "ocr_en": "There is an equilateral triangle ABC, E is a point on AC, and an equilateral triangle EFB is made with BE as the side, and F is outside the triangle ABC, close to BC; Connect the CF", "ocr_zh": "有等边三角形ABC，E为AC上一点，以BE为边做等边三角形EFB，F在三角形ABC外，靠近BC处；连接CF", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1269.jpg" }, { "index": 1270, "ocr_en": "E is the upper point of the edge BC of the square ABCD, and B is the vertex to make the square BHGF, F is in the square ABCD, G and H are outside the square ABCD, and the BF is less than BC; Connect AF with points A, F, E, G in a straight line", "ocr_zh": "E为正方形ABCD的边BC上一点，以B为顶点做正方形BHGF，F在正方形ABCD内，G、H在正方形ABCD外，BF小于BC；连接AF，点A、F、E、G在一条直线上", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1270.jpg" }, { "index": 1271, "ocr_en": "The two rectangles are crossed to form an X-shape, and the overlapping part is a quadrilateral ABCD; Connect BD; P is the upper point of the BD, which is connected to the AP and PC; The CD and another rectangle are angled at an acute angle sandwiched between the quadrilateral ABCD α", "ocr_zh": "两个长方形交叉摆成X型，重叠部分为四边形ABCD；连接BD；P为BD上一点，连接AP、PC；CD与另一长方形在四边形ABCD外所夹锐角为角α", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1271.jpg" }, { "index": 1272, "ocr_en": "In triangle ABC, the angle ACB is 90 degrees, and P is a point in the triangle, connecting AP, BP, CP", "ocr_zh": "在三角形ABC中，角ACB为90度，P为三角形中一点，连接AP、BP、CP", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1272.jpg" }, { "index": 1273, "ocr_en": "There is a point M in the rectangular ABCD and a point E on the edge BC, connecting AM, DM, EM", "ocr_zh": "在矩形ABCD中有一点M，边BC上有一点E，连接AM、DM、EM", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1273.jpg" }, { "index": 1274, "ocr_en": "In the isosceles right triangle ABC, the angle ACB is 90 degrees, and E is a point in the triangle, connecting AE, BE, CE", "ocr_zh": "在等腰直角三角形ABC中，角ACB为90度，E为三角形中一点，连接AE、BE、CE", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1274.jpg" }, { "index": 1275, "ocr_en": "There is a dot E in the square ABCD, which connects AE, BE, CE", "ocr_zh": "在正方形ABCD中有一点E，连接AE、BE、CE", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1275.jpg" }, { "index": 1276, "ocr_en": "In the triangle ABC, the angle BAC and the angle BCA are acute angles, and P is a point in the triangle, connecting AP, BP, CP", "ocr_zh": "在三角形ABC中，角BAC、角BCA为锐角，P为三角形中一点，连接AP、BP、CP", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1276.jpg" }, { "index": 1277, "ocr_en": "There is an obtuse triangle ABC, and the angle BCA is an obtuse angle; Rotate the triangle around point A at a certain angle counterclockwise to obtain the triangle ADE (assuming that the point obtained by rotation from point B is the point D), DE and BC intersect at the point P, connect AP, and the angle BAD is an acute angle", "ocr_zh": "有钝角三角形ABC，角BCA为钝角；将三角形绕点A逆时针旋转一定角度得到三角形ADE（假设由B点旋转后得到的点为点D），DE与BC交于点P，连接AP，角BAD为锐角", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1277.jpg" }, { "index": 1278, "ocr_en": "Within the acute triangle NGM, there is a little O", "ocr_zh": "在锐角三角形NGM内，有一点O", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1278.jpg" }, { "index": 1279, "ocr_en": "In the isosceles obtuse triangle ABC, the angle BAC is obtuse and AB is equal to AC; P is a point in the triangle and connects AP, BP, CP", "ocr_zh": "在等腰钝角三角形ABC中，角BAC为钝角，AB等于AC；P为三角形中一点，连接AP、BP、CP", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1279.jpg" }, { "index": 1280, "ocr_en": "In the isosceles right triangle ABC, the angle BAC is equal to 90 degrees, and there is a bit D on BC, connecting AD; Rotate AD around point A counterclockwise to obtain AE, and the angular DAC is smaller than the angular BAD and both are acute angles. Connect DE and CE; F is the midpoint of DE, connected to CF; The intersection of CF and BA at the point G was prolonged, respectively", "ocr_zh": "在等腰直角三角形ABC中，角BAC等于90度，BC上有点D，连接AD；把AD绕点A逆时针旋转一定角度得到AE，此时角DAC小于角BAD且二者均为为锐角；连接DE、CE；F为DE中点，连接CF；分别延长CF、BA相交于点G", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1280.jpg" }, { "index": 1281, "ocr_en": "In the isosceles right triangle ABC, the angle BAC is equal to 90 degrees, and there is a bit D on BC, connecting AD; Rotate AD around point A counterclockwise to obtain AE, and the angular DAC is greater than the angular BAD and both are acute angles. Connect DE and CE; F is the midpoint of DE, connected to CF", "ocr_zh": "在等腰直角三角形ABC中，角BAC等于90度，BC上有点D，连接AD；把AD绕点A逆时针旋转一定角度得到AE，此时角DAC大于角BAD且二者均为锐角；连接DE、CE；F为DE中点，连接CF", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1281.jpg" }, { "index": 1282, "ocr_en": "In the obtuse triangle ABC, the angle BAC is an obtuse angle and P is a point in the triangle, connecting AP, BP, CP", "ocr_zh": "在钝角三角形ABC中，角BAC为钝角，P为三角形中一点，连接AP、BP、CP", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1282.jpg" }, { "index": 1283, "ocr_en": "P is a point in the square ABCD, connecting AP, BP, DP, and extending the AP and CD at the point Q", "ocr_zh": "P为正方形ABCD内一点，连接AP、BP、DP，延长AP与CD交于点Q", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1283.jpg" }, { "index": 1284, "ocr_en": "In the planar Cartesian coordinate system, the origin is O, and the point A and point B are the points on the y-axis, x-axis, and positive semi-axis, respectively. With AO as one side, an equilateral triangle AOC is made in the second quadrant; M is the point on the bisector of the angular AOB, connecting AM and BM; N is a point in the triangle AOC, connecting ON and CN", "ocr_zh": "在平面直角坐标系中，原点为O，点A，点B分别是y轴，x轴正半轴上的点；以AO为一边，在第二象限中作等边三角形AOC；M为角AOB平分线上的点，连接AM、BM；N为三角形AOC中一点，连接ON、CN", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1284.jpg" }, { "index": 1285, "ocr_en": "In the obtuse triangle ABC, angle B is an obtuse angle; Rotate the triangle clockwise around point A to obtain the triangle ADE, the angle DAB is greater than the angle BAC and less than 90 degrees, and the auxiliary line BD is made", "ocr_zh": "钝角三角形ABC中，角B为钝角；将三角形绕A点顺时针旋转一定角度，得到三角形ADE，角DAB大于角BAC且小于90度，做辅助线BD", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1285.jpg" }, { "index": 1286, "ocr_en": "In the equilateral triangle ABC, P is a point in the triangle and connects AP, BP, CP", "ocr_zh": "在等边三角形ABC中，P为三角形中一点，连接AP、BP、CP", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1286.jpg" }, { "index": 1287, "ocr_en": "In the equilateral triangle ABC, P is a point in the triangle, connecting AP, BP, CP, and the angle APC=90 degrees", "ocr_zh": "在等边三角形ABC中，P为三角形中一点，连接AP、BP、CP，此时角APC=90度", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1287.jpg" }, { "index": 1288, "ocr_en": "In triangle ABC, P is a point in the triangle and connects AP, BP, CP", "ocr_zh": "在三角形ABC中，P为三角形中一点，连接AP、BP、CP", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1288.jpg" }, { "index": 1289, "ocr_en": "In the diamond-shaped ABCD, E is one of the points, connecting AE, BE, CE, DE, and the angle BEC = 90 degrees", "ocr_zh": "在菱形ABCD中，E为其中一点，连接AE、BE、CE、DE，角BEC=90度", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1289.jpg" }, { "index": 1290, "ocr_en": "There is an obtuse triangle BPC, and the angle BPC is obtuse; Make an auxiliary line: rotate the triangle BPC clockwise around point B to obtain the triangle BDE and connect the PD; Angular EBC is greater than angular EBD and less than 90 degrees; The triangle PBD is an equilateral triangle; There is a point A above point P, which connects AB and AC", "ocr_zh": "有一钝角三角形BPC，角BPC为钝角；做辅助线：将三角形BPC绕点B顺时针旋转一定角度得到三角形BDE，连接PD；角EBC大于角EBD且小于90度；三角形PBD为等边三角形；P点上方有一点A，连接AB、AC", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1290.jpg" }, { "index": 1291, "ocr_en": "In the right triangle ABC, the angle CAB=90 degrees, P is a point in the triangle, connecting AP, BP, CP", "ocr_zh": "在直角三角形ABC中，角CAB=90度，P为三角形中一点，连接AP、BP、CP", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1291.jpg" }, { "index": 1292, "ocr_en": "In the acute triangle ABC, there is a point P, which is used as an auxiliary line AP and CP to obtain an obtuse triangle APC, and the angle APC is an obtuse angle, which connects BP; Rotate the triangle APC around the point C clockwise at a certain angle to obtain the triangle CA'P', and A' and P' are outside the triangle ABC; B, P, P', A' are not collinear, triangle APC, A'P'C are marked in black", "ocr_zh": "在锐角三角形ABC中，有一点P，做辅助线AP、CP得到钝角三角形APC，角APC为钝角，连接BP；将三角形APC绕点C顺时针旋转一定角度得到三角形CA'P'，此时A'、P'均在三角形ABC外；做辅助线PP'；B、P、P'、A'不共线；三角形APC、A'P'C标黑", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1292.jpg" }, { "index": 1293, "ocr_en": "In the acute triangle ABC, there is a point P, which is used as an auxiliary line AP and CP to obtain an obtuse triangle APC, and the angle APC is an obtuse angle, which connects BP; Rotate the triangle APC around the point C clockwise at a certain angle to obtain the triangle CA'P', and A' and P' are outside the triangle ABC; B, P, P', A' collinear, triangle APC, A'P'C in black", "ocr_zh": "在锐角三角形ABC中，有一点P，做辅助线AP、CP得到钝角三角形APC，角APC为钝角，连接BP；将三角形APC绕点C顺时针旋转一定角度得到三角形CA'P'，此时A'、P'均在三角形ABC外；做辅助线PP'；B、P、P'、A'共线；三角形APC、A'P'C标黑", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1293.jpg" }, { "index": 1294, "ocr_en": "In the obtuse triangle ABC, angle A is an obtuse angle", "ocr_zh": "在钝角三角形ABC中，角A为钝角", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1294.jpg" }, { "index": 1295, "ocr_en": "P is a point in the square ABCD, and Q is the point above BC, connecting AP, QP, DP", "ocr_zh": "P为正方形ABCD内一点，Q为BC上一点，连接AP、QP、DP", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1295.jpg" }, { "index": 1296, "ocr_en": "In the acute triangle ABC, P is one point in the triangle and connects AP, BP, CP", "ocr_zh": "在锐角三角形ABC中，P为三角形中一点，连接AP、BP、CP", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1296.jpg" }, { "index": 1297, "ocr_en": "In triangle ABC, the angle ACB is greater than the angle ABC is greater than the angle CAB, and P is the point in the triangle, connecting AP, BP, CP", "ocr_zh": "在三角形ABC中，角ACB大于角ABC大于角CAB，P为三角形中一点，连接AP、BP、CP", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1297.jpg" }, { "index": 1298, "ocr_en": "In the acute triangle ABC, there is a point P, and the obtuse triangle BPA is obtained by connecting AP, BP, and CP, and the angular BPA is an obtuse angle; The triangle BA'P' is obtained by rotating the triangle BPA around the point B at a certain angle counterclockwise, and A' and P' are outside the triangle ABC, and the angle PBP' is less than 90 degrees. Do auxiliary line PP, A'C'", "ocr_zh": "在锐角三角形ABC中，有一点P，连接AP、BP、CP得到钝角三角形BPA，角BPA为钝角；将三角形BPA绕点B逆时针旋转一定角度得到三角形BA'P'，此时A'、P'均在三角形ABC外，且角PBP'小于90度；做辅助线PP、A'C'", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1298.jpg" }, { "index": 1299, "ocr_en": "There is a quadrilateral ABCD, which connects AC and BD to intersect at the point O; The parallel lines of BD and AC intersect F, where G is the midpoint of CD and P is the point above CF", "ocr_zh": "有四边形ABCD，连接AC、BD交于点O；分别过点C、D作BD、AC的平行线相交于F，点G是CD的中点，P为CF上一点", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1299.jpg" }, { "index": 1300, "ocr_en": "There is a quadrilateral ABCD, AD parallel to BC; There are some M and N on AD, and AM is less than AN; Connect the BM and BN", "ocr_zh": "有四边形ABCD，AD平行于BC；AD上有点M、N，AM小于AN；连接BM、BN", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1300.jpg" }, { "index": 1301, "ocr_en": "There is an acute triangle ABC, D is the midpoint of BC, and the straight line l passes through the point D; AE is perpendicular to E, BF perpendicular to F", "ocr_zh": "有锐角三角形ABC，D是BC中点，直线l经过点D；AE垂直l于E，BF垂直l于F", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1301.jpg" }, { "index": 1302, "ocr_en": "In the planar Cartesian coordinate system, the origin O is used as the center of the circle to make the circle O, and the linear line with two or three and four quadrants outside the circle intersects the negative half axis of the x-axis at point A, and the negative half-axis of the y-axis is at point B. D is the middle point of the second quadrant on the circle O, and C is the middle point of the first quadrant on the circle O, connecting OC, OD, CD; P is the midpoint of the CD and is in the first quadrant, connecting the AP and BP", "ocr_zh": "在平面直角坐标系中，以原点O为圆心做圆O，圆外有一过二、三、四象限的直线交x轴负半轴于A点，交y轴负半轴于B点；D为圆O上第二象限中一点，C为圆O上第一象限中一点，连接OC、OD、CD；P为CD的中点且在第一象限中，连接AP、BP", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1302.jpg" }, { "index": 1303, "ocr_en": "In the acute triangle ABC, the points D and E are on BC and AC respectively, and the connection AD and BE intersect with F", "ocr_zh": "在锐角三角形ABC中，点D、E分别在BC、AC上，连接AD、BE交于F", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1303.jpg" }, { "index": 1304, "ocr_en": "In the equilateral triangle ABC, P and Q are on AB and AQ is less than AP. D is on AC and AD is less than CD; Connect QD and CP", "ocr_zh": "在等边三角形ABC中，P、Q在AB上且AQ小于AP；D在AC上且AD小于CD；连接QD、CP", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1304.jpg" }, { "index": 1305, "ocr_en": "In the first quadrant of the planar Cartesian coordinate system, there is a rectangular ABCD, in which the points A and D are on the x-axis and y-axis positive semi-axis, respectively. Connect the CO", "ocr_zh": "在平面直角坐标系第一象限中，有矩形ABCD，其中点A、D分别在x轴、y轴正半轴上；连接CO", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1305.jpg" }, { "index": 1306, "ocr_en": "In the equilateral triangle ABC, the circle C is made with C as the center and the radius less than AC; P is on AB, and P is passed to make the tangent PQ of the circle C, the tangent point is Q and the tangent point is outside the triangle ABC", "ocr_zh": "在等边三角形ABC中，以C为圆心、小于AC的半径做圆C；P在AB上，过P做圆C的切线PQ，切点为Q且切点在三角形ABC外", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1306.jpg" }, { "index": 1307, "ocr_en": "D is a point outside the equilateral triangle ABC near BC, connecting AD, BD, CD", "ocr_zh": "D为等边三角形ABC外靠近BC处一点，连接AD、BD、CD", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1307.jpg" }, { "index": 1308, "ocr_en": "In the right triangle ABC, the angle C = 90 degrees; E is one point above BC and CE is less than BE; Connect AE, and cross the perpendicular line of E to make AE cross AB to F", "ocr_zh": "在直角三角形ABC中，角C=90度；E为BC上一点且CE小于BE；连接AE，过E做AE的垂线交AB于F", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1308.jpg" }, { "index": 1309, "ocr_en": "In the planar Cartesian coordinate system xOy, P is on the positive half axis of the y axis, A and B are on the positive half axis of the x-axis, and OA is less than AB and less than OB, and Q is the point in the first quadrant, connecting AP, AQ and PQ to form an equilateral triangle APQ; Connect to QB", "ocr_zh": "在平面直角坐标系xOy中，P在y轴正半轴上，A、B在x轴正半轴上且OA小于AB小于OB，Q为第一象限中一点，连接AP、AQ、PQ构成等边三角形APQ；连接QB", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1309.jpg" }, { "index": 1310, "ocr_en": "In the planar Cartesian coordinate system xOy, the line AB intersects with the positive semi-axes of the x and y axes at points A and B, respectively. Rotate the line segment AB clockwise around point A to obtain another line segment in the first quadrant, and the other endpoint in the line segment except A is on an inverse proportional function image. Under the inverse proportional function, there is a point D on the positive and semi-axis of the x-axis, and the auxiliary line is used to connect the point with the point D", "ocr_zh": "在平面直角坐标系xOy中，直线AB分别与x、y轴正半轴交于点A、B；将线段AB绕A点顺时针旋转一定角度，在第一象限中得到另一条线段，该线段中除A外另一端点在一反比例函数图像上；反比例函数下、x轴正半轴上有点D,做辅助线：连接该点与D点", "class": "Analytic Geometry", "difficulty": 3, "image_path": "fig_1310.jpg" }, { "index": 1311, "ocr_en": "In the planar Cartesian coordinate system xOy, the second quadrant has a circle C, and point C is the center of the circle; P is a point on the circle C, and A and B are connected to AP and PB on the negative and positive semi-axes of the x-axis, respectively", "ocr_zh": "在平面直角坐标系xOy中，第二象限有一圆C，C点是圆的圆心；P是圆C上一点，A、B分别在x轴负、正半轴上，连接AP、PB", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1311.jpg" }, { "index": 1312, "ocr_en": "In the plane Cartesian coordinate system xOy, there is a circle B, where point B is the center of circle B and on the negative half axis of the y axis, and there are three intersection points between the negative half axis of the circle B and the negative half axis of the y axis. P is a point on the circle B, a point in the fourth quadrant, and A is a point on the positive half axis of the x-axis, connecting AP; C is the midpoint of the AP and connects to the CO", "ocr_zh": "在平面直角坐标系xOy中有一圆B，B点是圆B的圆心且在y轴负半轴上，圆B与y轴负半轴共有三个交点；P是圆B上、第四象限中一点，A是x轴正半轴上一点，连接AP；C是AP的中点，连接CO", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1312.jpg" }, { "index": 1313, "ocr_en": "There is a right triangle ABC, the angle BAC = 90 degrees; E is a point on BC and BE is greater than EC, and AE is connected; D is outside the triangle ABC, a point close to AB, connect AD and DE, make a right-angled triangle ADE, angle DAE=90 degrees; F is the midpoint of DE and in triangle ABC, connect BF", "ocr_zh": "有直角三角形ABC，角BAC=90度；E为BC上一点且BE大于EC，连接AE；D为三角形ABC外，靠近AB处一点，连接AD、DE，做直角三角形ADE，角DAE=90度；F为DE中点且在三角形ABC中，连接BF", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1313.jpg" }, { "index": 1314, "ocr_en": "In the rectangular DEFG, with DE as the diameter, make a half circle in the rectangle, and have a little P on the half circle to connect GP and FP", "ocr_zh": "在矩形DEFG中，以DE为直径，在矩形内做一半圆，半圆上有一点P，连接GP、FP", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1314.jpg" }, { "index": 1315, "ocr_en": "In the planar Cartesian coordinate system xOy, the first and second quadrants each have an inverse proportional function, the vertex A of the rectangular OABC, on the function of the first quadrant, the vertices B and C are on the function of the second quadrant, and the vertex O coincides with the origin", "ocr_zh": "在平面直角坐标系xOy中，一、二象限各有一条反比例函数，矩形OABC的顶点A，在第一象限的函数上，顶点B、C在第二象限的函数上，顶点O与原点重合", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1315.jpg" }, { "index": 1316, "ocr_en": "In the planar Cartesian coordinate system xOy, point B is in the first quadrant, and point A is on the positive semi-axis of the x-axis, connecting OB and AB, and the angle OAB is an obtuse angle. Rotate the triangle OAB counterclockwise around the point O to obtain the triangle OA'B', where A' is on the y-axis positive semi-axis and B' is in the second quadrant", "ocr_zh": "在平面直角坐标系xOy中，点B在第一象限，点A在x轴正半轴上，连接OB、AB，此时角OAB为钝角；将三角形OAB绕点O逆时针旋转一定角度，得到三角形OA'B'，A'在y轴正半轴上，B'在第二象限中", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1316.jpg" }, { "index": 1317, "ocr_en": "Points A, B, and C are on the same line, B is between A and C, D and E are on the same side as AC, E is above A, and D is above C; Connect AE, BE, CD, BD, ED; AB is less than BC, angle A is equal to angle C is equal to 90 degrees; AB is a, BC is b, and ED is c", "ocr_zh": "点A、B、C在同一条直线上，B在A、C之间，D、E在AC同侧，E在A上方，D在C上方；连接AE、BE、CD、BD、ED；AB小于BC，角A等于角C等于90度；AB为a，BC为b，ED为c", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1317.jpg" }, { "index": 1318, "ocr_en": "In the planar Cartesian coordinate system, there is a straight line in the third and fourth quadrants parallel to the x-axis, and the line intersects the y-axis at the point G. B is a point on the straight line, C is a point on the positive half axis of the y-axis, and A is a point in the first quadrant; Connect AC, AB, CB; The AC is perpendicular to BC at the point C, and the acute angle between AB and the positive semi-axis of the x-axis is α", "ocr_zh": "在平面直角坐标系中，有一条位于三、四象限且平行于x轴的直线，直线与y轴交于点G；B为直线上一点，C为y轴正半轴上一点，A为第一象限中一点；连接AC、AB、CB；AC垂直BC于点C，AB与x轴正半轴所夹的锐角为α", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1318.jpg" }, { "index": 1319, "ocr_en": "In a planar Cartesian coordinate system, there is a point of B on the positive half axis of the y-axis, and a point A on the positive half axis of the x-axis, which is connected to AB; Rotate AB clockwise around point A to obtain AC, AC is in the first quadrant: AC is perpendicular to point A", "ocr_zh": "在平面直角坐标系中，y轴正半轴上有点B，x轴正半轴上有点A，连接AB；将AB绕A点顺时针旋转一定角度得到AC，AC在第一象限中：AC垂直AB于点A", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1319.jpg" }, { "index": 1320, "ocr_en": "The square ABCD and the right-angled triangle AEF have no other intersecting parts except point A, the angle EAF is equal to 90 degrees, and the angle FAB is less than 90 degrees; Connect BF and DE", "ocr_zh": "正方形ABCD和直角三角形AEF除点A外无其他相交部分，角EAF等于90度，角FAB小于90度；连接BF、DE", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1320.jpg" }, { "index": 1321, "ocr_en": "Triangle ABC, D on line segment AB, connect CD, make a square CDEF with CD as one side, connect BF", "ocr_zh": "三角形ABC，D在线段AB上，连接CD，以CD为一边做正方形CDEF，EF在CD上方且平行于CD，连接BF", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1321.jpg" }, { "index": 1322, "ocr_en": "In the coordinate axis xoy, the hyperbola is located in the first and third quadrants, point A is on the hyperbola and located in the first quadrant, connecting AO and extending the intersection hyperbola to point B, point C is located in the second quadrant, connecting BC", "ocr_zh": "坐标轴xoy中，双曲线位于一、三象限，点A在双曲线上且位于第一象限，连接AO并延长交第三象限的双曲线于点B，点C位于第二象限，连接BC", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1322.jpg" }, { "index": 1323, "ocr_en": "In the coordinate axis xoy, the hyperbola is located in the first quadrant, triangle ABC, point A is located on the positive half axis of the yz axis, point B is located on the positive half axis of the x-axis, and point C is located on the hyperbola", "ocr_zh": "坐标轴xoy中，双曲线位于第一象限，三角形ABC，点A位于yz轴的正半轴上，点B位于x轴的正半轴上，点C在双曲线上", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1323.jpg" }, { "index": 1324, "ocr_en": "Square ABCD, triangle EGP, connecting AC, point E on line segment AC, point P on the extension of AC, point F at the intersection of the extension of DC and GP, connecting EF", "ocr_zh": "正方形ABCD，点E在对角线AC上，点G在AB左边，点F是DC延长线上一点且位于C点下方，点P是AC与GF延长线的交点，EGP三点组成了三角形EGP，连接EF", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1324.jpg" }, { "index": 1325, "ocr_en": "Right angled triangle ABC, CA=CB, M is the midpoint of AB, point D is on BM, AE is perpendicular to CD at point E, passing through point B is an extension line of BF perpendicular to CD, perpendicular foot is F, connecting EM", "ocr_zh": "直角三角形ABC，CA=CB，M是AB中点，点D在BM上，作AE垂直CD于点E，过点B作BF垂直CD的延长线，垂足为F，连接EM", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1325.jpg" }, { "index": 1326, "ocr_en": "Coordinate system xoy, the hyperbola is located in the first quadrant, point A is on the hyperbola, point B is below the x-axis, and points A, O, and B form a triangle AOB", "ocr_zh": "坐标系xoy，双曲线位于第一象限，点A在双曲线上，点B在x轴下方，A、O、B三点组成三角形AOB，角AOB为直角三角形", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1326.jpg" }, { "index": 1327, "ocr_en": "Coordinate system xoy, the hyperbola is located in the first and third quadrants, point A is located on the hyperbola in the first quadrant, the hyperbola connecting AO and intersecting the third quadrant is located at point B, point C is located in the fourth quadrant, connecting AC and BC, and points A, B, and C form triangle ABC", "ocr_zh": "坐标系xoy，双曲线位于一、三象限，点A位于第一象限的双曲线上，连接AO并交第三象限的双曲线于点B，点C位于第四象限，连接AC、BC，点A、B、C组成三角形ABC", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1327.jpg" }, { "index": 1328, "ocr_en": "Quadrilateral ABCD, angles ABC and DCB are acute angles, point P is a point on BC, connecting AP and DP, and removing the line between AD", "ocr_zh": "四边形ABCD，角ABC和角DCB是锐角，点P是BC上一点，连接AP、DP，删掉AD之间的连线", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1328.jpg" }, { "index": 1329, "ocr_en": "Quadrilateral ABCD, angles ABC and DCB are right angles, point P is a point on BC, connecting AP and DP, and removing the line between AD", "ocr_zh": "四边形ABCD，角ABC和角DCB是直角，点P是BC上一点，连接AP、DP，删掉AD之间的连线", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1329.jpg" }, { "index": 1330, "ocr_en": "Coordinate system xoy, line AB passes through quadrants one, two, and four, intersecting with the x-axis and y-axis at points A and B, respectively. Points C and D are on the right side of line AB, connecting AD, BC, and CD", "ocr_zh": "坐标系xoy，直线AB过一、二、四象限，与x轴、y轴分别交于点A、B，点C、D在直线AB的右侧，连接AD、BC、CD", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1330.jpg" }, { "index": 1331, "ocr_en": "Triangle ABC, angle BAC is a right angle, AB=AC， Make squares ABDE and ACFG with sides AB and AC respectively. DE is parallel to AB and on the left side of AB, FG is parallel to AC and on the right side of AC. Connect EG and BC, pass through A to make AM perpendicular to BC at point M, and extend MA to intersect EG at point N", "ocr_zh": "三角形ABC，角BAC为直角，AB=AC，分别以AB、AC为边做正方形ABDE和正方形ACFG，DE平行AB且在AB左边，FG平行AC且在AC右边，连接EG、BC，过A作AM垂直BC于点M，延长MA交EG于点N", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1331.jpg" }, { "index": 1332, "ocr_en": "Triangle ABC, angle BAC is a right angle, length AB is slightly greater than AC. Square ABDE and square ACFG are made with AB and AC as sides, respectively. DE is parallel to AB and to the left of AB, FG is parallel to AC and to the right of AC, connecting EG and BC, passing through A to make AM perpendicular to BC at point M, extending MA to intersect EG at point N", "ocr_zh": "三角形ABC，角BAC为直角，长度AB略大于AC，分别以AB、AC为边做正方形ABDE和正方形ACFG，DE平行AB且在AB左边，FG平行AC且在AC右边，连接EG、BC，过A作AM垂直BC于点M，延长MA交EG于点N", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1332.jpg" }, { "index": 1333, "ocr_en": "Triangle ABC, angle BAC is not a right angle, length AB is greater than AC. Make squares ABDE and ACFG with AB and AC as sides respectively. DE is parallel to AB and to the left of AB, FG is parallel to AC and to the right of AC, connecting EG and BC. Make AM perpendicular to BC at point M by passing through A, and extend MA to intersect EG at point N", "ocr_zh": "三角形ABC，角BAC不是直角，长度AB大于AC，分别以AB、AC为边做正方形ABDE和正方形ACFG，DE平行AB且在AB左边，FG平行AC且在AC右边，连接EG、BC，过A作AM垂直BC于点M，延长MA交EG于点N", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1333.jpg" }, { "index": 1334, "ocr_en": "Triangle ABC, angle BAC is a right angle, length AB is less than AC. Make squares ABDE and ACFG with AB and AC as sides respectively. DE is parallel to AB and on the left side of AB, FG is parallel to AC and on the right side of AC, connecting EG and BC, passing through A to make AH perpendicular to BC at point H, extending HA to intersect EG at point I", "ocr_zh": "三角形ABC，角BAC是直角，长度AB小于AC，分别以AB、AC为边做正方形ABDE和正方形ACFG，DE平行AB且在AB左边，FG平行AC且在AC右边，连接EG、BC，过A作AH垂直BC于点H，延长HA交EG于点I", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1334.jpg" }, { "index": 1335, "ocr_en": "Square ABCD, connecting diagonal BD, M is a point on diagonal BD, passing through point M as MN perpendicular AM intersects BC at point N,", "ocr_zh": "正方形ABCD，连接对角线BD，M为对角线BD上一点，过点M作MN垂直AM交BC于点N，", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1335.jpg" }, { "index": 1336, "ocr_en": "Triangle ABC, l is a horizontal line, points D, A, and E are all on line l, and point A is between D and E, connecting BD and CE. Remove the line connecting BC", "ocr_zh": "三角形ABC，l为一条水平线，点D、A、E三点都在直线l上，且A点在D和E中间，连接BD,CE，去掉BC之间的连线", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1336.jpg" }, { "index": 1337, "ocr_en": "Square ABCD, connecting diagonal BD, M is a point on diagonal BD, passing through point M as MN, perpendicular AM, intersecting BC at point N, passing through point N as NH, perpendicular BD at point H", "ocr_zh": "正方形ABCD，连接对角线BD，M为对角线BD上一点，过点M作MN垂直AM交BC于点N，过点N作NH垂直BD于点H", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1337.jpg" }, { "index": 1338, "ocr_en": "Square ABCD, connecting diagonal BD, M is a point on diagonal BD, passing through point M as MN, perpendicular AM, intersecting BC at point N, connecting AN, O is the midpoint of AN, connecting MO and extending intersection AB at point P", "ocr_zh": "正方形ABCD，连接对角线BD，M为对角线BD上一点，过点M作MN垂直AM交BC于点N，连接AN，O为AN中点，连接MO并延长交AB于点P", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1338.jpg" }, { "index": 1339, "ocr_en": "Triangle ABC, angle BAC is a right angle, AB=AC， Line l passes through point A, BD is perpendicular l, CE is perpendicular l, and the vertical feet are D and E, respectively", "ocr_zh": "三角形ABC，角BAC为直角，AB=AC，直线l过点A，BD垂直l、CE垂直l，垂足分别为D、E", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1339.jpg" }, { "index": 1340, "ocr_en": "Equilateral triangle ABC, with BC as the side, make a quadrilateral BCDE outward. Extend AB by intersecting the extension lines of CE and DE at points F and N, CH perpendicular to AF at point H, EM perpendicular to AF at point M, and connect AE", "ocr_zh": "等边三角形ABC，以BC为边向外做四边形BCDE，延长AB分别交CE、DE的延长线于点F、N，CH垂直AF于点H，EM垂直AF于点M，连接AE", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1340.jpg" }, { "index": 1341, "ocr_en": "Quadrilateral ABCD, angle C is obtuse, point E is a point inside quadrilateral ABCD, connecting AE, BE, CE, DE, passing through E makes EF intersection AD at point F, extending FE intersection BC at point G", "ocr_zh": "四边形ABCD,角C为钝角，点E为四边形ABCD内一点，连接AE、BE、CE、DE，过E作EF交交AD于点F，延长FE交BC于点G", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1341.jpg" }, { "index": 1342, "ocr_en": "Quadrilateral ABCD, angles C and ADC are right angles, point E is on edge CD, point F is on the extension of edge AD, point G is on the right side of edge CD, connecting AE, BE, FE, GE, angles AEB and FEG are right angles, connecting BG and intersecting CD at point H", "ocr_zh": "四边形ABCD，角C和角ADC为直角，点E在边CD上，点F在边AD的延长线上，点G在边CD的右边，连接AE、BE、FE、GE，角AEB和角FEG为直角，连接BG交CD于点H", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1342.jpg" }, { "index": 1343, "ocr_en": "Quadrilateral ABCD, angles C and D are right angles, point E is on edge CD, connecting AE and BE", "ocr_zh": "四边形ABCD，角C和角D为直角，点E在边CD上，连接AE、BE", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1343.jpg" }, { "index": 1344, "ocr_en": "Square ABCD, E, and F are points on CD and BC respectively, connecting AE, AF, and EF. Rotate triangle ADE clockwise by 90 degrees to obtain triangle ABG, angle GAE is a right angle, passing through point B to form BM parallel AG angle AF at point M", "ocr_zh": "正方形ABCD，E、F分别是CD、BC上一点，连接AE、AF、EF，将三角形ADE顺时针旋转90度得到三角形ABG，角GAE为直角，过点B作BM平行AG角AF于点M", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1344.jpg" }, { "index": 1345, "ocr_en": "In squares ABCD, points E and F are points on edges BC and CD respectively, connecting AE, AF, and diagonal\nLine BD intersects points G and H respectively, connecting EH and RF", "ocr_zh": "正方形ABCD中，点E,F分别为边BC,CD上的点，连接AE,AF,与对角\n线BD分别交于点G,H,连接EH、RF", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1345.jpg" }, { "index": 1346, "ocr_en": "Square ABCD, E is a point on BC. Fold the side AB of the square along AE to AF, extend EF to intersect BC at point G, and connect AG and FC", "ocr_zh": "正方形ABCD，E是BC上一点，将正方形边AB沿AE折叠到AF，延长EF交BC于点G，连接AG、FC", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1346.jpg" }, { "index": 1347, "ocr_en": "Coordinate axis xoy, points A and B are on the positive and negative axes of the y-axis and x-axis, respectively. The inverse proportional function is located in the first quadrant, and point P is on the inverse proportional function, connecting AP and BP", "ocr_zh": "坐标轴xoy，点A、B分别在y轴和x轴的正半轴上，反比例函数位于第一象限，点P位于反比例函数上，连接AP、BP", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1347.jpg" }, { "index": 1348, "ocr_en": "Square ABCD, G is the midpoint of BC, fold triangle ABG along AG to AFG, extend GF to intersect DC at point E", "ocr_zh": "正方形ABCD，G是BC中点，将三角形ABG沿AG对折至AFG，延长GF交DC于点E", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1348.jpg" }, { "index": 1349, "ocr_en": "Square ABCD, point E is a point above AB, point M is a point outside of square ABCD and above AD, connecting AM, angle DAM is an acute angle, point F is on ray AM, connecting EF, CE, CF, CF intersects edge AD at point G, and connects EG", "ocr_zh": "正方形ABCD,点E为AB上一点，点M为正方形ABCD外一点且在AD上方，连接AM，角DAM为锐角，点F在射线AM上，连接EF、CE、CF，CF与边AD交于点G，连接EG", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1349.jpg" }, { "index": 1350, "ocr_en": "In squares ABCD, connect diagonal BD, and triangle ABC rotates counterclockwise around point A to triangle AB'C ', with a rotation angle less than 45 degrees. AB' and AC 'intersect diagonal BD at E and F, respectively", "ocr_zh": "正方形ABCD中，连接对角线BD，三角形ABC绕点A逆时针旋转到三角形AB'C'，旋转角小于45度，AB'、AC'分别交对角线BD于E、F", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1350.jpg" }, { "index": 1351, "ocr_en": "Rectangular shapes ABCD, E, and F are connected on edges BC and CD, respectively, connecting AE, AF, and EF", "ocr_zh": "矩形ABCD，E、F分别在边BC、CD上，连接AE、AF、EF", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1351.jpg" }, { "index": 1352, "ocr_en": "Square ABCD, E is a point on the side of CD, fold triangle ADE along AE to triangle AFE, extend EF intersection BC at point G, connect AG", "ocr_zh": "正方形ABCD，E为CD边上一点，将三角形ADE沿AE对折至三角形AFE，延长EF交边BC于点G，连接AG", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1352.jpg" }, { "index": 1353, "ocr_en": "Square ABCD, E, F are points on edges AB, BC, connecting DE, DF", "ocr_zh": "正方形ABCD,E、F为边AB、BC上一点，连接DE、DF", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1353.jpg" }, { "index": 1354, "ocr_en": "Right angled triangle EBF (where B is a right angle) has a point G on FE and a point D on the upper left corner of EF, connecting DG DF DE", "ocr_zh": "直角三角形EBF(B为直角)FE上有一点G，EF左上方有一点D，连接DG DF DE", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1354.jpg" }, { "index": 1355, "ocr_en": "Isosceles triangle ABC, angle BAC is a right angle, E and F are two points above the hypotenuse BC, connecting AE and AF, taking the middle point H of EF as the auxiliary line AH, angle EAH=α, angle FAH=β", "ocr_zh": "等腰三角形ABC，角BAC为直角，E、F为斜边BC上两点，连接AE、AF，取EF中间一点H，作辅助线AH，角EAH=α，角FAH=β", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1355.jpg" }, { "index": 1356, "ocr_en": "Isosceles triangle ABC,angle BAC is a right angle, E, F are two points on the hypotenuse BC, connecting AE, AF, D is a point on the right side of the side AC, CD is perpendicular to BC at point D, connecting AD, FD", "ocr_zh": "等腰三角形ABC，角EAC为直角，E、F为斜边BC上两点，连接AE、AF，D为边AC右边一点，CD垂直BC于点D，连接AD、FD", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1356.jpg" }, { "index": 1357, "ocr_en": "Square ABCD, connecting diagonal BD, E and F are points on edges BC and CD, connecting AE and AF, intersecting diagonal BD at points N and F, connecting EF", "ocr_zh": "正方形ABCD,连接对角线BD，E、F为边BC、CD上一点，连接AE、AF，交对角线BD分别于N、F两点，连接EF", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1357.jpg" }, { "index": 1358, "ocr_en": "Rectangle ABCD, BC is not much different from AB, E is a point on edge CD, triangle BCE is folded along BE to obtain triangle BEF, point F falls exactly on edge AD, extends EF, intersects the angle bisector of angle ABF at point M, BM intersects AD at point N", "ocr_zh": "矩形ABCD,BC与AB相差不大，E为边CD上一点，将三角形BCE沿BE翻折得到三角形BEF，F点恰好落到边AD上,延长EF，与角ABF的角平分线交于点M,BM交AD于点N", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1358.jpg" }, { "index": 1359, "ocr_en": "Rectangle ABCD, BC is not significantly different from AB, E is a point on edge CD, folding triangle BCE along BE yields triangle BEF, with point F falling exactly on edge AD", "ocr_zh": "矩形ABCD,BC与AB相差不大，E为边CD上一点，将三角形BCE沿BE翻折得到三角形BEF，F点恰好落到边AD上", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1359.jpg" }, { "index": 1360, "ocr_en": "Rectangle ABCD, BC is significantly larger than AB, and E is a point on edge CD. Folding triangle BCE along BE yields triangle BEF, with point F falling exactly on edge AD", "ocr_zh": "矩形ABCD,BC明显大于AB，E为边CD上一点，将三角形BCE沿BE翻折得到三角形BEF，F点恰好落到边AD上", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1360.jpg" }, { "index": 1361, "ocr_en": "Triangle ABC, D is a point on AC, AD is perpendicular to BC at point D. Fold triangle ACD along AC to form ACF, triangle ABD along AB to form ABG, and extend FC and GB to intersect at point H", "ocr_zh": "三角形ABC，D为AC上一点，AD垂直BC于点D，将三角形ACD沿AC折叠为ACF，将三角形ABD沿AB折叠为ABG，延长FC和GB相交于点H", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1361.jpg" }, { "index": 1362, "ocr_en": "Square ABCD, E on side CD, fold triangle ADE along AE to triangle AFE, extend EF to intersect side BC at point G, connect AG, CF", "ocr_zh": "正方形ABCD,E在边CD上，将三角形ADE沿AE对折至三角形AFE，延长EF交边BC于点G，连接AG、CF", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1362.jpg" }, { "index": 1363, "ocr_en": "Square ABCD, points E and F are the points on the extension lines of sides CB and DC, respectively, connecting AE, AF, and EF", "ocr_zh": "正方形ABCD,点E、F分别为边CB、DC延长线上的点，连接AE、AF、EF", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1363.jpg" }, { "index": 1364, "ocr_en": "Square ABCD, point A is in the lower left corner, clockwise are A, B, C, D, and points E and F are points on edges BC and CD respectively, connecting AE, AF, and EF", "ocr_zh": "正方形ABCD,点A在左下方，顺时针依次是A、B、C、D，点E、F分别为边BC、CD上一点，连接AE、AF、EF", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1364.jpg" }, { "index": 1365, "ocr_en": "Square ABCD, point A is in the upper left corner, clockwise are A, B, C, D, points E and F are points on edges BC and CD respectively, connecting AE, AF, and EF", "ocr_zh": "正方形ABCD,点A在左上角，顺时针依次是A、B、C、D，点E、F分别为边BC、CD上一点，连接AE、AF、EF", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1365.jpg" }, { "index": 1366, "ocr_en": "Isosceles triangle ABC, angle BAC is a right angle, M is a point on side BC, connect AM, rotate triangle ABM around point A to obtain triangle ACN, connect MN", "ocr_zh": "等腰三角形ABC，角BAC为直角，M是边BC上一点，连接AM，将三角形ABM绕点A旋转得到三角形ACN，连接MN", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1366.jpg" }, { "index": 1367, "ocr_en": "Quadrilateral ABCD, angles A and C are obtuse, angles B and D are acute", "ocr_zh": "四边形ABCD，角A、角C为钝角，角B、角D为锐角", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1367.jpg" }, { "index": 1368, "ocr_en": "Left figure: The zither quadrilateral ABCD, AD=AB, CD=CB, connects AC and BD, AC is perpendicular to BD, points P and Q are points on edges AB and AD respectively, connecting QP, CP, and CQ; Right figure: Quadrilateral ABCD, angles A and C are obtuse, angles B and D are acute", "ocr_zh": "左图：筝形四边形ABCD,AD=AB,CD=CB,连接AC、BD，AC垂直BD，点P、Q分别是边AB、AD上的点，连接QP、CP、CQ；右图：四边形ABCD，角A、角C为钝角，角B、角D为锐角", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1368.jpg" }, { "index": 1369, "ocr_en": "Triangle ABC, angle BAC is obtuse, AB=AC, D and E are points on the sides of BC, connect AD and AE, rotate triangle ABD around point A to obtain triangle ACD '\nConnect D'E. angle BAC to form an obtuse angle“", "ocr_zh": "三角形ABC,角BAC为钝角，AB=AC,D、E是BC边上的点，连接AD、AE，将三角形ABD绕点A旋转，得到三角形ACD′\n连接D'E.角BAC为钝角", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1369.jpg" }, { "index": 1370, "ocr_en": "Equilateral triangle ABC, point D on side BC, connecting AD", "ocr_zh": "等边三角形ABC，点D在边BC上，连接AD", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1370.jpg" }, { "index": 1371, "ocr_en": "Triangle ABC, AB=AC, D and E are points on the side of BC, connecting AD and AE. Rotate triangle ABD around point A to obtain triangle ACD ', connecting D'E. Figure 2 has a slightly smaller angle BAC compared to Figure 1", "ocr_zh": "三角形ABC,AB=AC,D、E是BC边上的点，连接AD、AE，将三角形ABD绕点A旋转，得到三角形ACD′\n连接D'E.图2和图1相比角BAC略小", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1371.jpg" }, { "index": 1372, "ocr_en": "Rectangle ABCD, point E on edge BC, point F on edge CD, connecting AE, AF, EF", "ocr_zh": "正方形ABCD,点E在边BC上，点F在边CD上，连接AE、AF、EF", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1372.jpg" }, { "index": 1373, "ocr_en": "Triangle ABC, AB=AC, D and E are points on the side of BC, connecting AD and AE. Rotate triangle ABD around point A to obtain triangle ACD ', connecting D'E. Figure 2 has a slightly smaller angle BAC compared to Figure 1, which is a right angle and DE is greater than BD", "ocr_zh": "三角形ABC,AB=AC,D、E是BC边上的点，连接AD、AE，将三角形ABD绕点A旋转，得到三角形ACD′\n连接D'E.图2和图1相比角BAC略小，角BAC为直角，且DE大于BD", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1373.jpg" }, { "index": 1374, "ocr_en": "Square ABCD, point E on edge BC, point F on edge CD, connecting AE, AF, EF", "ocr_zh": "正方形ABCD,点E在边BC上，点F在边CD上，连接AE、AF、EF", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1374.jpg" }, { "index": 1375, "ocr_en": "Square ABCD, point F is a point on the side of BC, connecting AF, and using AF as the diagonal to form square AEFG. Edge FG intersects with the diagonal AC of square ABCD at point H, connecting DG", "ocr_zh": "正方形ABCD，点F是BC边上一点，连接AF，以AF作对角线作正方形AEFG，边FG与正方形ABCD的对角线AC相交于点H，连接DG", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1375.jpg" }, { "index": 1376, "ocr_en": "Triangle ABC, angle ACB is obtuse, in the same plane, rotate triangle ABC counterclockwise around point A to obtain triangle ADE, connect EC", "ocr_zh": "三角形ABC，角ACB为钝角，在同一平面内，将三角形ABC绕点A逆时针旋转一定角度得到三角形ADE，连接EC", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1376.jpg" }, { "index": 1377, "ocr_en": "Line segment BE, point C is on line segment BC, BC is greater than CE, and equilateral triangles ABC and DCE are formed with BC and CE as sides respectively. AE and CD intersect at point N, BD and AC intersect at point M, AE and BD intersect at point O, and MN and OC are connected\n\"\n", "ocr_zh": "线段BE，点C在线段BC上，BC大于CE，分别以BC、CE为边作等边三角形ABC和等边三角形DCE，连接AE与CD相交于点N，连接BD与AC相交于点M，AE与BD相交于点O，连接MN、OC\n", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1377.jpg" }, { "index": 1378, "ocr_en": "Triangle ABC, with AB and AC as waist outward, forms isosceles right angled triangles ABE and ACD, angles BAE and CAE as right angles, connecting BD, CE, and ED, passing through point A as a straight line L, intersecting line segments DE and BC at points M and N, respectively", "ocr_zh": "三角形ABC，以AB、AC为腰向外做等腰直角三角形ABE、三角形ACD，角BAE和角CAE为直角，连接BD、CE、ED，过点A作直线L分别交线段DE、BC于点M、N", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1378.jpg" }, { "index": 1379, "ocr_en": "Square ABCD and Square AEFG, with side AB greater than side AE, connect BE, DG, DE", "ocr_zh": "正方形ABCD和正方形AEFG，边AB大于边AE，连接BE、DG、DE", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1379.jpg" }, { "index": 1380, "ocr_en": "Isometric triangle AOB (OA=OB) and isosceles triangle COD (OC=OD), OAOM=ON， AON is a sharp angle connecting AM and BN", "ocr_zh": "等腰直角三角形AOB和MON，点B在O点右侧，点N在O点左下方，角AOB和角MON为直角，OA=OB>OM=ON，角AON为锐角，连接AM、BN，B点上方有噪声点M", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1455.jpg" }, { "index": 1456, "ocr_en": "Isosceles right triangle AOB, angle AOB is a right angle", "ocr_zh": "等腰直角三角形AOB，角AOB为直角", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1456.jpg" }, { "index": 1457, "ocr_en": "Triangle ABC, point O is the midpoint of edge BC, connect AO, rotate triangle AOC clockwise around point O (rotation angle is obtuse) to obtain triangle EOF, connect AE and CF, extend AO to point D so that OD=OA, connect DE", "ocr_zh": "三角形ABC，点O为边BC上的中点，连接AO，将三角形AOC绕点O顺时针旋转（旋转角为钝角）得到三角形EOF，连接AE、CF，延长AO到点D使得OD=OA，连接DE", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1457.jpg" }, { "index": 1458, "ocr_en": "In triangle ABC, AB=AC， Point D is a point on the edge of BC, connecting AD, rotating AD counterclockwise around point A to obtain line segment AE, connecting BE, G is the midpoint of BE, connecting AG", "ocr_zh": "三角形ABC中，AB=AC，点D是BC边上一点，连接AD，将AD绕点A逆时针旋转得到线段AE，连接BE，G为BE的中点，连接AG，C的右侧有个噪声点B", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1458.jpg" }, { "index": 1459, "ocr_en": "In triangle ABC, AB=AC， Point D is a point on the edge of BC, connecting AD, rotating AD counterclockwise around point A to obtain line segment AE, connecting CE, angles BAC and DAE are right angles, connecting BE intersects AC at point F, BE is the angle bisector of angle ABC", "ocr_zh": "三角形ABC中，AB=AC，点D是BC边上一点，连接AD，将AD绕点A逆时针旋转得到线段AE，连接CE，角BAC和角DAE为直角，连接BE交AC于点F，BE是角ABC的角平分线", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1459.jpg" }, { "index": 1460, "ocr_en": "In triangle ABC, AB=AC， Point D is a point on the edge of BC, connecting AD, rotating AD counterclockwise around point A to obtain line segment AE, connecting BE, G is the midpoint of BE, connecting AG, DG, CE, angle BAC is obtuse, angle AEC is obtuse, BD is greater than CD", "ocr_zh": "三角形ABC中，AB=AC，点D是BC边上一点，连接AD，将AD绕点A逆时针旋转得到线段AE，连接BE，G为BE的中点，连接AG、DG、CE，角BAC为钝角，角AEC为钝角，BD大于CD，B上方有个噪声点E", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1460.jpg" }, { "index": 1461, "ocr_en": "Triangle ABC, point O is the midpoint of edge BC, connect AO, rotate triangle AOC clockwise around point O (rotation angle is obtuse) to obtain triangle EOF, connect AE and CF, angle BAC is a right angle, and AB=AC", "ocr_zh": "三角形ABC，点O为边BC上的中点，连接AO，将三角形AOC绕点O顺时针旋转（旋转角为钝角）得到三角形EOF，连接AE、CF，角BAC为直角，且AB=AC", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1461.jpg" }, { "index": 1462, "ocr_en": "Triangle ABC, point O is the midpoint of edge BC, connects AO, rotates triangle AOC clockwise around point O (rotation angle is obtuse) to obtain triangle EOF, connects AE and CF, angle BAC is a right angle, and AB and AC are not equal", "ocr_zh": "三角形ABC，点O为边BC上的中点，连接AO，将三角形AOC绕点O顺时针旋转（旋转角为钝角）得到三角形EOF，连接AE、CF，角BAC为直角，且AB与AC不相等", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1462.jpg" }, { "index": 1463, "ocr_en": "Line BC, point A is above line BC, angle ABC is an acute angle, point F is the middle point between B and C, connecting AF, forming an equilateral triangle AFE with AF as the side, point E is below line BC, passing through E to form ED, perpendicular to AB, intersecting AB at point D, and there is a noise point E above C", "ocr_zh": "直线BC，点A在直线BC上方，角ABC为锐角，点F为B、C中间一点，连接AF，以AF为边作等边三角形AFE，点E在直线BC的下方，过E作ED垂直AB交AB于点D，C上方有个噪声点E", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1463.jpg" }, { "index": 1464, "ocr_en": "Triangle ABC, angle ABC is an acute angle, AB=AC， Triangle CDE, CD=CE (CE greater than CA), BC=CD, A and E on the same side of BC, connected to BD", "ocr_zh": "三角形ABC，角ABC为锐角，AB=AC，三角形CDE，CD=CE（CE大于CA），BC=CD，A、E在BC同侧，连接BD", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1464.jpg" }, { "index": 1465, "ocr_en": "Triangle ABC, AB=AC, triangle CDE, CD=CE (CE greater than CA), BC=CD, point D is a point on the extension line of BC, E is a point on the extension line of CA, points B, C, E are not collinear, point P is a point on the line segment DE, connecting PB\n", "ocr_zh": "三角形ABC，AB=AC，三角形CDE，CD=CE（CE大于CA），BC=CD，点D为BC延长线上一点，E是CA延长线上一点，点B、C、E不共线，点P为线段DE上一点，连接PB", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1465.jpg" }, { "index": 1466, "ocr_en": "Line BC, point A is above line BC, angle ABC is an acute angle, point F is the middle point between B and C, connecting AF, forming an equilateral triangle AFE with AF as the side, A and E are on the same side of line BC, passing E to form ED, perpendicular AB, intersecting AB at point D, and there is also a noise point C below E", "ocr_zh": "直线BC，点A在直线BC上方，角ABC为锐角，点F为B、C中间一点，连接AF，以AF为边作等边三角形AFE，A、E在直线BC的同侧，过E作ED垂直AB交AB于点D，E下方还有个噪声点C", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1466.jpg" }, { "index": 1467, "ocr_en": "Triangle ABC, AB=AC, triangle CDE, CD=CE (CE is greater than CA), BC=CD, points B, C, and E are not collinear, point P is a point on line segment DE, connecting PB, and points A and E are on the same side of line BC", "ocr_zh": "三角形ABC，AB=AC，三角形CDE，CD=CE（CE大于CA），BC=CD，点B、C、E不共线，点P为线段DE上一点，连接PB，点A、E在直线BC的同侧", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1467.jpg" }, { "index": 1468, "ocr_en": "Line BC, point A is above line BC, angle ABC is an acute angle, point F is on the extension of line CB, connected to AF, forming an equilateral triangle AFE with AF as the side, crossing E to form ED, perpendicular AB, intersecting AB at point D", "ocr_zh": "直线BC，点A在直线BC上方，角ABC为锐角，点F在直线CB的延长线上，连接AF，以AF为边作等边三角形AFE，过E作ED垂直AB交AB于点D", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1468.jpg" }, { "index": 1469, "ocr_en": "Triangle ABC, angle C is a right angle, point D is a point within triangle ABC, connects BD, rotates BD counterclockwise around point D to obtain line segment BF (rotation angle is an acute angle), rotates line segment AB counterclockwise around point B to obtain line segment BE (rotation angle is an acute angle), connects AD, CD, AE, AF, BF, EF", "ocr_zh": "三角形ABC，角C为直角，点D为三角形ABC内一点，连接BD，将BD绕点D逆时针旋转得到线段BF（旋转角度为锐角），将线段AB绕点B逆时针旋转得到线段BE（旋转角为锐角），连接AD、CD、AE、AF、BF、EF", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1469.jpg" }, { "index": 1470, "ocr_en": "Triangle ABC, angle C is a right angle, point D is a point within triangle ABC, connects BD, rotates BD counterclockwise around point D to obtain line segment BF (rotation angle is an acute angle), rotates line segment AB counterclockwise around point B to obtain line segment BE (rotation angle is an acute angle), connects AD, CD, AE, AF, BF, EF, points M, N, and P are the midpoints of DF, AF, and AE, respectively, and connects MP and NP", "ocr_zh": "三角形ABC，角C为直角，点D为三角形ABC内一点，连接BD，将BD绕点D逆时针旋转得到线段BF（旋转角度为锐角），将线段AB绕点B逆时针旋转得到线段BE（旋转角为锐角），连接AD、CD、AE、AF、BF、EF，点M、N、P分别是DF、AF、AE的中点，连接MP、NP", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1470.jpg" }, { "index": 1471, "ocr_en": "Triangle BAC, Extend BC", "ocr_zh": "三角形BAC,延长BC", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1471.jpg" }, { "index": 1472, "ocr_en": "Triangle ABC, AB=AC, angle BAC is a right angle, passes through A as ray AM intersects ray BC at point D, rotates line segment AM counterclockwise around point A to obtain line segment AN (rotation angle is a right angle), passes through point C as CM parallel AM intersects line AN at point F, E is a point on AM, connects BE, and there is a noise point M above N", "ocr_zh": "三角形ABC，AB=AC，角BAC为直角，过A作射线AM交射线BC于点D，将线段AM绕点A逆时针旋转得到线段AN（旋转角为直角），过点C作CM平行AM交直线AN于点F，E为AM上一点，连接BE，N上方有一个噪声点M", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1472.jpg" }, { "index": 1473, "ocr_en": "Triangle ABC, AB=AC, angle BAC is an acute angle, passes through A as ray AM intersects ray BC at point D, rotates line segment AM counterclockwise around point A to obtain line segment AN (rotation angle is an acute angle), passes through point C as CM parallel AM intersects line AN at point F, E is a point on AM, connects BE，There is a noise point N below M", "ocr_zh": "三角形ABC，AB=AC，角BAC为锐角，过A作射线AM交射线BC于点D，将线段AM绕点A逆时针旋转得到线段AN（旋转角为锐角），过点C作CM平行AM交直线AN于点F，E为AM上一点，连接BE，M下方有个噪声点N", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1473.jpg" }, { "index": 1474, "ocr_en": "AB is the diameter of circle O, BC is the chord of circle O. Fold the BC arc along BC and intersect AB at point D. Then fold the BD arc along AB and intersect BC at point E, where E is the midpoint of arc BD", "ocr_zh": "AB为圆O的直径，BC为圆O的弦，将BC弧沿BC翻折交AB于点D，再将BD弧沿AB翻折交BC于点E且E是弧BD的中点", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1474.jpg" }, { "index": 1475, "ocr_en": "Rectangle ABCD, points E and F are points on edges AD and BC respectively, and EF is the axis of symmetry of rectangle ABCD. P is a point on edge CD. Fold triangle BCP along BP to obtain triangle BQP, where point Q falls exactly on EF, and make auxiliary line AP", "ocr_zh": "矩形ABCD，点E、F分别为边AD、BC上一点且EF是矩形ABCD的对称轴，P为边CD上一点，将三角形BCP沿BP折叠得到三角形BQP，点Q恰好落在EF上，做辅助线AP", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1475.jpg" }, { "index": 1476, "ocr_en": "Rectangle ABCD, points E and F are points on edges AD and BC respectively, and EF is the axis of symmetry of rectangle ABCD. M is a point on edge AD, and serves as an auxiliary line BM to intersect EF at point N. Fold triangle ABM along BM to obtain triangle A'BM, where point A 'falls exactly on EF. Connect BA' and extend it to intersect CD at point O", "ocr_zh": "矩形ABCD，点E、F分别为边AD、BC上一点且EF是矩形ABCD的对称轴，M为边AD上一点，作辅助线BM交EF于点N，将三角形ABM沿BM折叠得到三角形A'BM，点A'恰好落在EF上，连接BA'并延长交CD于点O", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1476.jpg" }, { "index": 1477, "ocr_en": "Rectangle ABCD, with side BC greater than side AB, points E and F on sides AD and BC respectively. Fold quadrilateral ABFE along EF to obtain A'GFE, with point B's corresponding point G falling exactly on the extension of side AD. Connect BG, intersect CD at point K, FG at point CD at point H. Line segments BE, BF, AB, and AG are dashed lines, while the rest of the line segments are solid lines", "ocr_zh": "矩形ABCD，边BC大于边AB，点E、F分别在边AD和边BC上，将四边形ABFE沿着EF折叠得到A'GFE，点B的对应点G恰好落在边AD的延长线上，连接BG，交CD于点K，FG交CD于点H，线段BE、BF、AB、AG为虚线，其余线段为实线", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1477.jpg" }, { "index": 1478, "ocr_en": "Triangle ABC, point D is a point on edge BC, connecting AD, folding triangle ABD along AD to obtain triangle AED, DE intersects AC at point G, connecting BE intersects AD at point F, line segments AB, BD, BF are dashed lines, and the rest of the line segments are solid lines. Triangle ADG is a shaded triangle", "ocr_zh": "三角形ABC，点D是边BC上一点，连接AD，将三角形ABD沿AD翻折得到三角形AED，DE交AC于点G，连接BE交AD于点F，线段AB、BD、BF为虚线，其余线段均为实线，三角形ADG为阴影三角形", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1478.jpg" }, { "index": 1479, "ocr_en": "In triangle ABC, point D is the midpoint of edge AC, connecting BD, and line segments CD and BC are dashed lines. Fold triangle BDC along BD to obtain triangle BDC ', which intersects AB at point E and connects AC'. Triangle BED is a shaded triangle", "ocr_zh": "三角形ABC中，点D是边AC上的中点，连接BD，线段CD、BC为虚线，将三角形BDC沿BD翻折得到三角形BDC'，DC'交AB于点E，连接AC'，三角形BED为阴影三角形", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1479.jpg" }, { "index": 1480, "ocr_en": "Rectangle ABCD, side AB is greater than BC, point P is a point on the side of BC, connect dashed line DP, fold triangle CDP along DP to obtain triangle EDP, point E falls exactly below line segment AB, and PE and DE intersect AB at points O and F, respectively", "ocr_zh": "矩形ABCD，边AB大于BC，点P为BC边上一点，连接虚线DP，将三角形CDP沿DP折叠得到三角形EDP，点E恰好落在线段AB下方，PE、DE分别交AB于点O、F", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1480.jpg" }, { "index": 1481, "ocr_en": "Rectangle ABCD, points E and F are located on edges AD and BC, respectively. Connect EF and fold quadrilateral CDEF along EF to obtain quadrilateral AD'EF. Point D is above line segment AD and connects AC, where AC intersects with EF at point O", "ocr_zh": "矩形ABCD，点E、F分别在边AD、BC上，连接EF，将四边形CDEF沿EF折叠得到四边形AD'EF，点D'在线段AD上方，连接AC，AC与EF交于点O", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1481.jpg" }, { "index": 1482, "ocr_en": "Rectangle ABCD, BC is greater than AB, points E and F are points on edges AD and BC, respectively. Folding ABFE along EF yields MGFE, and point B's corresponding point G falls exactly on line segment AD, connecting EF, BG, and BE. EF and BG intersect at point N, and line segments AB, AG, GF, and BF are dashed lines", "ocr_zh": "矩形ABCD，BC大于AB，点E、F分别是边AD、BC上的点，将ABFE沿EF折叠得到MGFE，点B的对应点G恰好落在线段AD上，连接EF、BG、BE，EF与BG交于点N，线段AB、AG、GF、BF为虚线", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1482.jpg" }, { "index": 1483, "ocr_en": "Triangle BEF, point G is on the right side of line segment EF, connects BG and EF at point O, crosses E to make dashed line AD parallel and equal to BC (points A and D are on both sides of point E), connects dashed line AB, crosses G to make dashed line CD parallel and equal to AB (points C and D are on both sides of point G), connects dashed line CF", "ocr_zh": "三角形BEF，点G在线段EF右侧，连接BG与EF交于点O，过E作虚线AD平行且等于BC（点A、D在点E两侧），连接虚线AB，过G作虚线CD平行且等于AB（点C、D在点G两侧），连接虚线CF", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1483.jpg" }, { "index": 1484, "ocr_en": "Right angled triangle ABC, angle BAC is a right angle, point D is the midpoint of edge BC, connects AD, folds triangle ACD along AD to obtain triangle AFD, point C corresponds to point F on the right side of line segment AB, line segment DF intersects with line segment AB at point E, and line segments AC, CD, AE are dashed lines", "ocr_zh": "直角三角形ABC，角BAC为直角，点D为边BC的中点，连接AD，将三角形ACD沿AD折叠得到三角形AFD，点C的对应点F落在线段AB的右侧，线段DF与线段AB交于点E，线段AC、CD、AE为虚线", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1484.jpg" }, { "index": 1485, "ocr_en": "Rectangle ABCD, connect diagonal BD, points E and F are on edges AD and CD respectively, connect BE and BF, fold rectangle ABCD along BE and BF so that the corresponding point M of point A and the corresponding point N of point C fall exactly on diagonal BD, connect EF", "ocr_zh": "矩形ABCD，连接对角线BD，点E、F分别在边AD、CD上，连接BE、BF，将矩形ABCD沿BE、BF折叠，使得点A的对应点M、点C的对应点N恰好落在对角线BD上，连接EF", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1485.jpg" }, { "index": 1486, "ocr_en": "Square ABCD, with points E and F on edges AD and BC, respectively. Fold the square along EF so that the corresponding point of point B falls exactly at point B 'on edge CD, and the corresponding point A' of point A falls above line segment AD", "ocr_zh": "正方形ABCD，点E、F分别在边AD、BC上，将正方形沿着EF翻折，使得点B的对应点恰好落在边CD上的点B'处，点A的对应点A'落在线段AD的上方", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1486.jpg" }, { "index": 1487, "ocr_en": "Equilateral triangle ABC, points D and E are points on sides AB and DC, respectively. Folding triangle BDE yields triangle FDE, where AC intersects with DF and EF at points G and H. Line segments DB, DF, and DE are dashed lines", "ocr_zh": "等边三角形ABC，点D、E分别是边AB、DC上的点，折叠三角形BDE得到三角形FDE，AC分别与DF、EF相交于点G、H，线段DB、DF、DE为虚线", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1487.jpg" }, { "index": 1488, "ocr_en": "Triangle ABC, angle B is an acute angle, point D is a point above BC, connecting AD", "ocr_zh": "三角形ABC，角B为锐角，点D为BC上一点，连接AD", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1488.jpg" }, { "index": 1489, "ocr_en": "Right angled triangle ABC, angle BAC is a right angle, point E is a point on edge AC, point D is a point on edge BC, connect DE, fold quadrilateral ABDE along DE to obtain quadrilateral FGDE, point G happens to fall on line segment AC, connect AF", "ocr_zh": "直角三角形ABC，角BAC为直角，点E为边AC上一点，点D为边BC上一点，连接DE，将四边形ABDE沿DE翻折得到四边形FGDE，点G恰好落在线段AC上，连接AF", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1489.jpg" }, { "index": 1490, "ocr_en": "Rectangle ABCD, point E is a point on edge AB, connected to dashed line DE, triangle ADE is folded along DE to obtain triangle A1DE, connected to dashed line A1E, triangle BA1E is folded along A1E to obtain triangle B1A1E, point B1 falls exactly on edge DE, and line segments A1D, A1E, DE, and B1A1 are all dashed lines", "ocr_zh": "矩形ABCD,点E为边AB上一点，连接虚线DE，将三角形ADE沿DE翻折得到三角形A1DE，连接虚线A1E，将三角形BA1E沿A1E翻折得到三角形B1A1E，点B1恰好落在边DE上，线段A1D、A1E、DE、B1A1均为虚线", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1490.jpg" }, { "index": 1491, "ocr_en": "Rectangle ABCD, points E and F are on sides BC and AD respectively, and quadrilateral ABEF is a square. Fold angle B inward so that point B falls exactly at point F, with the crease AE. Line segments AB and BE are dashed lines", "ocr_zh": "矩形ABCD，点E、F分别在边BC、AD上且四边形ABEF为正方形，将角B向内翻折使得点B恰好落在点F处，折痕为AE，线段AB、BE为虚线", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1491.jpg" }, { "index": 1492, "ocr_en": "Rectangle ABCD, points E and F are points on edges BC and CD, respectively. When passing through point E, make AE perpendicular to EF. Fold triangle ECF along EF to obtain triangle EC'F. Point C 'is inside the rectangle and connects AC'", "ocr_zh": "矩形ABCD，点E、F分别是边BC、CD上一点，过点E作AE垂直EF，将三角形ECF沿EF翻折得到三角形EC'F，点C'在矩形内部，连接AC'", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1492.jpg" }, { "index": 1493, "ocr_en": "Rectangle ABCD, points E and F are on sides BC and AD respectively, and quadrilateral ABEF is a square. Connect dashed line EF, fold angle C inward so that point C falls exactly at point F, and the intersection points of the fold with BC and CD are M and N, respectively. Connect MN, and line segments CN and CM are dashed lines", "ocr_zh": "矩形ABCD，点E、F分别在边BC、AD上且四边形ABEF为正方形，连接虚线EF，将角C向内翻折使得点C恰好落在点F处，折痕与BC、CD的交点分别为M、N，连接MN，线段CN、CM为虚线", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1493.jpg" }, { "index": 1494, "ocr_en": "Rectangle ABCD, with points G and E on sides BC and CD respectively, connecting AG, EG, and AE. Fold triangle ABG and triangle ECG along AG and EG respectively, so that points B and C fall exactly at the same point on line segment AE. This point is denoted as point F", "ocr_zh": "矩形ABCD，点G、E分别在边BC、CD上，连接AG、EG、AE，将三角形ABG和三角形ECG分别沿AG、EG折叠，使得点B、C恰好落在线段AE上的同一点处，该点记为点F", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1494.jpg" }, { "index": 1495, "ocr_en": "Rectangle ABCD, point E is the midpoint of edge AD, point F is the point above AB, connecting EF and CF. Fold triangle AEF along EF, and point A falls exactly at point G on line segment CE", "ocr_zh": "矩形ABCD，点E为边AD的中点，F为AB上一点，连接EF、CF，将三角形AEF沿EF折叠后点A恰好落在线段CE上的点G处", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1495.jpg" }, { "index": 1496, "ocr_en": "Rectangle ABCD, point E is a point on edge AB, connect dashed line CE, fold triangle BCE along CE so that point B's corresponding point F falls exactly on diagonal AC, connect DF, and points E, F, and D are exactly on the same straight line", "ocr_zh": "矩形ABCD，点E为边AB上一点，连接虚线CE，将三角形BCE沿CE对折使得点B的对应点F恰好落在对角线AC上，连接DF，点E、F、D恰好在同一直线上", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1496.jpg" }, { "index": 1497, "ocr_en": "Square ABCD, point E is a point on edge CD, connect AE, fold angle A inward so that point A's corresponding point G falls exactly on line segment AE, crease BF, point F is on edge AD, connect BG", "ocr_zh": "正方形ABCD，点E是边CD上一点，连接AE，将角A向内翻折使得点A的对应点G恰好落在线段AE上，折痕为BF，点F在边AD上，连接BG", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1497.jpg" }, { "index": 1498, "ocr_en": "Rectangle ABCD, point E is a point on edge BC, connect AE, fold triangle ADE along AE so that the corresponding point F of point D falls exactly on edge BC", "ocr_zh": "矩形ABCD，点E为边BC上一点，连接AE,将三角形ADE沿AE翻折使得点D的对应点F恰好落在边BC上", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1498.jpg" }, { "index": 1499, "ocr_en": "Rectangle ABCD, point E is a point on edge AB, connect dashed line DE, fold triangle ADE along DE so that the corresponding point F of point A falls exactly on edge BC, connect dashed line AF to intersect DE at point N, connect dashed line BN, and line segments EF and DF are also dashed lines", "ocr_zh": "矩形ABCD，点E为边AB上一点，连接虚线DE，将三角形ADE沿DE折叠，使得点A的对应点F恰好落在边BC上，连接虚线AF交DE于点N，连接虚线BN，线段EF、DF也为虚线", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1499.jpg" }, { "index": 1500, "ocr_en": "Figure 1: Rectangle ABCD; Figure 2: Rectangle ABCD, points E and F are the midpoints of edges BC and AD respectively, connected by dashed line EF; Figure 3: Rectangle ABCD, points E and F are the midpoints of edges BC and AD respectively, connected by dashed line EF and dashed line BD; Figure 4: Rectangle ABCD, points E and F are the midpoints of edges BC and AD respectively, connected by dashed line EF and dashed line BD, fold angle C inward so that it falls exactly on line segment BD, corresponding to point H, crease GE, point G on edge CD, and line segments CE and CG are dashed lines", "ocr_zh": "图①：矩形ABCD； 图②：矩形ABCD，点E、F分别为边BC、AD的中点，连接虚线EF；图③：矩形ABCD，点E、F分别为边BC、AD的中点，连接虚线EF，连接虚线BD； 图④：矩形ABCD，点E、F分别为边BC、AD的中点，连接虚线EF，连接虚线BD，将角C向内折叠使得其恰好落在线段BD上，其对应点为H，折痕为GE，点G在边CD上，线段CE、CG为虚线", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1500.jpg" }, { "index": 1501, "ocr_en": "Rectangle ABCD, AB is smaller than BC, fold angle A inward so that point A's corresponding point P falls inside the rectangle, crease DE, point E is on edge AB and happens to be the midpoint of edge AB, connect EP and extend intersection BC to point F", "ocr_zh": "矩形ABCD，AB小于BC，将角A向内折叠使得点A的对应点P落在矩形内部，折痕为DE，点E在边AB上且恰好是边AB的中点，连接EP并延长交BC于点F\n", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1501.jpg" }, { "index": 1502, "ocr_en": "Triangle ABC, angles C and B are acute angles, point E is the midpoint of edge AB, point F is on edge BC, connect EF, fold triangle AEF along EF to obtain triangle PEF, point P falls below edge BC, connect AP, and line segments AE and AF are dashed lines", "ocr_zh": "三角形ABC，角C、角B为锐角，点E为边AB的中点，点F在边BC上，连接EF，将三角形AEF沿EF折叠得到三角形PEF，点P落在边BC下方，连接AP，线段AE、AF为虚线", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1502.jpg" }, { "index": 1503, "ocr_en": "Triangle ABC, angles C and B are acute angles, point E is the midpoint of edge AB, point F is on edge BC, connect EF, fold triangle AEF along EF to obtain triangle PEF, point P happens to fall on edge BC, and line segments AE and AF are dashed lines", "ocr_zh": "三角形ABC，角C、角B为锐角，点E为边AB的中点，点F在边BC上，连接EF，将三角形AEF沿EF折叠得到三角形PEF，点P恰好落在边BC上，线段AE、AF为虚线", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1503.jpg" }, { "index": 1504, "ocr_en": "Triangle ABC, angles C and B are acute angles", "ocr_zh": "三角形ABC，角C、角B为锐角", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1504.jpg" }, { "index": 1505, "ocr_en": "Square ABCD, points E and F are located on edges AB and CD, respectively. Connect EF, fold BCFE along EF so that point B's corresponding point M falls exactly on edge AD, and point C's corresponding point N falls to the right of edge CD. Connect ME, MN, NF, MN intersects CD with point P", "ocr_zh": "正方形ABCD，点E、F分别在边AB、CD上，连接EF，将BCFE沿EF折叠使点B的对应点M恰好落在边AD上，点C的对应点N落在边CD右侧，连接ME、MN、NF，MN交CD于点P", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1505.jpg" }, { "index": 1506, "ocr_en": "Right angled triangle ABC, angle BAC is a right angle, point B is in the lower right corner, point P is a point on edge BC, connected to dashed line AP, fold triangle ABP along dashed line AP so that the corresponding point B 'of point B falls on the left side of edge AC, connected AB', CB ', BB', PB ', PB' is perpendicular to AC", "ocr_zh": "直角三角形ABC，角BAC为直角，点B在右下角，点P为边BC上一点，连接虚线AP，将三角形ABP沿虚线AP折叠使得点B的对应点B'落在边AC的左侧，连接AB'、CB'、BB'、PB'，PB'垂直AC", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1506.jpg" }, { "index": 1507, "ocr_en": "Right angled triangle ABC, angle BAC is a right angle, point B is in the lower right corner, AB=BC， Point P is a point on edge BC, BP is greater than CP. Connect the dashed line AP, fold the triangle ABP along the dashed line AP so that the corresponding point B 'of point B falls on the left side of edge AC, connect AB', PB ', and the point B' in the upper right corner is noise", "ocr_zh": "直角三角形ABC，角BAC为直角，点B在右下角，AB=BC，点P为边BC上一点，BP大于CP，连接虚线AP，将三角形ABP沿虚线AP折叠使得点B的对应点B'落在边AC的左侧，连接AB'、PB'，右上角的点B'为噪声", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1507.jpg" }, { "index": 1508, "ocr_en": "Right angled triangle ABC, angle BAC is a right angle, let the lower right corner be point B, point P be a point on edge BC, connect dashed line AP, fold triangle ABP along dashed line AP, let point B's corresponding point B 'fall on the left side of edge AC, connect AB', CB ', PB', PB 'is perpendicular to AC, and point B in the lower left corner is a noise point", "ocr_zh": "直角三角形ABC，角BAC为直角，设右下角为点B，点P为边BC上一点，连接虚线AP，将三角形ABP沿虚线AP折叠，设点B的对应点B'落在边AC的左侧，连接AB'、CB'、PB'，PB'垂直AC，左下角的点B为噪声点", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1508.jpg" }, { "index": 1509, "ocr_en": "Right angled triangle ABC, angle CAB is a right angle", "ocr_zh": "直角三角形ABC,角CAB为直角", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1509.jpg" }, { "index": 1510, "ocr_en": "Square ABCD, point G is a point on edge BC, connecting AG, passing through point D as DE, perpendicular AG to point E, BF perpendicular AG to point F, connecting BE, EF", "ocr_zh": "正方形ABCD，点G为边BC上一点，连接AG，过点D作DE垂直AG于点E，BF垂直AG于点F，连接BE、EF", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1510.jpg" }, { "index": 1511, "ocr_en": "Square ABCD, points E and F are located on sides AD and DC respectively, connecting BE, AF, and BF. BE and AF intersect at point G, and point H is the midpoint of BF, connecting GH", "ocr_zh": "正方形ABCD，点E、F分别在边AD、DC上，连接BE、AF、BF，BE与AF交于点G，点H为BF中点，连接GH", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1511.jpg" }, { "index": 1512, "ocr_en": "Square ABCD, points E and F are on sides CD and BC respectively, connecting BE and AF and intersecting at point G", "ocr_zh": "正方形ABCD，点E、F分别在边CD、BC上，连接BE、AF相交于点G", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1512.jpg" }, { "index": 1513, "ocr_en": "Square ABCD, connecting diagonal AC, points E and F on edges CD and BC respectively, connecting AE and DF, AE bisects angle CAD, DF intersects AE and AC at points G and M respectively, P is a point on AG, crossing point P as PN perpendicular AC, vertical foot N, connecting PM", "ocr_zh": "正方形ABCD，连接对角线AC，点E、F分别在边CD、BC上，连接AE、DF，AE平分角CAD，DF分别交AE、AC于点G、M，P为AG上一点，过点P作PN垂直AC，垂足为N，连接PM", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1513.jpg" }, { "index": 1514, "ocr_en": "Square ABCD, point G is a point on the BC side, connected to AG, passing through point B as BE perpendicular to point E, passing through point D as DF perpendicular to point F, connected to DE", "ocr_zh": "正方形ABCD，点G为BC边上一点，连接AG，过点B作BE垂直AG于点E，过点D作DF垂直AG于点F，连接DE", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1514.jpg" }, { "index": 1515, "ocr_en": "Square ABCD, connected to AC, point K on edge AD, connecting BK to AC at point P, passing through points A and C as perpendicular lines to BK, with legs M and N respectively. O is the center of square ABCD, connecting OM and ON,", "ocr_zh": "正方形ABCD，连接AC，点K在边AD上，连接BK交AC于点P，过点A、点C分别作BK的垂线，垂足分别为M、N，O是正方形ABCD的中心，连接OM、ON，", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1515.jpg" }, { "index": 1516, "ocr_en": "Square ABCD, point E is a point on edge CD, connected to BE, passing through A for AF, perpendicular to BE, intersecting BC at point F", "ocr_zh": "正方形ABCD，点E是边CD上一点，连接BE，过A作AF垂直BE交BC于点F", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1516.jpg" }, { "index": 1517, "ocr_en": "Square ABCD, point E is a point on edge CD, connecting BE, M is the midpoint of BE, passing through M makes FG perpendicular to BE, intersecting edges AD, BC with G, F respectively", "ocr_zh": "正方形ABCD，点E是边CD上一点，连接BE，M为BE的中点，过M作FG垂直BE分别交边AD、BC于G、F", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1517.jpg" }, { "index": 1518, "ocr_en": "Square ABCD, point E is a point on edge CD, connecting BE, M is the midpoint of BE, connecting CM, passing through point C as CG, perpendicular to BE, intersecting AD at point G, connecting EG, MG", "ocr_zh": "正方形ABCD，点E是边CD上一点，连接BE，M为BE的中点，连接CM，过点C作CG垂直BE交AD于点G，连接EG、MG", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1518.jpg" }, { "index": 1519, "ocr_en": "Rectangle ABCD, points G and F are located on edges CD and AB, respectively. Fold rectangle ADGF along GF to obtain quadrilateral EPGF. The corresponding point E of point A falls on edge BC, and the corresponding point P of point G falls on the right side of edge CD. Connect AE, intersect GF with point O, and connect CP", "ocr_zh": "矩形ABCD，点G、F分别在边CD、AB上，将矩形ADGF沿GF折叠使得到四边形EPGF，点A的对应点E落在边BC上，点G的对应点P落在边CD的右侧，连接AE交GF于点O，连接CP", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1519.jpg" }, { "index": 1520, "ocr_en": "Square ABCD, point E is the midpoint of side AB, connecting CE, passing through point B as BF, perpendicular CE to point G, intersecting AD at point F, connecting DG", "ocr_zh": "正方形ABCD，点E是AB边的中点，连接CE，过点B作BF垂直CE于点G，交AD于点F，连接DG", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1520.jpg" }, { "index": 1521, "ocr_en": "Square ABCD, point E is the midpoint of side AB, connecting CE, passing through point B as BF, perpendicular CE to point G, intersecting AD at point F, connecting DG, passing through point C as CM, perpendicular DG to point H, intersecting AD and BF at points M and N, respectively", "ocr_zh": "正方形ABCD，点E是AB边的中点，连接CE，过点B作BF垂直CE于点G，交AD于点F，连接DG，过点C作CM垂直DG于点H，分别交AD、BF于点M、N", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1521.jpg" }, { "index": 1522, "ocr_en": "Square ABCD, point E is a point on the AB side, connecting CE, passing through point B as BF, perpendicular CE to point G, intersecting AD at point F", "ocr_zh": "正方形ABCD，点E是AB边上一点，连接CE，过点B作BF垂直CE于点G，交AD于点F", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1522.jpg" }, { "index": 1523, "ocr_en": "Square ABCD, points E and Q are on sides BC and AB respectively, DQ is perpendicular to point O, connecting AE and DQ, points G and F are on sides CD and AB respectively, and GF is perpendicular to AE", "ocr_zh": "正方形ABCD，点E、Q分别在边BC、AB上，作DQ垂直AE于点O，连接AE、DQ，点G、F分别在边CD、AB上且GF垂直AE", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1523.jpg" }, { "index": 1524, "ocr_en": "In triangle ABC, AC=BC， Angle C is an obtuse angle, point D is the midpoint of the BC edge, passing through point E forms an acute angle EDF, DE and DF intersect AC and BC at points E and F respectively, and EF and AB are not parallel", "ocr_zh": "三角形ABC中，AC=BC，角C为钝角，点D为BC边上的中点，过点E作锐角EDF，DE、DF分别交AC、BC于E、F两点，EF与AB不平行", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1524.jpg" }, { "index": 1525, "ocr_en": "Equilateral triangle ABC, point D is on the left side of side AB, form triangle ABD, angle ADB is obtuse, connect CD, rotate triangle CBD clockwise around point C to obtain triangle CAE (CB happens to fall on CA), rotation angle is acute", "ocr_zh": "等边三角形ABC，点D在边AB的左侧，作三角形ABD，角ADB为钝角，连接CD，将三角形CBD绕点C顺时针旋转得到三角形CAE（CB恰好落在CA上），旋转角度为锐角", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1525.jpg" }, { "index": 1526, "ocr_en": "Square ABCD, connecting diagonal AC, point P is a point on AC, connecting BP, passing through point P makes PQ perpendicular BP, PQ intersects CD at point Q, point Q happens to be the midpoint of CD, connecting BQ intersects AC at point G", "ocr_zh": "正方形ABCD，连接对角线AC，点P是AC上一点，连接BP，过点P作PQ垂直BP，PQ交CD于点Q，点Q恰好为CD中点，连接BQ交AC于点G", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1526.jpg" }, { "index": 1527, "ocr_en": "In the coordinate system xoy, OC is the angle bisector of the first quadrant, point P is a point above OC, passing through P makes AP perpendicular BP, intersecting the positive half axis of the y-axis and the positive half axis of the x-axis at points B and A, respectively", "ocr_zh": "坐标系xoy中，OC为第一象限的角平分线，点P为OC上一点，过P作AP垂直BP，分别交y轴的正半轴、x轴的正半轴于B、A两点", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1527.jpg" }, { "index": 1528, "ocr_en": "Quadrilateral ABCD, AB=AD, angles A, B, and D are obtuse angles, angle C is acute angle, point E is a point on the extension line of BA, and the angle bisector of its outer angle EAD is taken as the extension line of the intersection AF and CD at point F", "ocr_zh": "四边形ABCD，AB=AD，角A、角B、角D为钝角，角C为锐角，点E为BA延长线上一点，作其外角EAD的角平分线AF交CD的延长线于点F", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1528.jpg" }, { "index": 1529, "ocr_en": "Quadrilateral ABCD, AB=AD, angles D and B complement each other, angles A and C complement each other, connect AC", "ocr_zh": "四边形ABCD，AB=AD，角B为锐角，其余三个角为钝角，连接AC", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1529.jpg" }, { "index": 1530, "ocr_en": "Points A, B, and C are located on circle O, connecting chords AB and BC. The angle bisector of angle ABC intersects circle O at point D, connecting AD and CD", "ocr_zh": "点A、B、C位于圆O上，连接弦AB、BC，角ABC的角平分线交圆O于点D，连接AD、CD", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1530.jpg" }, { "index": 1531, "ocr_en": "Quadrilateral ABCD, AB=AD, angles D and B complement each other, angles A and C complement each other, connecting BD and AC. Point A is above BD, and point C is below BD", "ocr_zh": "四边形ABCD，AB=AD，角D为锐角，角B为钝角，连接BD、AC，点A在BD上方，点C在BD下方", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1531.jpg" }, { "index": 1532, "ocr_en": "Triangle ABC, angle A is obtuse, AB is greater than AC, points D, E, and F are on sides BC, AB, and AC respectively, and triangle DEF is an equilateral triangle. There is also a noise point B in the lower right corner of point D", "ocr_zh": "三角形ABC，角A为钝角，AB大于AC，点D、E、F分别在边BC、AB和AC上，且三角形DEF为等边三角形，D点右下角还有个噪声点B", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1532.jpg" }, { "index": 1533, "ocr_en": "Triangle ABC, angle A is obtuse, AB=AC， Points D, E, and F are located on edges BC, AB, and AC, respectively. Point D is the midpoint of BC, and angle EDF is an acute angle", "ocr_zh": "三角形ABC，角A为钝角，AB=AC，点D、E、F分别在边BC、AB和AC上，点D为BC的中点，角EDF为锐角", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1533.jpg" }, { "index": 1534, "ocr_en": "Equilateral triangle ABC", "ocr_zh": "等边三角形ABC", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1534.jpg" }, { "index": 1535, "ocr_en": "Quadrilateral ABCD, AB=AD, angle D greater than angle B, connected AC, point A above BD, point C below BD", "ocr_zh": "四边形ABCD，AB=AD，角D大于角B，连接AC，点A在BD上方，点C在BD下方", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1535.jpg" }, { "index": 1536, "ocr_en": "Quadrilateral ABCD, BC=CD, angle D is greater than angle B, connected to AC, angle BAC is an acute angle, point D is above AC, point B is below AC", "ocr_zh": "四边形ABCD，BC=CD，角D大于角B，连接AC，角BAC为锐角，点D在AC上方，点B在AC下方", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1536.jpg" }, { "index": 1537, "ocr_en": "Triangle ABC, points D and E are located on sides AB and BC respectively, connecting DE", "ocr_zh": "三角形ABC，点D、E分别在边AB、BC上，连接DE", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1537.jpg" }, { "index": 1538, "ocr_en": "Isosceles triangle ABE, AE=BE, points C and D are on sides BE and AE respectively, connecting AC and BD and intersecting at point H, and AD=BC", "ocr_zh": "等腰三角形ABE，AE=BE，点C、D分别在边BE、AE上，连接AC、BD相交于点H，且AD=BC", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1538.jpg" }, { "index": 1539, "ocr_en": "In the coordinate system xoy, points A and D are on the positive half axes of the x-axis and y-axis, respectively. Points B and C are in the first quadrant, forming rectangles ABCD. AB is less than BC, and angle OAD is an acute angle. Point M is the midpoint of AD, connecting OM and CM", "ocr_zh": "坐标系xoy中，点A、D分别在x轴和y轴的正半轴上，点B、C在第一象限，作矩形ABCD，AB小于BC，角OAD为锐角，点M为AD的中点，连接OM、CM", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1539.jpg" }, { "index": 1540, "ocr_en": "In the coordinate system xoy, points A and D are on the positive half axes of the x-axis and y-axis, respectively. Points B and C are in the first quadrant, forming rectangles ABCD. AB is less than BC, and angle OAD is an acute angle", "ocr_zh": "坐标系xoy中，点A、D分别在x轴和y轴的正半轴上，点B、C在第一象限，作矩形ABCD，AB小于BC，角OAD为锐角", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1540.jpg" }, { "index": 1541, "ocr_en": "In the coordinate system xoy, point A3 is located at point O, A7 is located to the left of point A3, and to the right of point A3 are A1, A5, and A9 from left to right, with the same interval between each two points. A2 and A6 are located in the first quadrant, and A4 and A8 are located in the fourth quadrant, forming isosceles right angled triangles A1A2A3, A3A4A5, A5A6A7, and A7A8A9, respectively", "ocr_zh": "坐标系xoy中，点A3位于O点，A7位于A3点左侧，A3点右侧从左往右依次是A1、A5、A9，且每两个点之间间隔相同，A2、A6位于第一象限，A4、A8位于第四象限，分别作等腰直角三角形A1A2A3、A3A4A5、A5A6A7、A7A8A9", "class": "Analytic Geometry", "difficulty": 3, "image_path": "fig_1541.jpg" }, { "index": 1542, "ocr_en": "In the coordinate system xoy, points D, A, B, and C are located in the first, second, third, and fourth quadrants, respectively. Using these four points as rectangles ABCD, edge BC is greater than CD, edge CD is farther from the y-axis than edge AB is farther from the y-axis, and edge CB is farther from the x-axis than edge AD is farther from the x-axis", "ocr_zh": "坐标系xoy中，点D、A、B、C分别位于第一、二、三、四象限，以这四点作矩形ABCD，边BC大于CD，边CD到y轴的距离大于边AB到y轴的距离，边CB到x轴的距离大于边AD到x轴的距离", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1542.jpg" }, { "index": 1543, "ocr_en": "Coordinate system xoy, regular hexagon ABCDEF is located in the first quadrant, points A and B fall on the positive X-axis, and point F falls on the positive Y-axis", "ocr_zh": "坐标系xoy，正六边形ABCDEF位于第一象限，点A、B落在X轴正半轴上，点F落在y轴正半轴上", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1543.jpg" }, { "index": 1544, "ocr_en": "In the coordinate system xoy, point A is located on the positive half axis of the x-axis, point B is located in the first quadrant, and points A, O, and B form an equilateral triangle AOB. Rotate the triangle AOB counterclockwise around point O by a certain angle and enlarge each side by two times to obtain triangle OA1B1. Points A1 and B1 are located in the first and second quadrants, respectively. Rotate triangle OA1B1 counterclockwise around point O by a certain angle and enlarge each side by two times to obtain triangle OA2B2. Point A2 is located in the second quadrant, and point B2 is located on the negative half axis of x. Rotate triangle OA2B2 counterclockwise around point O by a certain angle and enlarge each side by two times to obtain triangle OA3B3. Point A3, B3 is located in the third quadrant", "ocr_zh": "坐标系xoy中，点A位于x轴正半轴上，点B位于第一象限，以A、O、B三点作等边三角形AOB，将三角形AOB绕点O逆时针旋转60度且每条边扩大原来的2倍得到三角形OA1B1，点A1、B1分别位于一、二象限，将三角形OA1B1绕点O逆时针旋转60度且每条边扩大原来的2倍得到三角形OA2B2，点A2位于第二象限，点B2位于x的负半轴上，将三角形OA2B2绕点O逆时针旋转60度且每条边扩大原来的2倍得到三角形OA3B3，点A3、B3位于第三象限", "class": "Analytic Geometry", "difficulty": 4, "image_path": "fig_1544.jpg" }, { "index": 1545, "ocr_en": "Coordinate system xoy, line l1 passes through quadrants one, two, and three, line l2 passes through the origin and quadrants one and three, and the two lines intersect at A1. The line passing through A1 forms the x-axis perpendicular line, with a vertical foot of B1. The parallel line passing through B1 forms l2 and intersects l1 with A2. The line passing through A2 forms the x-axis perpendicular line, with a vertical foot of B2. The line passing through B2 forms l2 and intersects l1 with A3. The line passing through A3 forms the x-axis perpendicular line, with a vertical foot of B3", "ocr_zh": "坐标系xoy，直线l1过一、二、三象限，直线l2过原点且过一、三象限，两直线相交于A1，过A1作x轴垂线，垂足为B1，过B1作l2的平行线交l1于A2，过A2作x轴的垂线垂足为B2，过B2作l2的平行线交l1于A3，过A3作x轴的垂线垂足为B3", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1545.jpg" }, { "index": 1546, "ocr_en": "Coordinate system xoy, points A and B are located on the positive half axes of the x-axis and y-axis, respectively. Connect AB, roll triangle ABO to the right to obtain triangle AB1C1, point B1 falls on the x-axis, point c1 falls on the first quadrant, roll triangle AB1C1 to the right to obtain triangle A1B1C2, and point C2 falls on the x-axis", "ocr_zh": "坐标系xoy，点A、B分别位于x轴和y轴的正半轴上，连接AB，将三角形ABO向右滚动得到三角形AB1C1，点B1落在x轴上，点c1落在第一象限，将三角形AB1C1向右滚动得到三角形A1B1C2，点C2落在x轴上", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1546.jpg" }, { "index": 1547, "ocr_en": "Coordinate system xoy, lines l1 and l2 pass through the origin and three quadrants, with a slope l2 greater than l1. The vertex A of triangle BAC is on l1, passing through A makes AB perpendicular to the x-axis and has a perpendicular foot of B, point C is on l2, passing through A makes a perpendicular line of l2, has a perpendicular foot of C1, intersects the x-axis with B1, passes through B1 makes A1B1 perpendicular to the x-axis, has a perpendicular foot of B1, intersects l1 at point A1, connects A1C1, A1B1, C1B1, passes through A1 makes a perpendicular line of l2, has a perpendicular foot of C2, intersects the x-axis with B2, passes through B2 makes A2B2 perpendicular to the x-axis, has a perpendicular foot of B2, intersects l1 at point A2, connects A2C2, A2B2, C2B2", "ocr_zh": "坐标系xoy，直线l1、l2过原点且过一三象限，斜率l2大于l1,三角形BAC的顶点A在l1上，过A作AB垂直x轴垂足为B，点C在l2上，过A作l2的垂线，垂足为C1，交x轴为B1，过B1作A1B1垂直x轴，垂足为B1，交l1于点A1，连接A1C1、A1B1、C1B1，过A1作l2的垂线，垂足为C2，交x轴为B2，过B2作A2B2垂直x轴，垂足为B2，交l1于点A2，连接A2C2、A2B2、C2B2", "class": "Analytic Geometry", "difficulty": 3, "image_path": "fig_1547.jpg" }, { "index": 1548, "ocr_en": "In the Cartesian coordinate system xoy, there are several points with integer coordinates in the order of (1,0), (2,0), (2,1), (1,1), (1,2), and (2,2) There are a total of 18 points, which are connected in sequence to form a line, with a right arrow at the top", "ocr_zh": "直角坐标系xoy中，有若干个横纵坐标均为整数的点，其顺序依次为（1，0），（2，0），（2，1），（1，1），（1，2），（2，2）...一共18个点，将这18个点依次连接成线，最顶部有一个向右的箭头", "class": "Analytic Geometry", "difficulty": 4, "image_path": "fig_1548.jpg" }, { "index": 1549, "ocr_en": "In the Cartesian coordinate system xoy, point A is located in the first quadrant, point C is located on the positive half axis of the X-axis, point B is located at point O, and triangle ABC is an equilateral triangle", "ocr_zh": "直角坐标系xoy中，点A位于第一象限，点C位于X轴正半轴上，点B位于O点，三角形ABC为等边三角形", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1549.jpg" }, { "index": 1550, "ocr_en": "In the Cartesian coordinate system xoy, points B and C are located on the positive half axes of the x-axis and y-axis, respectively. Point A is located in the first quadrant, and the quadrilateral ABOC is a square. With point B as the center and BA as the radius, it forms an arc AA1 intersecting the positive half axis of the x-axis as A1. With point O as the center and QA1 as the radius, it forms an arc A1A2 in the fourth quadrant, intersecting the negative half axis of the y-axis as A2. With point C as the center and CA2 as the radius, it forms an arc A2A3 in the fourth quadrant. The center angle of arc A2A3 is 90 degrees, and point A3 falls in the second quadrant, connecting CA3. With point A as the center and AA3 as the radius, it forms an arc A4A3. The central angle of point A4A3 is 90 degrees, and point A4 falls in the first quadrant, connecting AA4", "ocr_zh": "直角坐标系xoy中，点B、C分别位于x轴和y轴的正半轴上，点A位于第一象限，且四边形ABOC为正方形，以点B为圆心，BA为半径作弧AA1交x轴的正半轴为A1，以点O为圆心、QA1为半径在第四象限作圆弧A1A2，交y轴的负半轴于A2，以点C为圆心、CA2为半径在第四象限作圆弧A2A3，弧A2A3所对的圆心角为90度，点A3落在第二象限，连接CA3，以点A为圆心、AA3为半径作圆弧A4A3，弧A4A3所对的圆心角为90度，点A4落在第一象限，连接AA4", "class": "Analytic Geometry", "difficulty": 4, "image_path": "fig_1550.jpg" }, { "index": 1551, "ocr_en": "In the Cartesian coordinate system xoy, the two sides OA1 and OC1 of the square OA1B1C1 are located on the positive half axis of the x-axis and y-axis, respectively. Point B1 is located in the first quadrant and connects to OB1, forming a square OB1B2C2 on its left side with OB1 as the edge. Point B2 is located on the positive half axis of the y-axis and connects to OB2, forming a square OB2B3C3 on its left side with OB2 as the edge. Point C3 is located on the negative half axis of the x-axis and connects to OB3, forming a square OB3B4C4 on its left side with OB3 as the edge. Point B4 is located on the negative half axis of the x-axis and connects to OB4", "ocr_zh": "直角坐标系xoy中，正方形OA1B1C1的两边OA1、OC1分别落在x轴和y轴的正半轴上，点B1落在第一象限，连接OB1，以OB1为边在其左侧作正方形OB1B2C2，点B2落在y轴的正半轴上，连接OB2，以OB2为边在其左侧作正方形OB2B3C3，点C3落在x轴的负半轴上，连接OB3，以OB3为边在其左侧作正方形OB3B4C4，点B4落在x轴的负半轴上，连接OB4", "class": "Analytic Geometry", "difficulty": 4, "image_path": "fig_1551.jpg" }, { "index": 1552, "ocr_en": "In the Cartesian coordinate system xoy, A1 is located at the origin. The coordinates obtained by moving from point A1 are A2 (1,0), A3 (1,1), A4 (-1,1), A5 (-1,1), A6 (2,1), A7 (2,2)... The final point is A16", "ocr_zh": "直角坐标系xoy中，A1位于原点，从点A1出发依次移动到达的坐标分别是A2(1,0)、A3(1,1)、A4（-1，1）、A5(-1,-1)、A6(2,-1)、A7(2,2)...最终点为A16", "class": "Analytic Geometry", "difficulty": 4, "image_path": "fig_1552.jpg" }, { "index": 1553, "ocr_en": "In the Cartesian coordinate system xoy, point A1 is located on the positive half axis of the x-axis, forming a right angled triangle OA1A2 with OA1 as the right angled side, point A2 falls in the first quadrant, forming a right angled triangle OA2A3 with OA2 as the right angled side, point A3 falls in the second quadrant, forming a right angled triangle OA3A4 with OA3 as the right angled side, and point A4 falls on the negative half axis of the x-axis", "ocr_zh": "直角坐标系xoy中，点A1位于x轴正半轴上，以OA1为直角边作直角三角形OA1A2，点A2落在第一象限，再以OA2为直角边作直角三角形OA2A3，点A3落在第二象限，再以OA3为直角边作直角三角形OA3A4，点A4落在x轴的负半轴上", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1553.jpg" }, { "index": 1554, "ocr_en": "In the Cartesian coordinate system xoy, points A1, A2, A3, and A4 are located on the positive half axis of the x-axis, and ray OB1 passes through the origin and is in the first quadrant. Points B1, B2, and B3 are located on ray OB1, connecting A1B1, A2B1, A2B2, A3B2, A3B3, and A4B3. Triangles A1B1A2, A2B2A3, and A3B3A4 are all equilateral triangles", "ocr_zh": "直角坐标系xoy中，点A1、A2、A3、A4分别位于x轴的正半轴上，射线OB1过原点且在第一象限，点B1、B2、B3位于射线OB1上，连接A1B1、A2B1、A2B2、A3B2、A3B3、A4B3，三角形A1B1A2、A2B2A3、A3B3A4均为等边三角形", "class": "Analytic Geometry", "difficulty": 3, "image_path": "fig_1554.jpg" }, { "index": 1555, "ocr_en": "In the Cartesian coordinate system xoy, line l1 passes through the origin and three quadrants, line l2 passes through the origin and two or four quadrants, point A1 is in the fourth quadrant and on line l2, the perpendicular line passing through point A1 as the x-axis intersects l1 at point A2, the perpendicular line passing through point A2 as the y-axis intersects l2 at point A3, the perpendicular line passing through point A3 as the x-axis intersects l1 at point A4, the perpendicular line passing through point A4 as the y-axis intersects l2 at point A5, and the perpendicular line passing through point A5 as the x-axis intersects l1 at point A6", "ocr_zh": "直角坐标系xoy中，直线l1过原点且过一三象限，直线l2过原点且过二四象限，点A1在第四象限且在直线l2上，过点A1作x轴的垂线交l1于点A2，过点A2作y轴的垂线交l2于点A3，过点A3作x轴的垂线交l1于点A4，过点A4作y轴的垂线交l2于点A5，过点A5作x轴的垂线交l1于点A6", "class": "Analytic Geometry", "difficulty": 3, "image_path": "fig_1555.jpg" }, { "index": 1556, "ocr_en": "In the Cartesian coordinate system xoy, lines l1 and l2 pass through the origin and are in the first quadrant. The angle between l2 and the positive half axis of the x-axis is greater than the angle between l1 and the positive half axis of the x-axis. Square shapes A1B1C1A2, A2B2C2A3, A3B3C3A4, A4B4C4A5 are constructed between lines l1 and k2 in sequence. Points A1, A2, A3, A4, A5 are on line l1, points B1, B2, B3, B4 are on line l2, points C1, C2, C3, C4 are between two lines, connecting C1B2, C2B3, C3B4", "ocr_zh": "直角坐标系xoy中，直线l1、l2过原点且在第一象限，l2与x轴正半轴的夹角大于l1与x轴正半轴的夹角，在直线l1和k2之间依次构造正方形A1B1C1A2、A2B2C2A3、A3B3C3A4、A4B4C4A5，点A1、A2、A3、A4、A5在直线l1上，点B1、B2、B3、B4在直线l2上，点C1、C2、C3、C4在两直线之间，连接C1B2、C2B3、C3B4", "class": "Analytic Geometry", "difficulty": 4, "image_path": "fig_1556.jpg" }, { "index": 1557, "ocr_en": "In the Cartesian coordinate system xoy, the side OA of the isosceles right triangle OAA1 is on the positive half axis of the x-axis, and the point A1 is in the first quadrant. Taking A1 as the right angled vertex, OA1 as a right angled side forms the isosceles right angled triangle OA1A2, A2 as the right angled vertex, OA2 as a right angled side forms the isosceles right angled triangle OA2A3, A3 as the right angled vertex, OA3 as a right angled side forms the isosceles right angled triangle OA3A4, A4 as the right angled vertex, OA4 as a right angled side forms the isosceles right angled triangle OA4A5, A5 as the right angled vertex, OA5 as a right angled side forms the isosceles right angled triangle OA5A6, A6 as the right angled vertex, and OA6 as a right angled side forms the isosceles right angled triangle OA3A6. isosceles right angled triangle OA6A7,", "ocr_zh": "直角坐标系xoy中，等腰直角三角形OAA1的边OA在x轴的正半轴上，点A1在第一象限，以A1为直角顶点，OA1为一直角边作等腰直角三角形OA1A2，以A2为直角顶点，OA2为一直角边作等腰直角三角形OA2A3，以A3为直角顶点，OA3为一直角边作等腰直角三角形OA3A4，以A4为直角顶点，OA4为一直角边作等腰直角三角形OA4A5，以A5为直角顶点，OA5为一直角边作等腰直角三角形OA5A6，以A6为直角顶点，OA6为一直角边作等腰直角三角形OA6A7，", "class": "Analytic Geometry", "difficulty": 4, "image_path": "fig_1557.jpg" }, { "index": 1558, "ocr_en": "Cartesian coordinate system xoy, line l passes through the origin and three quadrants, point A1 is on the positive half axis of the x-axis, point A1 is A1B1 perpendicular to the x-axis intersects line l at B1, draw an arc intersecting the x-axis positive half axis at A2 with origin O as the center OB1 as the radius, draw an arc intersecting the x-axis positive half axis at B2 perpendicular to point A2, draw an arc intersecting the x-axis positive half axis at A3 perpendicular to origin O as the center OB2 as the radius, draw an arc intersecting the x-axis positive half axis at point A3 perpendicular to the x-axis at point B3, draw an arc intersecting the x-axis positive half axis at A4 with origin O as the center OB3 as the radius", "ocr_zh": "直角坐标系xoy，直线l过原点且过一三象限，点A1在x轴正半轴上，过点A1作A1B1垂直x轴交直线l于B1，以原点O为圆心OB1为半径画弧交x轴正半轴于A2，过点A2作A2B2垂直x轴交直线l于B2，以原点O为圆心OB2为半径画弧交x轴正半轴于A3，过点A3作A3B3垂直x轴交直线l于B3，以原点O为圆心OB3为半径画弧交x轴正半轴于A4", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1558.jpg" }, { "index": 1559, "ocr_en": "In the Cartesian coordinate system xoy, points A and C are located on the positive y-axis and negative x-axis, respectively, with OC greater than OA. Point B is located in the second quadrant and forms a rectangle OABC. The inverse proportional function is located in the second quadrant and connects OB to the inverse proportional function at point D", "ocr_zh": "直角坐标系xoy，点A、C分别位于y轴正半轴、x轴负半轴上，且OC大于OA，点B位于第二象限，作矩形OABC，反比例函数位于第二象限，连接OB交反比例函数于点D", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1559.jpg" }, { "index": 1560, "ocr_en": "In the Cartesian coordinate system xoy, the inverse proportional function is located in the first quadrant, point A is the point on the inverse proportional function, passing through A is AB, perpendicular to the X-axis is point B, and C is the midpoint of OB, connecting OA and AC", "ocr_zh": "直角坐标系xoy中，反比例函数位于第一象限，点A为反比例函数上一点，过A作AB垂直X轴垂足为点B，C是OB的中点，连接OA、AC", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1560.jpg" }, { "index": 1561, "ocr_en": "Cartesian coordinate system xoy, the edge OB of the quadrilateral AOBD coincides with the positive half axis of the x-axis, points A and D are located in the first quadrant and AD is parallel to the x-axis, DB is perpendicular to the x-axis, connecting OD and AB, OD and AB intersect at point M, and the inverse proportional function passes through point M exactly", "ocr_zh": "直角坐标系xoy，四边形AOBD的边OB与x轴正半轴重合，点A、D位于第一象限且AD平行x轴，DB垂直x轴，连接OD、AB，OD、AB交于点M，反比例函数恰好经过点M", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1561.jpg" }, { "index": 1562, "ocr_en": "Cartesian coordinate system xoy, point B is located in the second quadrant, passing through B to form a perpendicular line on the x-axis with a foot of A, connecting OB, and the inverse proportional function is located in the second quadrant intersecting OB at point C", "ocr_zh": "直角坐标系xoy，点B位于第二象限，过B作x轴垂线垂足为A，连接OB，反比例函数位于第二象限与OB交于点C", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1562.jpg" }, { "index": 1563, "ocr_en": "Cartesian coordinate system xoy, vertex D of diamond ABCD is located in the second quadrant, and the remaining vertices are in the first quadrant. AB is parallel to the x-axis, AO is perpendicular to AD, passing through A makes AE perpendicular to CD, and the vertical foot is E, connecting OE. The inverse proportional function is located in the first quadrant and passes through point E, intersects AB at point F, and connects OF", "ocr_zh": "直角坐标系xoy，菱形ABCD的顶点D位于第二象限，其余顶点在第一象限，AB平行x轴，AO垂直AD，过A作AE垂直CD垂足为E，连接OE，反比例函数位于第一象限且过点E，与AB交于点F，连接OF", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1563.jpg" }, { "index": 1564, "ocr_en": "Cartesian coordinate system xoy, vertices A and B of parallelogram OABC are in the first quadrant, vertex C is on the positive half axis of the y-axis, the inverse proportional function passes through point A and intersects BC with point D, CD is greater than BD", "ocr_zh": "直角坐标系xoy，平行四边形OABC的顶点A、B在第一象限，顶点C在y轴的正半轴上，反比例函数经过点A且交BC与点D，CD大于BD", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1564.jpg" }, { "index": 1565, "ocr_en": "In the Cartesian coordinate system xoy, vertex B of triangle AOB is located on the positive x-axis, point A is located in the first quadrant and on the hyperbola, OA=AB， Points C and D are the midpoints of edges AB and OB, respectively, connecting CD. Point E is a point on CD, connecting AE and OE", "ocr_zh": "直角坐标系xoy，三角形AOB的顶点B位于x轴正半轴上，点A位于第一象限且在双曲线上，OA=AB，点C、D分别为边AB、OB的中点，连接CD，点E为CD上一点，连接AE、OE", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1565.jpg" }, { "index": 1566, "ocr_en": "In the Cartesian coordinate system xoy, the vertices A, B, and C of the isosceles triangle ABC are located in the first, third, and fourth quadrants, respectively. AB passes through the origin O, BC is parallel to the x-axis, and the hyperbola is located in the third quadrant and passes through points A and B. Passing through C forms a CD parallel to the y-axis and intersects with the hyperbola in the first quadrant at point D, connecting BD", "ocr_zh": "直角坐标系xoy，等腰三角形ABC的顶点A、B、C分别位于第一、三、四象限，AB过原点O，BC平行x轴，双曲线位于一三象限且分别过A、B两点，过C作CD平行y轴交第一象限的双曲线于点D，连接BD", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1566.jpg" }, { "index": 1567, "ocr_en": "In the coordinate system xoy, a point A is taken on the hyperbola located in the second and fourth quadrants. A perpendicular AD is drawn from A to the x-axis, with the foot of the perpendicular being D. A point M is taken on the positive half-axis of the x-axis, and a point N is taken on the negative half-axis of the y-axis. Line MN is drawn, and a point P is taken on MN. A perpendicular PE is drawn from P to the x-axis, with the foot of the perpendicular being E. The line MN intersects the inverse proportion function in the fourth quadrant at points B and C.", "ocr_zh": "坐标轴xoy中，在位于第二象限的双曲线上取点A，作AD垂直于x轴，垂足为D，在x轴正半轴上取点M，y轴负半轴取点N，连接MN，在MN上取点P，作PE垂直于x轴，垂足为E。MN交第四象限的反比函数于点B、C", "class": "Analytic Geometry", "difficulty": 3, "image_path": "fig_1567.jpg" }, { "index": 1568, "ocr_en": "In the coordinate system xoy, points D and E are located on the positive half-axis of the x-axis and the negative half-axis of the y-axis, respectively. The dashed line MN is in the second quadrant. The triangle CBA is the axis-symmetric figure of triangle ODE with MN as axis the of symmetry, and point B is located on the hyperbola in the second quadrant.", "ocr_zh": "坐标轴xoy中，点D、点E分别位于x轴正半轴和y轴负半轴，虚线MN在第二象限，以MN为对称轴作三角形ODE的轴对称图形三角形CBA，且点B位于第二象限的双曲线上", "class": "Analytic Geometry", "difficulty": 3, "image_path": "fig_1568.jpg" }, { "index": 1569, "ocr_en": "In the coordinate system xoy, four points P1, P2, P3, and P4 are taken on the hyperbola located in the first quadrant. Perpendicular lines P1A1, P2A2, P3A3, and P4A4 are drawn from these points to the x-axis, with the feet of the perpendiculars being A1, A2, A3, and A4, respectively. Then, perpendicular lines are drawn from P1, P2, P3, and P4 to the y-axis, P1A1, P2A2, and P3A3, respectively.", "ocr_zh": "坐标轴xoy中，在位于第一象限的双曲线上取四个点P1、P2、P3、P4，过这四个点分别作P1A1、P2A2、P3A3、P4A4垂直于x轴，垂足分别为A1、A2、A3、A4，再过P1、P2、P3、P4分别作y轴、P1A1、P2A2、P3A3的垂线", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1569.jpg" }, { "index": 1570, "ocr_en": "In the coordinate system xoy, a line passing through the origin intersects the hyperbola in the first and third quadrants at points A and B, respectively, with point A in the first quadrant and point B in the third quadrant. Point C is another point on the hyperbola located in the first quadrant. Line BC is extended to intersect the x-axis at point D. Point G is a point on the y-axis, and lines BG and CG are drawn.", "ocr_zh": "坐标轴xoy中，过原点直线和一三象限的双曲线分别相交于A、B两点，其中A点在第一象限，B点在第三象限。点C是双曲线上一点且在第一象限。连接BC并延长，交x轴于点D，G为y轴上一点，连接BG，CG", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1570.jpg" }, { "index": 1571, "ocr_en": "In the coordinate system xoy, a line passing through the origin intersects the hyperbola in the first and third quadrants at points A and B, respectively, with point A in the first quadrant and point B in the third quadrant. Point C is another point on the hyperbola located in the first quadrant.", "ocr_zh": "坐标轴xoy中，过原点直线和一三象限的双曲线分别相交于A、B两点，其中A点在第一象限，B点在第三象限。点C是双曲线上一点且在第一象限。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1571.jpg" }, { "index": 1572, "ocr_en": "In the coordinate plane with axes x and y (xoy), a straight line intersects a hyperbola located in the second and fourth quadrants at points A and B, respectively. Point A lies in the second quadrant, and point B lies in the fourth quadrant. Line AB intersects the x-axis at point C and the y-axis at point D. Connect point A to the origin O.", "ocr_zh": "坐标轴xoy中，一条直线与二四象限的双曲线分别相交于A、B两点。A在第二象限，B在第四象限。直线AB与x轴相交于点C，与y轴相交于点D，连接AO。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1572.jpg" }, { "index": 1573, "ocr_en": "In the coordinate plane with axes x and y (xoy), take a point A on the branch of a hyperbola located in the first quadrant. Then take a point C on the hyperbola to the right of point A. From point A, draw a line perpendicular to the y-axis. From point C, draw a line perpendicular to the x-axis; let the foot of the perpendicular be point H. Line segment CH intersects the line perpendicular to the y-axis through point A at point D. Connect points O and D.", "ocr_zh": "坐标轴xoy中，在位于第一象限的双曲线上取点A，在双曲线上点A的右侧取点C，过点A作y轴的垂线，过x轴作C点的垂线，垂足为点H，CH交过点A的y轴的垂线于点D，连接OD。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1573.jpg" }, { "index": 1574, "ocr_en": "In the coordinate plane with axes x and y (xoy), a straight line intersects the inverse proportional function located in the first and third quadrants at points A and B, where point A lies in the first quadrant and point B lies in the third quadrant.", "ocr_zh": "坐标轴xoy中，直线交位于一三象限的反比例函数于点A、B，其中点A位于第一象限，点B位于第三象限。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1574.jpg" }, { "index": 1575, "ocr_en": "In the coordinate plane with axes x and y (xoy), a straight line passing through the first, second, and third quadrants intersects the x-axis at point A and the y-axis at point B. Line MN intersects the inverse proportional function in the second quadrant at points C and D. A circle is drawn with diameter AB and center at point P. Draw a line MN parallel to line AB and tangent to the circle centered at P; this line intersects the x-axis and y-axis at points M and N, respectively.", "ocr_zh": "坐标轴xoy中，过一二三象限的直线交x轴于点A，交y轴于点B。直线MN交第二象限反比例函数于点C、D。以AB为直径作圆，圆心为点P。作直线MN平行于直线AB且与圆p相切，分别交x轴、y轴于点M、N。", "class": "Analytic Geometry", "difficulty": 4, "image_path": "fig_1575.jpg" }, { "index": 1576, "ocr_en": "In the coordinate plane with axes x and y (xoy), take a point A on the branch of a hyperbola located in the first quadrant. Using point A as the vertex, construct a square ABCD such that AB is perpendicular to the y-axis at point E, and AD is perpendicular to the x-axis at point F. Connect points B to F and C to E; they intersect at point G.", "ocr_zh": "坐标轴xoy中，在位于第一象限的双曲线上取点A，以点A为顶点作正方形ABCD，AB垂直于y轴于点E，AD垂直于x轴于点F。连接BF，CE，相交于点G。", "class": "Analytic Geometry", "difficulty": 3, "image_path": "fig_1576.jpg" }, { "index": 1577, "ocr_en": "In the coordinate plane with axes x and y (xoy), take point A on the positive x-axis and point C on the positive y-axis. Connect points A and C, and construct rectangle OABC with vertices at O, A, and C. Line segment AB intersects the inverse proportional function in the first quadrant at point E, and line segment BC intersects the inverse proportional function in the first quadrant at point D.", "ocr_zh": "坐标轴xoy中，在x轴正半轴上取点A，y轴正半轴上取点C，连接AC，过顶点O、A、C作矩形OABC。AB交第一象限的反比例函数于点E，BC交第一象限的反比例函数于点D。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1577.jpg" }, { "index": 1578, "ocr_en": "In the coordinate plane with axes x and y (xoy), a straight line passing through the origin intersects the hyperbola in the second and fourth quadrants at points A and B, where point A lies in the second quadrant and point B lies in the fourth quadrant. Take a point C on the positive y-axis, and connect points A to C and B to C.", "ocr_zh": "坐标轴xoy中，过原点的直线与二四象限的双曲线交于点A、B，其中点A在第二象限，点B在第四象限。在y轴正半轴取点C，连接AC，BC。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1578.jpg" }, { "index": 1579, "ocr_en": "In the coordinate plane with axes x and y (xoy), a straight line passing through the first, second, and fourth quadrants intersects the x-axis at point A and the y-axis at point B. Line segment AB intersects the inverse proportional function in the second quadrant at point C.", "ocr_zh": "坐标轴xoy中，过一二四象限的直线交x轴、y轴分别于点A、点B。直线AB交第二象限的反比例函数于点C。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1579.jpg" }, { "index": 1580, "ocr_en": "In the coordinate plane with axes x and y (xoy), let point B be a point in the first quadrant. Draw AB perpendicular to the y-axis with foot A, and draw BC perpendicular to the x-axis with foot C. Connect points O and B, and take a point M on line segment OB. Draw a hyperbola in the first quadrant passing through point M, which intersects AB at point D and BC at point E. Connect points D and E, and extend the line to intersect the x-axis at point F. Take a point G on the positive x-axis, and connect points B and G.", "ocr_zh": "坐标轴xoy中，点B为第一象限中一点，作AB垂直于y轴，垂足为A，作BC垂直于x轴，垂足为C。连接OB，在OB上取点M，过点M作第一象限双曲线，交AB于点D，交BC于点E。连接DE并延长，交x轴于点F。在x轴正半轴上取点G，连接BG。", "class": "Analytic Geometry", "difficulty": 4, "image_path": "fig_1580.jpg" }, { "index": 1581, "ocr_en": "In the coordinate plane with axes x and y (xoy), let M be a point on the inverse proportional function in the first quadrant. Using point M as the center, draw a circle that intersects the positive x-axis at points A and B, and intersects the positive y-axis at point C. Connect points M to A, M to B, and M to C.", "ocr_zh": "坐标轴xoy中，M为第一象限的反比例函数上的一点，以点M为圆心作圆M，交x轴正半轴于点A、B，交y轴正半轴为点C，连接MA、MB、MC。", "class": "Analytic Geometry", "difficulty": 3, "image_path": "fig_1581.jpg" }, { "index": 1582, "ocr_en": "In the coordinate plane with axes x and y (xoy), take point A on the positive y-axis and point C on the positive x-axis. Construct rectangle OABC with vertices at O, A, and C. Line segment AB intersects the inverse proportional function in the first quadrant at point D, and line segment BC intersects the inverse proportional function in the first quadrant at point E. Connect points D and E.", "ocr_zh": "坐标轴xoy中，在y轴正半轴上取点A，x轴正半轴上取点C，过顶点O、A、C作矩形OABC。AB交第一象限的反比例函数于点D，BC交第一象限的反比例函数于点E。连接DE。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1582.jpg" }, { "index": 1583, "ocr_en": "In the coordinate plane with axes x and y (xoy), take point A on the positive y-axis and point B on the positive x-axis. Connect points A and B. On the branch of a hyperbola in the first quadrant, take point P. Connect points P and A, and extend the line to intersect the negative x-axis at point C. Connect points P and B, and extend the line to intersect the negative y-axis at point D. Connect points C and D.", "ocr_zh": "坐标轴xoy中，在y轴正半轴上取点A，x轴正半轴上取点B，连接AB。在第一象限的双曲线上取点P，连接PA并延长，交x轴负半轴于点C，连接PB并延长，交y轴负半轴于点D，连接CD。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1583.jpg" }, { "index": 1584, "ocr_en": "In the coordinate plane with axes x and y (xoy), take point A on the branch of a hyperbola located in the first quadrant. Connect points A and O (the origin), and extend the line to intersect the branch of the hyperbola in the third quadrant at point B. From point A, draw AE perpendicular to the y-axis, with foot E. From point E, draw EP parallel to line AB, where point P lies on the hyperbola in the third quadrant. Line segment PE intersects the negative x-axis at point F.", "ocr_zh": "坐标轴xoy中，在第一象限的双曲线上取点A，连接AO并延长，交第三象限的双曲线于点B。过点A作AE垂直于y轴，垂足为E。过点E作EP平行于AB，P在第三象限的双曲线上。PE交x轴负半轴于点F。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1584.jpg" }, { "index": 1585, "ocr_en": "In the coordinate plane with axes x and y (xoy), take point A on the branch of a hyperbola located in the first quadrant. Connect points A and O (the origin), and extend the line to intersect the branch of the hyperbola in the third quadrant at point B. From point A, draw AE perpendicular to the y-axis, with foot E. From point B, draw BF perpendicular to the x-axis, with foot F. Connect points E to F and B to E.", "ocr_zh": "坐标轴xoy中，在第一象限的双曲线上取点A，连接AO并延长，交第三象限的双曲线于点B。过点A作AE垂直于y轴，垂足为E。过点B作BF垂直于x轴，垂足为F。连接EF，BE。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1585.jpg" }, { "index": 1586, "ocr_en": "ABCD is a parallelogram. Point E lies on segment AB. Connect points E and D, and extend the line to point F. Using CE and EF as adjacent sides, construct parallelogram CEFG. Connect points E and G; their line intersects segment CD at point O.", "ocr_zh": "ABCD为平行四边形，点E为AB上一点，连接ED并延长至点F。以CE、EF为邻边作平行四边形CEFG，连接EG交CD于点O。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1586.jpg" }, { "index": 1587, "ocr_en": "ABCD is a rectangle. Point E lies on side AB. Connect points D and E, and extend the line to point F. Connect points C and F, intersecting AB at point G. Connect points B and F.", "ocr_zh": "ABCD为矩形，点E为边AB上一点，连接DE并延长至点F，连接CF交AB于点G，连接BF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1587.jpg" }, { "index": 1588, "ocr_en": "ABCD is a parallelogram. Point E lies on side BC. Connect points A and E, and extend the line to point F. Connect points C and F. Point G is on segment AE. Connect points B and G.", "ocr_zh": "ABCD为平行四边形，点E为BC上一点，连接AE并延长至点F，连接CF。点G是AE上一点，连接BG。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1588.jpg" }, { "index": 1589, "ocr_en": "ABCD is a square. Point E lies on side BC. Connect points A and E, and connect points B and D; they intersect at point P. Point F lies on side CD. Connect points A and F. Line segment AF intersects BD at point M, and intersects DE at point N. Connect points P and N.", "ocr_zh": "ABCD为正方形，点E为BC上一点，连接AE、BD并相交于点P。F为CD上一点，连接AF。AF交BD于点M，AF交DE于点N，连接PN。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1589.jpg" }, { "index": 1590, "ocr_en": "In triangle ABC, there is a rectangle DEFG such that points D and E lie on side AB, point G lies on side AC, and point F lies on side BC.", "ocr_zh": "三角形ABC中有一矩形DEFG，其中点D、E在AB上，点G在AC上，点F在BC上。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1590.jpg" }, { "index": 1591, "ocr_en": "ABCD is a quadrilateral. Connect points A and C, and points B and D; the two lines intersect at point O.", "ocr_zh": "ABCD为四边形，连接AC、BD相交于点O。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1591.jpg" }, { "index": 1592, "ocr_en": "In triangle ABC, point D lies on side BC. Connect points A and D.", "ocr_zh": "三角形ABC中，点D为BC上一点，连接AD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1592.jpg" }, { "index": 1593, "ocr_en": "In triangle ABC, points M and N lie on sides AC and AB, respectively. Connect points B and M, and points C and N; they intersect at point O. Connect points M and N.", "ocr_zh": "三角形ABC中，M、N分别为AC、AB上的点，连接BM、CN相交于点O。连接MN。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1593.jpg" }, { "index": 1594, "ocr_en": "Using segment MN as the diameter and O as the origin, construct a semicircle. Point P lies on the semicircle. Draw PQ perpendicular to MN, with foot Q. Connect points M to P and N to P. Point E lies on segment MP. Connect points E and N; their line intersects PQ at point F.", "ocr_zh": "以MN为直径，O为原点作半圆O，点P为半圆上一点，作PQ垂直于MN，垂足为点Q。连接MP、NP。点E为MP上一点，连接EN交PQ于点F。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1594.jpg" }, { "index": 1595, "ocr_en": "In triangle ABC, points D and E lie on sides BC and AC, respectively. Connect points B and E, and points A and D; they intersect at point F.", "ocr_zh": "在三角形ABC中，点D、点E分别为BC、AC上一点，连接BE、AD相交于点F。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1595.jpg" }, { "index": 1596, "ocr_en": "In triangle ABC, point D lies on side AB and point E lies on side BC. Connect points C and D, and points A and E; they intersect at point F.", "ocr_zh": "在三角形ABC中。点D为AB上一点，点E为BC上一点，连接CD、AE交于点F。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1596.jpg" }, { "index": 1597, "ocr_en": "Line DE is parallel to BC and is longer than BC. Connect points B and D, and points C and E; they intersect at point A.", "ocr_zh": "直线DE与BC平行，且DE比BC长，连接BD、CE相交于点A.", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1597.jpg" }, { "index": 1598, "ocr_en": "In equilateral triangle ABC, points A₁, B₁, and C₁ lie on sides BC, AC, and AB, respectively, and triangle A₁B₁C₁ is also equilateral. Points A₂, B₂, and C₂ lie on segments B₁C₁, A₁C₁, and A₁B₁, respectively, and triangle A₂B₂C₂ is equilateral.", "ocr_zh": "在等边三角形ABC中，A1、B1、C1分别为BC、AC、AB上的点，且三角形A1B1C1为等边三角形。A2、B2、C2分别为B1C1、A1C1、A1B1上的点，且三角形A1B1C1为等边三角形。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1598.jpg" }, { "index": 1599, "ocr_en": "In triangle ABC, there is a square DEGF such that points D and E lie on side BC, point G lies on side AB, and point F lies on side AC.", "ocr_zh": "在三角形ABC中有一正方形DEGF，其中点D、E在BC上，点G在AB上，点F在AC上。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1599.jpg" }, { "index": 1600, "ocr_en": "In the coordinate plane with axes x and y (xoy), a line AB passing through the first, second, and fourth quadrants intersects the positive x-axis at point A and the positive y-axis at point B. Point C lies on the positive x-axis, and point P lies on the positive y-axis. Connect points C and P, and points A and P.", "ocr_zh": "坐标轴xoy中，过一二四象限的直线AB交x轴正半轴于点A，交y轴正半轴于点B。点C为x轴正半轴上一点，点P为y轴正半轴上一点，连接CP、AP。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1600.jpg" }, { "index": 1601, "ocr_en": "In triangle ACD, point B lies on side CD. Connect points A and B. Point P lies on side AD, and point Q lies on segment AB. Connect points P and Q.", "ocr_zh": "在三角形ACD中，点B为CD上一点，连接AB。点P为AD上一点，点Q为AB上一点，连接PQ。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1601.jpg" }, { "index": 1602, "ocr_en": "Using side AB of triangle ABC as the diameter, construct circle O with midpoint O of AB as the center. Point C lies on circle O. Point D is another point on circle O. Connect points B and D, and points C and D. Line segment CD intersects AB at point E.", "ocr_zh": "以三角形ABC的边AB为直径，AB中点O为圆心作圆O，点C在圆O上。点D为圆O上一点，连接BD、CD。CD交AB于点E。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1602.jpg" }, { "index": 1603, "ocr_en": "In square ABCD, points E and F are given such that triangles ABE and CDF are right triangles. From point D, draw DG perpendicular to DF, with foot at D, and extend DG to intersect the extension of line EF at point G. Connect points C and G; line CG intersects the extension of BE at point H.", "ocr_zh": "在正方形ABCD中有点E、F，三角形ABE和三角形CDF为直角三角形。过点D作DG垂直于DF，垂足为点D，且交EF延长线于点G。连接CG，交BE延长线于点H。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_1603.jpg" }, { "index": 1604, "ocr_en": "In triangle ABC, draw BD perpendicular to AB with foot at B, where point D lies on side AC.", "ocr_zh": "在三角形ABC中，作BD垂直于AB，垂足为B，点D在AC上。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1604.jpg" }, { "index": 1605, "ocr_en": "In square ABCD, points E and F lie on diagonal AC. Connect points D and E, and extend the line to intersect AB at point M. Connect points D and F, and extend the line to intersect BC at point N. Connect points M and N.", "ocr_zh": "在正方形ABCD中，点E、F是对角线AC上两点，连接DE并延长，交AB于点M，连接DF并延长，交BC于点N，连接MN。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1605.jpg" }, { "index": 1606, "ocr_en": "In square ABCD, point E lies on side CD. Connect points A and E, and extend the line to intersect the extension of BC at point F. Point G lies on side BC. Connect points D and G; line DG intersects AF at point H. Points M and N lie on segments BD and AF, respectively. Connect points M and N, and connect points C and M.", "ocr_zh": "在正方形ABCD中，点E为CD上一点，连接AE并延长，交BC的延长线于点F。点G为BC上一点，连接DG交AF于点H。点M、N分别为BD、AF上两点，连接MN，CM。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1606.jpg" }, { "index": 1607, "ocr_en": "In triangle ABC, point D lies on side AB. Connect points C and D. Point E lies on segment CD. Connect points B and E.", "ocr_zh": "在三角形ABC中，点D为AB上一点，连接CD，点E为CD上一点，连接BE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1607.jpg" }, { "index": 1608, "ocr_en": "In square ABCD, point P lies on side AB. The diagonals AC and BD intersect at point O. Connect points O and P. Points M and N lie on sides AD and BC, respectively. Connect points M and P; line MP intersects AO at point E. Connect points N and P; line NP intersects OB at point F.", "ocr_zh": "在正方形ABCD中，点P在AB上，对角线AC与BD相交于点O，连接OP。点M、N分别为AD、BC上的点，连接MP交AO于点E，连接NP交OB于点F。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1608.jpg" }, { "index": 1609, "ocr_en": "In triangles ABC and CDE, connect points A and D, and points B and E.", "ocr_zh": "在三角形ABC和三角形CDE中，连接AD和BE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1609.jpg" }, { "index": 1610, "ocr_en": "In the coordinate plane with axes x and y (xoy), point B lies on the graph of an inverse proportional function in the first quadrant. Through point A, draw line AB parallel to the x-axis, intersecting the positive y-axis at point D. Point C lies on the negative x-axis. Connect points B and C; line BC intersects the y-axis at point E.", "ocr_zh": "坐标轴xoy中，点B在第一象限的反比例函数图像上，过点A作AB平行于x轴，交y轴正半轴于点D。点C为x轴负半轴上一点，连接BC交y轴于点E。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1610.jpg" }, { "index": 1611, "ocr_en": "In rectangle ABCD, point E lies on side BC. Connect points A and E, and points D and E. Point F lies on segment AE. Connect points D and F. Extend line BF; it intersects DE at point O and CD at point G.", "ocr_zh": "矩形ABCD中，点E在BC上，连接AE、DE。点F在AE上，连接DF。连接BF并延长交DE于点O，交CD于点G。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1611.jpg" }, { "index": 1612, "ocr_en": "In triangle ABC, construct square ABEF on side AB.", "ocr_zh": "在三角形ABC中，以AB为边作正方形ABEF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1612.jpg" }, { "index": 1613, "ocr_en": "In triangle ABC, point O lies on side AB. Connect points O and D; line OD intersects AC at point G. Connect points B and D; line BD intersects AC at point E. Connect points O and C; line OC intersects BD at point F.", "ocr_zh": "在三角形ABC中，点O为AB上一点，连接OD交AC于点G，连接BD交AC于点E，连接OC交BD于点F。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1613.jpg" }, { "index": 1614, "ocr_en": "In triangle ABC, points D and E lie on sides AB and BC, respectively. Connect points D and E. Take a point F on segment DE. Connect points A and F, and extend the line to intersect BC at point G.", "ocr_zh": "在三角形ABC中，点D、点E分别在AB、BC上，连接DE，在DE上取一点F，连接AF并延长交BC于点G。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1614.jpg" }, { "index": 1615, "ocr_en": "In triangles ABC and AB'C', connect points B and B', and points C and C'.", "ocr_zh": "三角形ABC和三角形AB'C'，连接BB'、CC'。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1615.jpg" }, { "index": 1616, "ocr_en": "In triangle ABC, draw BD perpendicular to BC with foot at B. Connect points C and D; line CD intersects AB at point M. From point M, draw MN perpendicular to BC with foot at N.", "ocr_zh": "在三角形ABC中，作BD垂直于BC，垂足为B，连接CD交AB于点M，过点M作MN垂直于BC，垂足为N。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1616.jpg" }, { "index": 1617, "ocr_en": "In triangle ABC, draw AD perpendicular to BC with foot at D. Squares EFGH and GHMN lie within triangle ABC, where points F, G, and N lie on side BC, point E lies on side AB, and point M lies on side AC.", "ocr_zh": "在三角形ABC中，作AD垂直于BC，垂足为D。正方形EFGH和正方形GHMN在三角形ABC中，且点F、G、N在BC上，点E在AB上，点M在AC上。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1617.jpg" }, { "index": 1618, "ocr_en": "In rhombus ABCD, from point D, draw DE perpendicular to the extension of BC, with foot E. Connect points A and E; line AE intersects CD at point G. Connect points A and E; line AE intersects BD at point F. Point H lies on side CD. Connect points F and H, and points F and C.", "ocr_zh": "在菱形ABCD中，过点D作DE垂直于BC的延长线，垂足为E。连接AE交CD于点G。连接AE交BD于点F。H为CD上一点，连接FH、FC。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1618.jpg" }, { "index": 1619, "ocr_en": "In square ABCD, point E lies on side CD. Connect points B and E, and construct rectangle BFEG using BE as the diagonal. Line EF intersects BD at point H. Connect points A and F.", "ocr_zh": "在正方形ABCD中，点E是CD上一点，连接BE并以BE为对角线作矩形BFEG。EF交BD于点H，连接AF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1619.jpg" }, { "index": 1620, "ocr_en": "In quadrilateral ABCD, point E lies on side BC. Connect points A and E such that AE is parallel to CD. Connect points D and E such that DE is parallel to AB. Draw CF parallel to AD, intersecting AE at point F. Connect points B and F; extend BF to intersect AD at point M.", "ocr_zh": "在四边形ABCD中，点E为BC上一点，连接AE平行于CD，连接DE平行于AB，作CF平行于AD交AE于点F，连接BF并延长交AD于点M。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1620.jpg" }, { "index": 1621, "ocr_en": "In rectangle ABCD, extend segment BA to point E. Connect points C and E; line CE intersects BD at point G and intersects AD at point F.", "ocr_zh": "在矩形ABCD中，延长BA至点E，连接CE交BD于点G，CE交AD于点F。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1621.jpg" }, { "index": 1622, "ocr_en": "In rectangle ABCD, extend segment BA to point E. Connect points C and E; line CE intersects BD at point G and intersects AD at point F. Connect points A and G.", "ocr_zh": "在矩形ABCD中，延长BA至点E，连接CE交BD于点G，CE交AD于点F。连接AG。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1622.jpg" }, { "index": 1623, "ocr_en": "In quadrilateral ABCD, point E lies on side BC. Connect points A and E such that AE is parallel to CD. Connect points D and E such that DE is parallel to AB. Draw CF parallel to AD, intersecting AE at point F.", "ocr_zh": "在四边形ABCD中，点E为BC上一点，连接AE平行于CD，连接DE平行于AB，作CF平行于AD交AE于点F.", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1623.jpg" }, { "index": 1624, "ocr_en": "In quadrilateral ABCD, take point P on side BC and connect points A and P. Take point E on side CD and connect points P and E.", "ocr_zh": "在四边形ABCD中，在BC上取点P，连接AP，在CD上取点E，连接PE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1624.jpg" }, { "index": 1625, "ocr_en": "In triangle ABC, extend CB to point N. Take point M on side AB, and connect points C and M. Extend CM to intersect AN at point P. Connect points B and P.", "ocr_zh": "在三角形ABC中，延长CB至点N，在AB上取点M，连接CM并延长，交AN于点P，连接BP。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1625.jpg" }, { "index": 1626, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function is symmetric about the line x=1, and its vertex lies in the first quadrant.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数关于直线x=1轴对称，且顶点在第一象限。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1626.jpg" }, { "index": 1627, "ocr_en": "In the coordinate plane xoy, there is a downward-opening quadratic function that is symmetric about the line x=−1, and its vertex lies in the second quadrant. The parabola intersects the positive y-axis at the point(0,2).", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数关于直线x=-1轴对称，且顶点在第二象限，二次函数与y轴正半轴交于点（0，2）", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1627.jpg" }, { "index": 1628, "ocr_en": "In the coordinate plane with axes x and y (xoy), an upward-opening quadratic function is symmetric about the line x=1, and its vertex lies in the fourth quadrant.", "ocr_zh": "坐标轴xoy中，开口朝上的二次函数关于直线x=1轴对称，且顶点在第四象限。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1628.jpg" }, { "index": 1629, "ocr_en": "In the coordinate plane with axes x and y (xoy), an upward-opening quadratic function is symmetric about the line x=1, and its vertex lies in the fourth quadrant. The quadratic function intersects the positive x-axis at the point (3,0).", "ocr_zh": "坐标轴xoy中，开口朝上的二次函数关于直线x=1轴对称，且顶点在第四象限，二次函数与x轴正半轴交于点（3，0）", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1629.jpg" }, { "index": 1630, "ocr_en": "In the coordinate plane with axes x and y (xoy), an upward-opening quadratic function has its vertex in the fourth quadrant. The function intersects the negative x-axis at the point \n(−1,0), and the x-coordinate of its intersection with the positive x-axis lies between 1 and 2.", "ocr_zh": "坐标轴xoy中，开口朝上的二次函数顶点在第四象限。二次函数与x轴负半轴交于点（-1，0）与x轴正半轴的交点横坐标在1到2之间。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1630.jpg" }, { "index": 1631, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function is symmetric about the linex=−2, and its vertex lies in the second quadrant. The x-coordinate of the function’s intersection with the negative x-axis lies between −4 and −3, and the x-coordinate of its intersection with the positive x-axis lies between−1 and 0.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数关于直线x=-2轴对称，且顶点在第二象限.二次函数与x轴负半轴交点横坐标在-4到-3之间，与x轴正半轴的交点横坐标在-1到0之间。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1631.jpg" }, { "index": 1632, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B, and intersects the positive y-axis at point C. Point D lies above the x-axis on the negative x-axis side of the function, and point E lies on the negative y-axis. Connect points A and D, A and E, and D and E.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，与y轴正半轴交于点C。点D为二次函数在x轴负半轴上方一点，点E为y轴负半轴一点，连接AD、AE、DE。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1632.jpg" }, { "index": 1633, "ocr_en": "In the coordinate plane with axes x and y (xoy), an upward-opening quadratic function intersects the x-axis at points A and B, and intersects the negative y-axis at point C. Point P lies on the graph of the quadratic function in the fourth quadrant. From point P, draw PH perpendicular to the x-axis, with foot H. Line PH intersects segment BC at point M.", "ocr_zh": "坐标轴xoy中，开口朝上的二次函数与x轴交于A、B两点，与y轴负半轴交于点C。点P为第四象限内二次函数图像上一点，作PH垂直于x轴，垂足为H，PH与BC交于点M。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1633.jpg" }, { "index": 1634, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function has its vertex in the first quadrant. A linear function passing through the first, second, and fourth quadrants intersects the quadratic function at points A and B, intersects the positive y-axis at point D, and intersects the positive x-axis at point C.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数顶点在第一象限。过一二四象限的一次函数交二次函数图像于A、B两点，交y轴正半轴于点D，交x轴正半轴于点C。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1634.jpg" }, { "index": 1635, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B, and intersects the positive y-axis at point C. Connect points A and C, and points B and C. Point P lies on the graph of the quadratic function in the fourth quadrant. Connect points P and A, and extend the line. Point Q lies on the extension of line PA.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，与y轴正半轴交于点C，连接AC、BC。点P为第四象限内二次函数图像上一点，连接PA并延长，点Q在PA延长线上。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1635.jpg" }, { "index": 1636, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B, and intersects the positive y-axis at point C. Connect points A and C, and points B and C. Point P lies on the graph of the quadratic function in the first quadrant. Connect points A and P; line AP intersects BC at point F. From point P, draw PQ perpendicular to the x-axis, with foot Q. Line PQ intersects BC at point H.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，与y轴正半轴交于点C，连接AC、BC。点P为第一象限内二次函数图像上一点，连接AP交BC于点F。作PQ垂直于x轴，垂足为Q，PQ交BC于点H。", "class": "Analytic Geometry", "difficulty": 3, "image_path": "fig_1636.jpg" }, { "index": 1637, "ocr_en": "In the coordinate plane with axes x and y (xoy), an upward-opening quadratic function intersects the x-axis at points A and B, and intersects the negative y-axis at point C. Draw DE perpendicular to BC, with foot E on BC and point D on the x-axis.", "ocr_zh": "坐标轴xoy中，开口朝上的二次函数与x轴交于A、B两点，与y轴负半轴交于点C。作DE垂直于BC，垂足E在BC上，点D在x轴上。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1637.jpg" }, { "index": 1638, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B, and intersects the positive y-axis at point C. Connect points B and C. Point P lies on the graph of the quadratic function in the first quadrant. Draw PM perpendicular to the x-axis, with foot M. Line PM intersects BC at point Q. Draw PN perpendicular to BC, with foot N.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，与y轴正半轴交于点C，连接BC。点P为第一象限内二次函数图像上一点，作PM垂直于x轴，垂足为M，PM交BC于点Q。作PN垂直于BC，垂足为N。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1638.jpg" }, { "index": 1639, "ocr_en": "In the coordinate plane with axes x and y (xoy), an upward-opening quadratic function intersects the x-axis at points A and B, and intersects the negative y-axis at point C. The axis of symmetry of the quadratic function lies to the right of the y-axis, and its vertex is in the fourth quadrant.", "ocr_zh": "坐标轴xoy中，开口朝上的二次函数与x轴交于A、B两点，与y轴负半轴交于点C。二次函数的对称轴在y轴右侧，顶点在第四象限。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1639.jpg" }, { "index": 1640, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B, and intersects the positive y-axis at point C. The axis of symmetry of the quadratic function lies to the right of the y-axis, and its vertex is point D. Connect points A and D; line AD intersects the positive y-axis at point E.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，与y轴正半轴交于点C.二次函数的对称轴在y轴右侧，顶点为点D。连接AD交y轴正半轴于点E", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1640.jpg" }, { "index": 1641, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B, and intersects the positive y-axis at point C. Connect points A and C, and points B and C. Point M lies on the x-axis to the right of point A. Draw MN perpendicular to the x-axis, with foot M. Line MN intersects BC at point N, and MN intersects the quadratic function’s graph at point D.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，与y轴正半轴交于点C，连接AC、BC。点M在A点右侧的x轴上，作MN垂直于x轴，垂足为M，交BC于点N，MN交二次函数图像于点D。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1641.jpg" }, { "index": 1642, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B, and intersects the positive y-axis at point C. The axis of symmetry of the quadratic function is x=−1, and its vertex is point D. Connect points B and C, and extend the line.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，与y轴正半轴交于点C.二次函数的对称轴为x=-1，顶点为点D。连接BC并延长。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1642.jpg" }, { "index": 1643, "ocr_en": "Left figure:\nIn the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and C, and intersects the positive y-axis at point B. Connect points A and B, and points B and C. Point P lies on the graph of the quadratic function in the first quadrant. Draw PD perpendicular to the x-axis, with foot D. Line PD intersects AB at point E. Draw PF perpendicular to PE, with foot P. Using FP and PE as adjacent sides, construct rectangle PEGF.\n\nRight figure:\nIn the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and C, and intersects the positive y-axis at point B. Connect points A and B, and points B and C. Point P is the vertex of the quadratic function. Draw PD perpendicular to the x-axis, with foot D. Line PD intersects AB at point E.", "ocr_zh": "左图：坐标轴xoy中，开口朝下的二次函数与x轴交于A、C两点，与y轴正半轴交于点B，连接AB、BC。点P为第一象限内二次函数图像上的一点，作PD垂直于x轴，垂足为D，PD交AB于点E。作PF垂直于PE，垂足为P，以FP、PE为邻边作矩形PEGF。右图：坐标轴xoy中，开口朝下的二次函数与x轴交于A、C两点，与y轴正半轴交于点B，连接AB、BC。点P为二次函数顶点，作PD垂直于x轴，垂足为D，PD交AB于点E。", "class": "Analytic Geometry", "difficulty": 4, "image_path": "fig_1643.jpg" }, { "index": 1644, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B, and intersects the positive y-axis at point C. Its axis of symmetry lies to the right of the y-axis, and its vertex is point D. Connect points A and C, and extend the line.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，与y轴正半轴交于点C，对称轴在y轴右侧，顶点为D。连接AC并延长。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1644.jpg" }, { "index": 1645, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B, and intersects the positive y-axis at point C. Its axis of symmetry lies to the left of the y-axis and intersects the x-axis at point F. Connect points A and C, and draw line", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，与y轴正半轴交于点C，对称轴在y轴左侧，交x轴于点F。连接AC，作直线m平行于AC。在直线AC上方抛物线上取一点E，过点E作EH垂直于直线m，垂足为H，连接AE、EC、CH、AH。", "class": "Analytic Geometry", "difficulty": 4, "image_path": "fig_1645.jpg" }, { "index": 1646, "ocr_en": "In the coordinate plane with axes x and y (xoy), an upward-opening quadratic function has its vertex at the origin O. Point A lies on the parabola in the second quadrant. Point M lies on the positive x-axis. Line AM intersects the positive y-axis at point C, and intersects the parabola in the first quadrant at point B.", "ocr_zh": "坐标轴xoy中，开口朝上的二次函数顶点为原点o，点A为第二象限内抛物线上一点，点M为x轴正半轴上一点，直线AM交y轴正半轴于点C，交第一象限内的抛物线于点B。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1646.jpg" }, { "index": 1647, "ocr_en": "In the coordinate plane with axes x and y (xoy), an upward-opening quadratic function intersects the positive x-axis at point B, and intersects the negative y-axis at point A. Its axis of symmetry lies to the right of the y-axis, and its vertex is point C. The line AB intersects the parabola’s axis of symmetry at point E.", "ocr_zh": "坐标轴xoy中，开口朝上的二次函数与x轴正半轴交于点B，与y轴负半轴交于点A，对称轴在y轴右侧，顶点为点C。直线AB与抛物线对称轴交于点E。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1647.jpg" }, { "index": 1648, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B, and intersects the positive y-axis at point C. Connect points A and C, and points B and C.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，与y轴正半轴交于点C，连接AC、BC。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1648.jpg" }, { "index": 1649, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and C, and intersects the positive y-axis at point B. Its axis of symmetry lies to the right of the y-axis. Connect points A and B. Point D lies on the parabola above line AB. Connect points B and D.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A、C两点，与y轴正半轴交于点B，对称轴在y轴右侧，连接AB。点D为直线AB上方抛物线上一点，连接BD。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1649.jpg" }, { "index": 1650, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B, and intersects the positive y-axis at point C. Its axis of symmetry lies to the right of the y-axis. Connect points B and C.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，与y轴正半轴交于点C，对称轴在y轴右侧，连接BC。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1650.jpg" }, { "index": 1651, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B, and intersects the positive y-axis at point C. Point P lies on the parabola in the second quadrant.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，与y轴正半轴交于点C，点P在第二象限内的抛物线上。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1651.jpg" }, { "index": 1652, "ocr_en": "Left figure:\nIn the coordinate plane with axes x and y (xoy), a line passing through the first, second, and fourth quadrants intersects the positive x-axis at point A and the positive y-axis at point B. A downward-opening parabola passes through points A and B. Point D lies on the parabola in the first quadrant. Draw DC perpendicular to the x-axis, with foot C. Line CD intersects AB at point E. Connect points B and D.\n\nRight figure:\nBased on the left figure, take point F on the parabola in the first quadrant, and point G on line AB such that DEGF forms a parallelogram.", "ocr_zh": "左侧：坐标轴xoy中，过一二四象限的直线交x轴正半轴于点A，交y轴正半轴于点B，过点A、B作开口向下的抛物线。点D为第一象限内的抛物线上一点，作DC垂直于x轴，垂足为C，CD交AB于E，连接BD。右侧：在左侧图的基础上，在第一象限内抛物线上取点F，在AB上取点G，使DEGF为平行四边形。", "class": "Analytic Geometry", "difficulty": 3, "image_path": "fig_1652.jpg" }, { "index": 1653, "ocr_en": "In the coordinate axis xoy, the straight line passing one, two, four quadrants has a positive half-axis at point A, a positive half-axis at point B, and the passing points A and B are parabolas with openings downward. Point D is a point on the parabola in the first quadrant, with a DC perpendicular to the x-axis, a vertical foot being C, a CD intersecting AB at E, taking point F on the parabola in the first quadrant, taking point G on AB, making DEGF a parallelogram.", "ocr_zh": "坐标轴xoy中，过一二四象限的直线交x轴正半轴于点A，交y轴正半轴于点B，过点A、B作开口向下的抛物线。点D为第一象限内的抛物线上一点，作DC垂直于x轴，垂足为C，CD交AB于E，在第一象限内抛物线上取点F，在AB上取点G，使DEGF为平行四边形。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1653.jpg" }, { "index": 1654, "ocr_en": "In the coordinate axis xoy, the quadratic function with the opening facing down intersects the x axis at two points A and B, the negative half axis of the y axis intersects at point C, and the straight line passing through one, three, four quadrants passes through points B and C.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，与y轴负半轴交于点C，过一三四象限的直线经过点B、C。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1654.jpg" }, { "index": 1655, "ocr_en": "In the coordinate plane with axes x and y (xoy), an upward-opening quadratic function intersects the x-axis at points A and B, and intersects the negative y-axis at point C. Connect points A and C, and extend the line. Connect points B and C, and extend the line.", "ocr_zh": "坐标轴xoy中，开口朝上的二次函数与x轴交于A、B两点，与y轴负半轴交于点C。连接AC并延长，连接BC并延长。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1655.jpg" }, { "index": 1656, "ocr_en": "Left figure:\nIn the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B, and intersects the positive y-axis at point C. Connect points A and C, and points B and C. Take point P on the parabola in the first quadrant. Connect points A and P; line AP intersects BC at point D. Connect points C and P.\n\nRight figure:\nIn the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B, and intersects the positive y-axis at point C. Connect points A and C, and points B and C. The axis of symmetry of the parabola lies to the right of the y-axis and intersects the x-axis at point E. From point E, draw EF perpendicular to BC, with foot F.", "ocr_zh": "左侧：坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，与y轴正半轴交于点C，连接AC、BC。在第一象限内的抛物线上取点P，连接AP交BC于点D，连接CP。右侧：坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，与y轴正半轴交于点C，连接AC、BC。抛物下对称轴在y轴右侧，与x轴交于点E，过点E作EF垂直于BC，垂足为F。", "class": "Analytic Geometry", "difficulty": 4, "image_path": "fig_1656.jpg" }, { "index": 1657, "ocr_en": "In the coordinate plane with axes x and y (xoy), an upward-opening quadratic function intersects the x-axis at points A and B, and intersects the positive y-axis at point C. The axis of symmetry lies to the right of the y-axis, and the vertex is point M. Point N is the reflection of vertex M across the x-axis. Connect points A and N, and extend the line to intersect the graph of the parabola at point D.", "ocr_zh": "坐标轴xoy中，开口朝上的二次函数与x轴交于A、B两点，与y轴正半轴交于点C，对称轴在y轴右侧，顶点为M。点N是顶点M关于x轴的对称点，连接AN并延长交抛物线图像于点D。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1657.jpg" }, { "index": 1658, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B, and intersects the positive y-axis at point C. Connect points B and C, and extend the line. The axis of symmetry of the parabola lies to the right of the y-axis. Point P lies on the parabola in the first quadrant.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，与y轴正半轴交于点C，连接BC并延长，对称轴在y轴右侧。点P为第一象限内抛物线上一点。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1658.jpg" }, { "index": 1659, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B, and intersects the positive y-axis at point C. Connect points A and C.\n\nLeft figure:\nPoint P lies on the parabola above line AC. Draw PD perpendicular to the x-axis, with foot D. Line PD intersects AC at point E. Draw PF perpendicular to AC, with foot F.\n\nRight figure:\nPoint P lies on the parabola above line AC. The axis of symmetry of the parabola lies to the left of the y-axis.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，与y轴正半轴交于点C，连接AC。左侧：点P是AC上方抛物线上一点，作PD垂直于x轴，垂足为D，PD交AC于点E。作PF垂直于AC，垂足为F。右侧：点P是AC上方抛物线上一点，抛物线对称轴在y轴左侧。", "class": "Analytic Geometry", "difficulty": 4, "image_path": "fig_1659.jpg" }, { "index": 1660, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B, and intersects the positive y-axis at point C. A line passing through the first, second, and third quadrants intersects the parabola at points A and D, and intersects line segment BC at point E. Point M lies on segment AB. From point M, draw MF perpendicular to the x-axis, with foot M. Line MF intersects the parabola at point F. Segment FM intersects line AD at point G and intersects line BE at point H.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，与y轴正半轴交于点C。过一二三象限的直线交抛物线于A、D两点，交BC于点E。点M在AB上，过点M作MF垂直于x轴，垂足为M，直线MF交抛物线于点F，FM交AD于点G，交BE于点H。", "class": "Analytic Geometry", "difficulty": 3, "image_path": "fig_1660.jpg" }, { "index": 1661, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B, and intersects the positive y-axis at point C. Connect points B and C. The axis of symmetry lies to the right of the y-axis, and the vertex is point D.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，与y轴正半轴交于点C，连接BC，对称轴在y轴右侧，顶点为D。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1661.jpg" }, { "index": 1662, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B, and intersects the negative y-axis at point C. Connect points A and C, and points B and C.\n\n\n\n\n\n\n\n\n", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，与y轴负半轴交于点C，连接AC、BC。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1662.jpg" }, { "index": 1663, "ocr_en": "In the coordinate plane with axes x and y (xoy), a line AB passes through the first, second, and fourth quadrants, intersecting the positive x-axis at point A and the positive y-axis at point B. A downward-opening parabola passes through points A and B, with vertex M and its axis of symmetry located to the right of the y-axis. The parabola intersects line AB at point N. Point P is a point on line AB. From point P, draw PC perpendicular to the x-axis, with foot C. Line PC intersects the parabola at point D.", "ocr_zh": "坐标轴xoy中，过一二四象限的直线AB交x轴正半轴于点A，交y轴正半轴于点B。过AB两点作开口向下的抛物线，顶点为M，对称轴在y轴右侧，交AB于点N。点P为AB上一点，过点P作PC垂直于x轴，垂足为C，交抛物线于点D。", "class": "Analytic Geometry", "difficulty": 3, "image_path": "fig_1663.jpg" }, { "index": 1664, "ocr_en": "In the coordinate plane with axes x and y (xoy), an upward-opening quadratic function intersects the x-axis at points A and B, and intersects the negative y-axis at point C. The axis of symmetry lies to the left of the y-axis. Connect points A and C.", "ocr_zh": "坐标轴xoy中，开口朝上的二次函数与x轴交于A、B两点，与y轴负半轴交于点C，对称轴在y轴左侧，连接AC。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1664.jpg" }, { "index": 1665, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at point C and the y-axis at point E. Its vertex is point A, and its axis of symmetry lies to the right of the y-axis, intersecting the x-axis at point B. Point P lies on the axis of symmetry. From point P, draw PD perpendicular to line AB, intersecting AC at point D, with foot P. From point D, draw a line parallel to the y-axis that intersects the parabola at point Q. Connect points A and Q, and points C and Q.\n\n\n\n\n\n\n\n\n\n询问 ChatGPT", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交点C，于y轴交于点E，顶点为A，对称轴在y轴右侧，交x轴于点B。点P为对称轴上一点，作PD垂直于AB交AC于点D，垂足为P。过点D作y轴平行线，交抛物线于点Q，连媒介AQ、CQ。", "class": "Analytic Geometry", "difficulty": 3, "image_path": "fig_1665.jpg" }, { "index": 1666, "ocr_en": "In the coordinate plane with axes x and y (xoy), an upward-opening quadratic function passes through the origin O. A line passing through the first, second, and third quadrants intersects the parabola at points A and B, where point A is in the second quadrant and point B is in the first quadrant. Construct point A', the reflection of point A about the origin O. Connect points A' and B.", "ocr_zh": "坐标轴xoy中，开口朝上的二次函数过原点O，过点一二三象限的直线交抛物线于点A、B，其中点A在第二象限，点B在第一象限。作点A关于原点O的对称点A'，连接A'B。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1666.jpg" }, { "index": 1667, "ocr_en": "In the coordinate plane with axes x and y (xoy), an upward-opening quadratic function passes through the origin O. A line passing through the first, second, and third quadrants intersects the parabola at points A and B, where point A lies in the second quadrant and point B lies in the first quadrant.", "ocr_zh": "坐标轴xoy中，开口朝上的二次函数过原点O，过点一二三象限的直线交抛物线于点A、B，其中点A在第二象限，点B在第一象限。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1667.jpg" }, { "index": 1668, "ocr_en": "In the coordinate plane with axes x and y (xoy), an upward-opening quadratic function intersects the x-axis at points A and B, and intersects the negative y-axis at point C. Point P lies on the x-axis. From point P, draw a line perpendicular to the x-axis, with foot P. This line intersects the parabola at point N. Line PN intersects line AC at point M.", "ocr_zh": "坐标轴xoy中，开口朝上的二次函数与x轴交于A、B两点，与y轴负半轴交于点C.点P在x轴上，过点P作x轴垂线，垂足为P，交抛物线于点N，PN交直线AC于点M。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1668.jpg" }, { "index": 1669, "ocr_en": "In the coordinate plane with axes x and y (xoy), a line AB intersects the positive x-axis at point A and the negative y-axis at point B. An upward-opening quadratic function passes through points A and B, with its axis of symmetry located to the right of the y-axis. Point C is the reflection of point B about the axis of symmetry. Connect points A and C, B and C, and A and B.", "ocr_zh": "坐标轴xoy中，直线AB交x轴正半轴于点A，交y轴负半轴于点B，开口向上的二次函数图像过点A、B，对称轴在y轴右侧，点B关于对称轴的对称点为C，连接AC、BC、AB。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1669.jpg" }, { "index": 1670, "ocr_en": "In the coordinate plane with axes x and y (xoy), a line AB passing through the first, second, and fourth quadrants intersects the positive x-axis at point A and the positive y-axis at point B. An upward-opening parabola passes through the origin O and point A, with vertex at point M. Point E lies on the parabola below line AB. Connect points B and E, and points A and E.", "ocr_zh": "坐标轴xoy中，过一二四象限的直线AB交x轴正半轴于点A，交y轴正半轴于点B，开口向上的抛物线过原点O和点A，顶点为M。点E是直线AB下方抛物线上一点，连接BE、AE。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1670.jpg" }, { "index": 1671, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B, and intersects the negative y-axis at point C. Connect points A and C, and points B and C.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，与y轴负半轴交于点C，连接AC、BC。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1671.jpg" }, { "index": 1672, "ocr_en": "In the coordinate plane with axes x and y (xoy), an upward-opening quadratic function intersects the x-axis at points A and B, and intersects the negative y-axis at point C. A line passing through point A intersects the parabola again at point D. Line AD intersects the y-axis at point E.", "ocr_zh": "坐标轴xoy中，开口朝上的二次函数与x轴交于A、B两点，与y轴负半轴交于点C，过点A的直线交抛物线于点D，直线AD交y轴于点E。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1672.jpg" }, { "index": 1673, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B, and intersects the positive y-axis at point C. Connect points B and C.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，与y轴正半轴交于点C，连接BC。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1673.jpg" }, { "index": 1674, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B, and intersects the positive y-axis at point C. A line passing through point A intersects the parabola again at point D. Line AD intersects the y-axis at point E.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，与y轴正半轴交于点C，过点A的直线交抛物线于点D，直线AD交y轴于点E。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1674.jpg" }, { "index": 1675, "ocr_en": "Left figure:\nIn the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects a line passing through the first, second, and fourth quadrants at points B and C. Point A lies on the parabola in the first quadrant. The parabola’s axis of symmetry is to the right of the y-axis, and its vertex is point M. Point D lies on the parabola above line BC. From point D, draw a perpendicular line to the x-axis, intersecting BC at point E.\n\nRight figure:\nIn the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects a line passing through the first, second, and fourth quadrants at points B and C. Point A lies on the parabola in the first quadrant. The parabola’s axis of symmetry is to the right of the y-axis, and its vertex is point M.", "ocr_zh": "左图：坐标轴xoy中，开口朝下的二次函数与过一二四象限的直线交于B、C两点，点A为第一象限内抛物线上一点，抛物线对称轴在y轴右侧，顶点为M。点D是直线BC上方抛物线上一点，过点D作x轴垂线交BC于点E。右图：坐标轴xoy中，开口朝下的二次函数与过一二四象限的直线交于B、C两点，点A为第一象限内抛物线上一点，抛物线对称轴在y轴右侧，顶点为M。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1675.jpg" }, { "index": 1676, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects a line passing through the first, second, and fourth quadrants at points B and C. Point A lies on the parabola in the first quadrant. The parabola’s axis of symmetry lies to the right of the y-axis, and its vertex is point M.\n\n\n\n\n\n\n\n\n\n询问 ChatGPT", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与过一二四象限的直线交于B、C两点，点A为第一象限内抛物线上一点，抛物线对称轴在y轴右侧，顶点为M。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1676.jpg" }, { "index": 1677, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects a line passing through the first, second, and fourth quadrants at points B and C. Point A lies on the parabola in the first quadrant. The parabola’s axis of symmetry lies to the right of the y-axis, and its vertex is point M. Point D is a point on the parabola above line BC. From point D, draw a line perpendicular to the x-axis that intersects BC at point E.\n\n\n\n\n\n\n\n\n\n询问 ChatGPT", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与过一二四象限的直线交于B、C两点，点A为第一象限内抛物线上一点，抛物线对称轴在y轴右侧，顶点为M。点D是直线BC上方抛物线上一点，过点D作x轴垂线交BC于点E。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1677.jpg" }, { "index": 1678, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B, and intersects the positive y-axis at point C. Connect points B and C.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，与y轴正半轴交于点C，连接BC。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1678.jpg" }, { "index": 1679, "ocr_en": "In the coordinate plane with axes x and y (xoy), an upward-opening quadratic function intersects a line passing through the second, third, and fourth quadrants at points A and C. The parabola intersects the x-axis at points A and B, and its vertex lies in the fourth quadrant.", "ocr_zh": "坐标轴xoy中，开口朝上的二次函数与过二三四象限的直线交于A、C两点，抛物线交x轴于A、B两点，抛物线顶点在第四象限。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1679.jpg" }, { "index": 1680, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects a line passing through the first, second, and fourth quadrants at points A and B. The parabola intersects the x-axis at points A and C. Its axis of symmetry lies to the right of the y-axis. Point D lies on the negative y-axis. Connect points A and D; line AD intersects the axis of symmetry at point E. Connect points B and E.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与过一二四象限的直线交于A、B两点，抛物线交x轴于A、C两点，对称轴在y轴右侧。点D为y轴负半轴一点，连接AD交对称轴于点E，连接BE。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1680.jpg" }, { "index": 1681, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B, and intersects the positive y-axis at point C. Connect points B and C. Point D lies on the parabola in the first quadrant.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，与y轴正半轴交于点C，连接BC，点D在第一象限内的抛物线上。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1681.jpg" }, { "index": 1682, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B, and intersects the positive y-axis at point C. Connect points B and C, and extend the line. The axis of symmetry of the parabola lies to the right of the y-axis. Point P lies on the parabola in the first quadrant.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，与y轴正半轴交于点C，连接BC并延长。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1682.jpg" }, { "index": 1683, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B, and intersects the positive y-axis at point C. Connect points A and C, and points B and C.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，与y轴正半轴交于点C，连接AC、BC。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1683.jpg" }, { "index": 1684, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B, and intersects the positive y-axis at point C. Connect points B and C. Point P lies on the parabola in the second quadrant. Connect points A and P, and extend the line to intersect the y-axis at point E. Connect points B and P, and extend the line to intersect the y-axis at point F.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，与y轴正半轴交于点C，连接BC。点P为第二象限内二次函数图像上一点，连接AP并延长交y轴于点E，连接BP交y轴于点F。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1684.jpg" }, { "index": 1685, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B, and intersects the positive y-axis at point C. Connect points B and C. The vertex of the parabola is point D. Connect points B and D. Point E is a point on the positive x-axis. Point F lies on the parabola. Connect points D and F, and points B and F.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，与y轴正半轴交于点C，连接BC，抛物线顶点为D，连接BD。点E为x轴正半轴一点。F为抛物线上一点，连接DF、BF。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1685.jpg" }, { "index": 1686, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B, and intersects the positive y-axis at point C. Connect points B and C. The axis of symmetry lies to the right of the y-axis, and the vertex is point D.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，与y轴正半轴交于点C，连接BC，对称轴在y轴右侧，顶点为D。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1686.jpg" }, { "index": 1687, "ocr_en": "In the coordinate plane with axes x and y (xoy), an upward-opening quadratic function intersects the x-axis at points A and C. Point B lies on the parabola, and the vertex of the parabola is point D. Connect points B and C, and points C and D. Point P lies on the parabola below line BC. Connect points B and P, and points C and P.", "ocr_zh": "坐标轴xoy中，开口朝上的二次函数与x轴交于A、C两点，点B为抛物线上一点，抛物线顶点为D，连接BC、CD。点P为BC下方抛物线上一点，连接BP、CP。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1687.jpg" }, { "index": 1688, "ocr_en": "In the coordinate plane with axes x and y (xoy), an upward-opening quadratic function intersects the y-axis at point A and the x-axis at point C. Its vertex is point D. From point A, draw a line parallel to the x-axis that intersects the parabola at point B. Connect points B and C, C and D, and B and D.", "ocr_zh": "坐标轴xoy中，开口朝上的二次函数与y轴交于点A，交x轴于点C，顶点为D，过点A作x轴平行线交抛物线于点B，连接BC、CD、BD。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1688.jpg" }, { "index": 1689, "ocr_en": "In the coordinate plane with axes x and y (xoy), an upward-opening quadratic function intersects the x-axis at points A and B, and intersects the positive y-axis at point C. Connect points B and C, and extend the line. The extended line intersects the parabola again at point D.", "ocr_zh": "坐标轴xoy中，开口朝上的二次函数与x轴交于A、B两点，与y轴正半轴交于点C，连接BC并延长，交抛物线于点D。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1689.jpg" }, { "index": 1690, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B, and intersects the positive y-axis at point C. Connect points A and C, and points B and C. The axis of symmetry lies to the right of the y-axis. Point D lies on the parabola in the first quadrant. From point D, draw DE perpendicular to the x-axis, with foot E. Line DE intersects AC at point F.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，与y轴正半轴交于点C，连接AC、BC，对称轴在y轴右侧。点D为第一象限内抛物线上一点，过点D作DE垂直于x轴，垂足为E，DE交AC于F。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1690.jpg" }, { "index": 1691, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B, and intersects the positive y-axis at point C. Connect points A and C, and points B and C. Point P is a point on the parabola in the first quadrant. Connect points C and P, and points B and P.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，与y轴正半轴交于点C，连接AC、BC，点P为第一象限内抛物线上一点，连接CP、BP。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1691.jpg" }, { "index": 1692, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B, and the positive y-axis at point C. Connect points A and C, and points B and C. The axis of symmetry lies to the right of the y-axis, with vertex at point D. The axis of symmetry intersects line BC at point E. Point P lies on the parabola in the first quadrant. Connect points C and P, and points B and P.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，与y轴正半轴交于点C，连接AC、BC，对称轴在y轴右侧，顶点为D，对称轴与BC交于点E。点P为第一象限内抛物线上一点，连接CP、BP。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1692.jpg" }, { "index": 1693, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B, and intersects the positive y-axis at point C. Connect points A and C, and points B and C. The axis of symmetry lies to the right of the y-axis, with vertex at point D. The axis of symmetry intersects line BC at point E. Point P is a point on the parabola in the first quadrant. Connect points C and P, and points B and P.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，与y轴正半轴交于点C，连接AC、BC，对称轴在y轴右侧，顶点为D，对称轴与BC交于点E。点P为第一象限内抛物线上一点，连接CP、BP。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1693.jpg" }, { "index": 1694, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B, and intersects the positive y-axis at point C. Connect points B and C. The axis of symmetry lies to the right of the y-axis, and the vertex is point D. The axis of symmetry intersects line BC at point E. Point P is a point on the parabola in the first quadrant. From point P, draw a line perpendicular to the x-axis, intersecting line BC at point F.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，与y轴正半轴交于点C，连接BC，对称轴在y轴右侧，顶点为D，对称轴与BC交于点E。点P为第一象限内抛物线上一点，过点P作x轴垂线，交BC与点F。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1694.jpg" }, { "index": 1695, "ocr_en": "Left figure:\nIn the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B, and intersects the positive y-axis at point C. Its vertex is point D. Connect points A and C, A and D, and C and D. Point F lies on segment AD. Connect points C and F.\n\nRight figure:\nBased on the left figure, connect points O (the origin) and F.", "ocr_zh": "左图：坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，与y轴正半轴交于点C，顶点为D，连接AC、AD、CD。点F在AD上，连接CF。右图：在左图基础上连接OF。", "class": "Analytic Geometry", "difficulty": 3, "image_path": "fig_1695.jpg" }, { "index": 1696, "ocr_en": "In the coordinate plane with axes x and y (xoy), an upward-opening quadratic function passes through the origin O and intersects a line passing through the first, second, and fourth quadrants at points A and B. The parabola also intersects the positive y-axis at point C. Rotate point A counterclockwise 90 degrees about point C to get point D. Connect points A and D, C and D, and B and D.\n\nBelow line AB, draw a line parallel to AB that intersects the parabola at points E and F. Point G lies on the positive y-axis. Connect points E and G, and points F and G.", "ocr_zh": "坐标轴xoy中，过原点O且开口朝上的二次函数与过一二四象限的直线交于点A、B，与y轴正半轴交于点C。将点A绕点C逆时针旋转90度，连接AD、CD、BD。在AB下方作AB的平行线交抛物线于点E、F，点G为y轴正半轴一点，连接EG、FG。", "class": "Analytic Geometry", "difficulty": 4, "image_path": "fig_1696.jpg" }, { "index": 1697, "ocr_en": "In the coordinate plane with axes x and y (xoy), an upward-opening quadratic function passing through the origin O intersects a line passing through the first, second, and fourth quadrants at points A and B. The quadratic function intersects the positive y-axis at point C. Rotate point A 90 degrees counterclockwise about point C to get point D. Connect points A and D, C and D, and B and D.", "ocr_zh": "坐标轴xoy中，过原点O且开口朝上的二次函数与过一二四象限的直线交于点A、B，与y轴正半轴交于点C。将点A绕点C逆时针旋转90度，连接AD、CD、BD。", "class": "Analytic Geometry", "difficulty": 3, "image_path": "fig_1697.jpg" }, { "index": 1698, "ocr_en": "In the coordinate plane with axes x and y (xoy), an upward-opening quadratic function intersects the x-axis at points O (the origin) and B. Point A lies on the parabola. Through point A, draw a line parallel to the x-axis intersecting the y-axis at point C. Point P is another point on the parabola to the right of point A. From point P, draw a perpendicular line to line AC, with foot at point D. Connect points O and A, and points A and P.", "ocr_zh": "坐标轴xoy中，开口朝上的二次函数与x轴交于原点O、B两点，点A为抛物线上一点，过点A作x轴平行线交y轴于点C。点P为点A右侧抛物线上一点，过点P作PD垂直于AC，垂足为D，连接OA、AP。", "class": "Analytic Geometry", "difficulty": 3, "image_path": "fig_1698.jpg" }, { "index": 1699, "ocr_en": "In the coordinate plane with axes x and y (xoy), an upward-opening quadratic function intersects the x-axis at the origin O and another point B. Point A lies on the parabola. From point A, draw a line parallel to the x-axis, intersecting the y-axis at point C. Connect points O and A.", "ocr_zh": "坐标轴xoy中，开口朝上的二次函数与x轴交于原点O、B两点，点A为抛物线上一点，过点A作x轴平行线交y轴于点C，连接OA。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1699.jpg" }, { "index": 1700, "ocr_en": "In the coordinate plane with axes x and y (xoy), a line passing through the first, third, and fourth quadrants intersects the positive x-axis at point A and the negative y-axis at point B. Draw an upward-opening parabola passing through points A and B, which intersects the x-axis at point C.", "ocr_zh": "坐标轴xoy中，过点一三四象限的直线交x轴正半轴于点A，交y轴负半轴于点B，经过A、B两点作开口向上的抛物线，交x轴于点C。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1700.jpg" }, { "index": 1701, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at point A and the y-axis at point B. The axis of symmetry is \n𝑥\n=\n−\n1\nx=−1. Point P lies on the parabola between points A and B. Connect points A and B, A and P, and B and P.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A，与y轴交于点B，对称轴为x=-1。点P为点A、B之间抛物线上一点，连接AB、AP、BP。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1701.jpg" }, { "index": 1702, "ocr_en": "In the coordinate plane with axes x and y (xoy), an upward-opening quadratic function intersects the x-axis at points A and B. From point A, draw a line that intersects the positive y-axis at point C and intersects the parabola at point D, and its vertex lies in the fourth quadrant. Point E lies on the parabola below line AD. Connect points A and E, and points C and E.", "ocr_zh": "坐标轴xoy中，开口朝上的二次函数与x轴交于A、B两点，过点A作直线交y轴正半轴于点C，交抛物线于点D，抛物线顶点在第四象限。点E为直线AD下方的抛物线上一点，连接AE、CE。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1702.jpg" }, { "index": 1703, "ocr_en": "In the coordinate plane with axes x and y (xoy), an upward-opening quadratic function intersects the x-axis at points A and B, and intersects the negative y-axis at point C. Connect points A and C. Point P lies on the parabola in the third quadrant. Connect points B and P, and points C and P. Point D lies on segment AC. From point D, draw a perpendicular line DE to segment AC, with foot at D, and intersecting the x-axis at point E.", "ocr_zh": "坐标轴xoy中，开口朝上的二次函数与x轴交于A、B两点，与y轴负半轴交于点C，连接AC。点P为第三象限抛物线上一点，连接BP、CP。点D为AC上一点，过点D作DE垂直与AC，垂足为D，交x轴于点E。", "class": "Analytic Geometry", "difficulty": 3, "image_path": "fig_1703.jpg" }, { "index": 1704, "ocr_en": "In the coordinate plane with axes x and y (xoy), an upward-opening quadratic function intersects the x-axis at points A and B, and intersects the negative y-axis at point C. Connect points A and C. Point P lies on the parabola in the third quadrant. Connect points B and P, and points C and P.\n", "ocr_zh": "坐标轴xoy中，开口朝上的二次函数与x轴交于A、B两点，与y轴负半轴交于点C，连接AC。点P为第三象限抛物线上一点，连接BP、CP。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1704.jpg" }, { "index": 1705, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B, and intersects the positive y-axis at point C. The axis of symmetry lies to the right of the y-axis. Connect points A and C.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，与y轴正半轴交于点C，对称轴在y轴右侧，连接AC。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1705.jpg" }, { "index": 1706, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B, and intersects the positive y-axis at point C. Connect points A and C, and B and C. To the right of the y-axis, draw a line \n𝑙\nl perpendicular to the x-axis, with foot E. Line \n𝑙\nl intersects the parabola at point P and intersects line segment BC at point D. Connect points A and P. From point P, draw \n𝑃\n𝐹\nPF perpendicular to BC, with foot at F.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，与y轴正半轴交于点C，连接AC、BC。在y轴右侧作x轴垂线l，垂足为E，交抛物线于点P，交BC于点D，连接AP。过点P作PF垂直于BC，垂足为F。", "class": "Analytic Geometry", "difficulty": 4, "image_path": "fig_1706.jpg" }, { "index": 1707, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B, and intersects the positive y-axis at point C. Connect points A and C, and points B and C. Point D is a point on the parabola above line BC. Connect points C and D, and points O and D, where O is the origin. Line OD intersects line BC at point F.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，与y轴正半轴交于点C，连接AC、BC。点D是直线BC上方抛物线上一一点，连接CD、OD，OD交BC于点F。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1707.jpg" }, { "index": 1708, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B, and intersects the positive y-axis at point C. Connect points A and C, and points B and C. Point D lies on the parabola in the first quadrant. Connect points C and D.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，与y轴正半轴交于点C，连接AC、BC。点D为第一象限内抛物线上一点，连接CD。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1708.jpg" }, { "index": 1709, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B, and intersects the positive y-axis at point C. Connect points A and C, and points B and C. Point D is a point on the parabola in the first quadrant. Connect points C and D.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，与y轴正半轴交于点C，连接AC、BC。点D为第一象限内抛物线上一点，连接CD。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1709.jpg" }, { "index": 1710, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B. Its axis of symmetry lies to the right of the y-axis, and its vertex is point D.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，对称轴在y轴右侧，顶点为D。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1710.jpg" }, { "index": 1711, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B. Its axis of symmetry lies to the right of the y-axis. Points D and E lie on the axis of symmetry. Connect points C and D, and points A and E.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，对称轴在y轴右侧，点D、E在对称轴上，连接CD、AE。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1711.jpg" }, { "index": 1712, "ocr_en": "Left figure:\nIn the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B. Its axis of symmetry lies to the right of the y-axis. Points D and E lie on the axis of symmetry. Connect points C and D, and points A and E.\n\nRight figure:\nIn the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B. Its axis of symmetry lies to the right of the y-axis. Point P lies on the parabola. Connect points A and P, and points C and P.", "ocr_zh": "左侧：坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，对称轴在y轴右侧，点D、E在对称轴上，连接CD、AE。右侧：坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，对称轴在y轴右侧，点P为抛物线上一点，连接AP、CP。", "class": "Analytic Geometry", "difficulty": 4, "image_path": "fig_1712.jpg" }, { "index": 1713, "ocr_en": "In the coordinate plane with axes x and y (xoy), a downward-opening quadratic function intersects the x-axis at points A and B. Its axis of symmetry lies to the right of the y-axis. Point P is a point on the parabola. Connect points A and P, and points C and P.", "ocr_zh": "坐标轴xoy中，开口朝下的二次函数与x轴交于A、B两点，对称轴在y轴右侧，点P为抛物线上一点，连接AP、CP。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1713.jpg" }, { "index": 1714, "ocr_en": "In the coordinate plane with axes x and y (xoy), an upward-opening quadratic function intersects the x-axis at points A and B. A line passing through point B intersects the parabola again at point C.", "ocr_zh": "坐标轴xoy中，开口朝上的二次函数与x轴交于A、B两点，过点B的直线交抛物线于点C。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1714.jpg" }, { "index": 1715, "ocr_en": "Left side:\nIn the coordinate plane with axes x and y (xoy), an upward-opening quadratic function intersects the x-axis at points A and B, and intersects the negative y-axis at point C.\n\nRight side:\nBased on the left figure, draw a line passing through the origin that intersects the parabola at two points. Connect each of these two points to point C.", "ocr_zh": "左侧：坐标轴xoy中，开口朝上的二次函数与x轴交于A、B两点，与y轴负半轴交于点C。右侧：在左图基础上，过原点作直线交抛物线于两点，分别与C相连。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1715.jpg" }, { "index": 1716, "ocr_en": "In the coordinate plane with axes x and y (xoy), point P lies on an upward-opening quadratic function. Point F lies on the positive y-axis, and point M lies in the first quadrant. Connect points M and F, points F and P, and points P and M.", "ocr_zh": "坐标轴xoy中，点P为开口朝上的二次函数上一点，点F为y轴正半轴一点，点M为第一象限内一点，连接MF、FP、PM。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1716.jpg" }, { "index": 1717, "ocr_en": "In the coordinate plane with axes x and y (xoy), an upward-opening quadratic function intersects the x-axis at the origin O and at point A. Point B lies on the parabola in the first quadrant. Connect points O and B.", "ocr_zh": "坐标轴xoy中，开口朝上的二次函数与x轴交于原点O、A两点，点B为第一象限内抛物线上一点，连接OB。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1717.jpg" }, { "index": 1718, "ocr_en": "In the coordinate plane with axes x and y (xoy), an upward-opening quadratic function intersects the negative y-axis at point B, and its vertex is point A. The axis of symmetry lies to the right of the y-axis. Point F lies on the axis of symmetry, and point P lies on the parabola. Point D lies in the first quadrant. From point C on the negative y-axis, draw a line \n𝑙\nl perpendicular to the y-axis, with foot at C.", "ocr_zh": "坐标轴xoy中，开口朝上的二次函数与y轴负半轴交于点B，抛物线顶点为点A，对称轴在y轴右侧，F为对称轴上一点，点P为抛物线上一点，点D为第一象限内一点。过y轴负半轴点C作y轴垂线l，垂足为C。", "class": "Analytic Geometry", "difficulty": 3, "image_path": "fig_1718.jpg" }, { "index": 1719, "ocr_en": "In the coordinate plane with axes x and y (xoy), an upward-opening quadratic function intersects the x-axis at points A and B, and intersects the negative y-axis at point C. Its axis of symmetry lies to the left of the y-axis. A line passing through point A intersects the parabola again at point E, intersects the y-axis at point D, and intersects the axis of symmetry at point F.", "ocr_zh": "坐标轴xoy中，开口朝上的二次函数与x轴交于A、B两点，与y轴负半轴交于点C，对称轴在y轴左侧。经过点A的直线交抛物线于点E，交y轴于点D，交对称轴于点F。", "class": "Analytic Geometry", "difficulty": 3, "image_path": "fig_1719.jpg" }, { "index": 1720, "ocr_en": "With O as the center, draw circle O with diameter AB. Points C and D lie on circle O. From point D, draw DE perpendicular to AC, with foot E. Line DE intersects the extension of AB at point F. Connect points A and D. The line OE intersects AD at point G.", "ocr_zh": "以O为圆心，AB为直径作圆O，点C、D为圆O上两点，过点D作DE垂直于AC，垂足为E，DE交AB延长线于点F，连接AD、OE交于点G。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1720.jpg" }, { "index": 1721, "ocr_en": "In circle O, points A, C, and D lie on the circle. Connect points A and D, and extend the segment AD to point B. Connect points B and C, and points C and D.", "ocr_zh": "在圆O中，点A、C、D在圆O上，连接AD并延长至点B，连接BC、CD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1721.jpg" }, { "index": 1722, "ocr_en": "In circle O, points A, C, D, and E lie on the circle, with points D and E on opposite sides of segment AC. Connect points A and C, A and D, A and E, C and D, and C and E.\n\nUsing segment AD as the axis of symmetry, reflect triangle ADE to obtain triangle ADQ. Connect points C and Q.", "ocr_zh": "在圆O中，点A、C、D、E在圆O上且D、E位于AC两侧，连接AC、AD、AE、CD、CE。以AD为对称轴作三角形ADE的对称图形ADQ，连接CQ。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1722.jpg" }, { "index": 1723, "ocr_en": "In the coordinate plane with axes x and y (xoy), a line passing through the first, second, and third quadrants intersects the x-axis at point A and the y-axis at point B. Point P lies on line AB in the second quadrant. A circle \n𝐶\nC is drawn through points A, P, and the origin O. Connect points P and C, and extend line PC to intersect circle \n𝐶\nC again at point Q. Connect points O and Q.", "ocr_zh": "坐标轴xoy中，过一二三象限的直线交x轴于点A，交y轴于点B。点P在第二象限内直线AB上，过点A、P、O作圆C，连接PC并延长，交圆C于点Q，连接OQ。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1723.jpg" }, { "index": 1724, "ocr_en": "In quadrilateral ABCD, point E lies on segment CD, and AE is perpendicular to DE with foot at E. Point O lies on segment AE. With O as the center and OE as the radius, draw circle O, which is tangent to line AB at point G and intersects line BC at point F. Connect points O and G.", "ocr_zh": "在四边形ABCD中，点E为CD上一点，且AE垂直于DE，垂足为E。点O为AE上一点，以O为圆心，OE为半径作圆O与AB相切与点G，圆O交BC于点F，连接OG。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1724.jpg" }, { "index": 1725, "ocr_en": "Circle O is the circumcircle of triangle ABC. From point B, draw line BD perpendicular to AC, with foot D. Extend BD to intersect circle O again at point E. From point A, draw line AF perpendicular to CE, with foot F. Connect points A and E, and extend line BC. The extended line BC and line AE intersect at point G.", "ocr_zh": "圆O为三角形ABC外接圆，过点B作BD垂直于AC，垂足为D，延长BD交圆O于点E，过点A作AF垂直于CE，垂足为F，连接AE、BC并延长，相交于点G。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1725.jpg" }, { "index": 1726, "ocr_en": "With O as the center, draw a circle with diameter AC. From a point P outside the circle, draw the tangents PA and PB to circle O. Connect points O and P; line OP intersects the circle at point D and intersects line AB at point E. Connect points A and D, points B and C, and points O and B.", "ocr_zh": "以O为圆心，AC为直径作圆，过圆外一点P作圆O的切线PA、PB，连接OP交圆O于点D，交AB于点E，连接AD、BC、OB。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1726.jpg" }, { "index": 1727, "ocr_en": "With O as the center, draw circle O with diameter AB. Points C and E lie on circle O. Connect points A and C, and points A and E. From point C, draw line CD perpendicular to AE, with foot D. Connect points B and E, and points B and C. Extend segment AB to point M. From point B, draw line BP intersecting the extension of AC at point P. From point B, draw segment BQ. Connect points A and Q.", "ocr_zh": "以O为圆心，AB为直径作圆O，点C、E为圆O上两点，连接AC，AE。过点C作CD垂直于AE，垂足为D。连接BE、BC，延长AB至点M，过点B作BP交AC的延长线于点P，过点B作线段BQ，连接AQ。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_1727.jpg" }, { "index": 1728, "ocr_en": "In triangle ABC, with O as the center, draw circle O with diameter AB. Circle O intersects BC at point D. From point D, draw DE perpendicular to AC, with foot E. Draw MN perpendicular to AB, with foot G. Points M and N lie on circle O.", "ocr_zh": "在三角形ABC中，以O为圆心，AB为直径作圆O，圆O与BC交于点D，过点D作DE垂直于AC，垂足为E。作MN垂直于AB，垂足为G，M、N在圆O上。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1728.jpg" }, { "index": 1729, "ocr_en": "With O as the center, draw circle O with diameter AB. Points C and E lie on circle O. Connect points A and C, and points A and E. From point C, draw line CD perpendicular to AE, with foot D. Connect points B and E, and points B and C. Extend segment AB to point M. From point B, draw line BP intersecting the extension of AC at point P. From point B, draw segment BQ. Connect points A and Q.", "ocr_zh": "以O为圆心，AB为直径作圆O，点C、E为圆O上两点，连接AC，AE。过点C作CD垂直于AE，垂足为D。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1729.jpg" }, { "index": 1730, "ocr_en": "In triangle ABC, point D lies on segment BC. Connect points A and D. With O as the center, draw circle O passing through points A and D. The circle intersects segment AC at point F and segment AB at point E. Connect points D and F.", "ocr_zh": "在三角形ABC中，点D在BC上，连接AD，以O为圆心，过点A、D作圆O，交AC于点F，交AB于点E，连接DF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1730.jpg" }, { "index": 1731, "ocr_en": "Quadrilateral ABCD", "ocr_zh": "四边形ABCD", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1731.jpg" }, { "index": 1732, "ocr_en": "With O as the center, draw the circumcircle \n𝑂\nO of triangle \n𝐴\n𝐵\n𝐶\nABC using segment \n𝐵\n𝐶\nBC as the diameter. Point \n𝐷\nD lies on circle \n𝑂\nO. Connect points \n𝐴\nA and \n𝐷\nD, \n𝐵\nB and \n𝐷\nD, and \n𝐶\nC and \n𝐷\nD. From point \n𝐷\nD, draw the tangent to circle \n𝑂\nO, which intersects the extension of segment \n𝐴\n𝐶\nAC at point \n𝑃\nP.", "ocr_zh": "以O为圆心，BC为直径作三角形ABC的外接圆O，点D在圆O上，连接AD、BD、CD，过点D作圆O的切线交AC的延长线于点P。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1732.jpg" }, { "index": 1733, "ocr_en": "With O as the center, draw circle O with diameter AB. Point P lies on segment AB. From point P, draw a line perpendicular to AB, intersecting circle O at points C and D, with foot at P. Connect points A and D. Point Q lies on the arc BC of the circle. From point A, draw AH perpendicular to line DQ, with foot at H.", "ocr_zh": "以O为圆心，AB为直径作圆O，点P为AB上一点，过点P作AB的垂线交圆O于C、D两点，垂足为P，连接AD。点Q为弧BC上一点，过点A作AH垂直于DQ，垂足为H。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1733.jpg" }, { "index": 1734, "ocr_en": "With O as the center, draw circle O with diameter AB. Point P lies on segment AB. From point P, draw a line perpendicular to AB, intersecting circle O at points C and D, with foot at P. Connect points A and D. Point Q lies on the arc BC of circle O. From point A, draw line AH perpendicular to line DQ, with foot at H.", "ocr_zh": "以O为圆心，AB为直径作圆O，点P为AB上一点，过点P作AB的垂线交圆O于C、D两点，垂足为P，连接AD。点Q为弧BC上一点，过点A作AH垂直于DQ，垂足为H。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1734.jpg" }, { "index": 1735, "ocr_en": "In quadrilateral ABCD, with A as the center and radius equal to segment AD, draw circle A. The extension of segment CD intersects circle A at point F, and the extension of segment DA intersects circle A at point E. Connect points B and F, and connect points A and E; lines BF and AE intersect at point G.", "ocr_zh": "四边形ABCD中，以A为圆心，AD为半径作圆A，CD的延长线交圆A于点F，DA的延长线交圆A于点E，连接BF交AE于点G。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1735.jpg" }, { "index": 1736, "ocr_en": "With O as the center, draw circle O with diameter AB. Point D lies on circle O. Extend segment BA beyond A to point C. Connect points C and D, and points A and D. From point D, draw line DE that bisects angle ADB, intersecting circle O again at point E. Connect points B and E.", "ocr_zh": "以O为圆心，AB为直径作圆O，点D在圆O上，延长BA至点C，连接CD、AD。过点D作DE平分角ADB，交圆O于点E，连接BE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1736.jpg" }, { "index": 1737, "ocr_en": "With O as the center, draw circle O with diameter AB. Point D lies on circle O. Extend segment BA beyond A to point C. Connect points C and D, and points A and D. From point D, draw line DE that bisects angle ADB, intersecting circle O again at point E. Connect points B and E.", "ocr_zh": "以O为圆心，AB为直径作圆O，点D在圆O上，延长BA至点C，连接CD、AD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1737.jpg" }, { "index": 1738, "ocr_en": "In the rhombus ABCD, point O lies on segment BD. From O, draw OE perpendicular to AB, with foot E. With O as the center and OE as the radius, draw circle O, which intersects segment CD at point H. Extend EO to intersect circle O again at point F. Line EF intersects CD at point G. Connect points O and B, and points O and H.", "ocr_zh": "在菱形ABCD中，点O为BD上一点，作OE垂直于AB，垂足为E，以O为圆心，OE为半径作圆O，交CD于点H，延长EO交圆O于点F，EF交CD于点G，连接OB、OH。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1738.jpg" }, { "index": 1739, "ocr_en": "With O as the center, draw a semicircle O with diameter AB. Points M and N lie on segment AB, and points P and Q lie on arc AB such that quadrilateral MNPQ is a square. Point C lies on arc PQ. Connect points B and C, and extend line BC to intersect the extension of MQ at point D. Connect points A and C; line AC intersects MQ at point E. Connect points O and Q.", "ocr_zh": "以O为圆心，AB为直径作半圆O，点M、N在AB上，点P、Q在弧AB上，且四边形MNPQ为正方形，点C为弧PQ上，连接BC并延长交MQ的延长线于点D，连接AC交MQ于点E，连接OQ。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1739.jpg" }, { "index": 1740, "ocr_en": "Let the center of the circle be point O. With O as the center and AB as the diameter, draw circle O. Points C and D lie on the circle. Connect BD and OC; they intersect at point E. Line BD intersects the line through point C, CF, at point F. Connect OD, AD, AC, and CD. From point E, draw EG parallel to AB, intersecting AC at point G. Point M lies on segment AC.", "ocr_zh": "假设圆心为O点，以O为圆心，AB为直径作圆O，点C、D为圆O上两点，连接BD交OC于点E，直线BD交过点C的直线CF于点F。连接OD、AD、AC、CD，过点E作EG平行于AB，交AC于点G，点M在AC上。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_1740.jpg" }, { "index": 1741, "ocr_en": "With O as the center, draw the circumcircle O of quadrilateral ABCD using AD as the diameter. From point C, draw CE perpendicular to AB, with foot E. Connect points A and C.", "ocr_zh": "以O为圆心，AD为直径作四边形ABCD的外接圆O，过点C作CE垂直于AB，垂足为E，连接AC。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1741.jpg" }, { "index": 1742, "ocr_en": "With O as the center, draw the circumcircle O of quadrilateral ABCD using AD as the diameter. From point C, draw CE perpendicular to AB, with foot at E. Connect points O and C, and points A and C.", "ocr_zh": "以O为圆心，AD为直径作四边形ABCD的外接圆O，过点C作CE垂直于AB，垂足为E，连接OC、AC。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1742.jpg" }, { "index": 1743, "ocr_en": "In the coordinate plane with axes x and y (xoy), with M as the center and OM as the radius, draw circle M. The circle intersects the x-axis at point A and the y-axis at point B. Connect points A and B. A line through point M intersects circle M at points D and E, and intersects the x-axis at point C. Point P lies on segment AC. Connect points A and E, P and E, B and D, and O and D.", "ocr_zh": "坐标轴xoy中，以M为圆心，OM为半径作圆M，交x轴于点A，交y轴于点B，连接AB。过点M的直线CM交圆M于点D、E，交x轴于点C。点P在AC上，连接AE、PE、BD、OD。", "class": "Analytic Geometry", "difficulty": 4, "image_path": "fig_1743.jpg" }, { "index": 1744, "ocr_en": "BD is a chord of circle O with center O. Point A lies on circle O. Using points A, B, and D as vertices, construct parallelogram ABCD.", "ocr_zh": "BD是以O为圆心的圆上一条弦，点A在圆O上，以A、B、D为顶点作平行四边形ABCD。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1744.jpg" }, { "index": 1745, "ocr_en": "BD is a chord on a circle with center O.", "ocr_zh": "BD是以O为圆心的圆上一条弦。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1745.jpg" }, { "index": 1746, "ocr_en": "In rhombus ABCD, point M lies on segment AB, and point N lies on segment AD. Fold triangle AMN along segment MN so that point A falls at point P.", "ocr_zh": "在菱形ABCD中，点M为AB上一点，点N为AD上一点，沿MN将三角形AMN翻折，点A落在点P处。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1746.jpg" }, { "index": 1747, "ocr_en": "In rectangle ABCD, point E lies on segment AB, and point F lies on diagonal AC. Using EF as the axis of symmetry, reflect triangle AEF to obtain its symmetric image triangle GEF. Connect points C and G. Line EG intersects diagonal AC at point H.", "ocr_zh": "在矩形ABCD中，点E为AB上一点，点F为对角线AC上一点，以EF为对称轴作三角形AEF的轴对称图形GEF，连接CG，EG交AC于点H。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1747.jpg" }, { "index": 1748, "ocr_en": "Rectangle ABCD", "ocr_zh": "矩形ABCD", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1748.jpg" }, { "index": 1749, "ocr_en": "In rectangle ABCD, points M and N lie on segments AB and AD respectively. Fold triangle AMN along segment MN so that point A coincides with point P.", "ocr_zh": "在矩形ABCD中，M为AB上一点，N为AD上一点，沿MN将三角形AMN翻折，点A落在点P处。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1749.jpg" }, { "index": 1750, "ocr_en": "In triangles ABC and ADE, extend segment CE to intersect BD at point P.", "ocr_zh": "在三角形ABC、ADE中，延长CE交BD于点P。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1750.jpg" }, { "index": 1751, "ocr_en": "In triangle ABC, rotate triangle ABC counterclockwise about vertex B to obtain triangle A₁BC₁. Extend segment A₁C₁ to intersect AC at point H and AB at point G.", "ocr_zh": "在三角形ABC中，将三角形ABC绕顶点B逆时针旋转，得到三角形A1BC1，延长A1C1交AC于点H，交AB于点G。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1751.jpg" }, { "index": 1752, "ocr_en": "In triangle ABC, point P lies on segment BC, and point Q lies on segment AC. Connect points A and P, and points P and Q.", "ocr_zh": "在三角形ABC中，点P在BC上，点Q在AC上，连接AP、PQ。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1752.jpg" }, { "index": 1753, "ocr_en": "Point C lies on segment AB. Point M lies on segment AC. Point N lies on segment BC.", "ocr_zh": "点C为线段AB上一点，点M为线段AC上一点，点N为线段BC上一点。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1753.jpg" }, { "index": 1754, "ocr_en": "Point B lies on segment AC. Point M lies on segment AB. Point N lies on segment BC.", "ocr_zh": "点B为线段AC上一点，点M为线段AB上一点，点N为线段BC上一点。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1754.jpg" }, { "index": 1755, "ocr_en": "In triangles OAB and OCD, angle AOC is denoted as angle 1.", "ocr_zh": "在三角形OAB和三角形OCD中，角AOC为角1", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1755.jpg" }, { "index": 1756, "ocr_en": "Line \n𝑙\n1\nl \n1\n​\n is parallel to line \n𝑙\n2\nl \n2\n​\n . Points A and B lie on \n𝑙\n1\nl \n1\n​\n ; points D and E lie on \n𝑙\n2\nl \n2\n​\n ; point C lies between the two lines. Angle \n𝐵\n𝐶\n𝐷\nBCD is angle 2; angle \n𝐶\n𝐵\n𝐴\nCBA is angle 4; the adjacent supplementary angle to angle 4 is angle 1; angle \n𝐶\n𝐷\n𝐸\nCDE is angle 3.", "ocr_zh": "直线l1平行于直线l2，点A、B在l1上，点D、E在l2上，点C在两直线之间，角BCD为角2，角CBA为角4，角4的邻补角为角1，角CDE为角3.", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1756.jpg" }, { "index": 1757, "ocr_en": "AD is parallel to CE. Point F lies on the extension of DA, and point G lies on the extension of EC. Angle DAB is angle 1, angle BCG is angle 2, and angle BAF is angle 3.", "ocr_zh": "AD平行于CE，F在DA延长线上，G在EC延长线上，角DAB为角1，角BCG为角2，角BAF为角3.", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1757.jpg" }, { "index": 1758, "ocr_en": "Line AB is parallel to EF. Point C lies between lines AB and EF. Connect points B and F, and points C and F. Draw line CD parallel to AB.", "ocr_zh": "直线AB平行于EF，点C在AB、EF之间，连接BF、CF，作CD平行于AB。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1758.jpg" }, { "index": 1759, "ocr_en": "Line AB is parallel to line CD. Point E lies between lines AB and CD. Connect points B and E, and points D and E. Draw line EF parallel to AB.", "ocr_zh": "直线AB平行于CD，点E在AB、CD之间，连接BE、DE，作EF平行于AB。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1759.jpg" }, { "index": 1760, "ocr_en": "Lines AD and BC intersect.\nAngle ABC = 20°,\nAngle BAD = 30°,\nAngle BCD = 40°,\nAngle ADC = \n𝛼\nα.", "ocr_zh": "AD于BC相交，角ABC为20度，角BAD为30度，角BCD为40度，角ADC为α", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1760.jpg" }, { "index": 1761, "ocr_en": "In triangle ABC, points E and M lie on segments AC and BC respectively. Fold triangle CEM along segment EM onto triangle DEM. Extend segment DM to intersect CE at point F. Angle AED is angle 1, angle BMD is angle 2, and the acute angle between EC and BC is angle 3.", "ocr_zh": "三角形ABC中，点E、M分别为AC、BC上的点，沿EM将CEM折叠至DEM，延长DM交CE于F。角AED为角1，角BMD为角2，EC与BC的锐角夹角为角3.", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1761.jpg" }, { "index": 1762, "ocr_en": "In triangle ABC, points D and E lie on segments AC and BC respectively. Fold triangle CED along segment DE onto triangle C''ED. Extend segment C''E to intersect CD at point F. Angle BEC'' is angle 1, angle ADC'' is angle 2, the angle between DC'' and BC is angle 3, angle AEC'' is angle 4, and angle CEF is angle 5.", "ocr_zh": "三角形ABC中，点D、E分别为AC、BC上的点，沿DE将CED折叠至C''ED，延长C''E交CD于F。角BEC''为角1，角ADC''为角2，DC''与BC的锐角夹角为角3，角AEC''为角，角CEF为角5.", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1762.jpg" }, { "index": 1763, "ocr_en": "In pentagon ABCDE, extend segments BC and ED to intersect at point H. In triangle CDH, segment CP bisects angle HCD, and segment PD bisects angle CDH. Points F and G lie on segments BH and EH, respectively", "ocr_zh": "在五边形ABCDE中，延长BC、ED相交于点H，在三角形CDH中，CP平分角HCD，PD平分角CDH。点F、G分别在BH、EH上。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1763.jpg" }, { "index": 1764, "ocr_en": "In the coordinate plane \n𝑥\n𝑜\n𝑦\nxoy, there is triangle ABC, where point A lies on the positive y-axis and point C lies on the positive x-axis. From point B, draw BD perpendicular to the x-axis, with foot at point D.", "ocr_zh": "坐标轴xoy中有三角形ABC，其中点A在y轴正半轴上，点C在x轴正半轴上，过点B作BD垂直于x轴，垂足为D。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1764.jpg" }, { "index": 1765, "ocr_en": "In triangles OAB and OCD, connect points A and C, and points B and D; they intersect at point M. From point O, draw line OG perpendicular to AC, with foot at G. From point O, draw line OH perpendicular to BD, with foot at H.", "ocr_zh": "在三角形OAB和三角形OCD中，连接AC、BD相交于点M，过点O作OG垂直于AC，垂足为G，过点O作OH垂直于BD，垂足为H。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1765.jpg" }, { "index": 1766, "ocr_en": "In triangles OAB and OMN, lines AO and BN intersect at point J. From point O, draw line OH perpendicular to MN, with foot at H.", "ocr_zh": "在三角形OAB和三角形OMN中，AO交BN于点J，过点O作OH垂直于MN，垂足为H。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1766.jpg" }, { "index": 1767, "ocr_en": "In triangles OAB and OMN, connect points A and M.", "ocr_zh": "在三角形OAB和三角形OMN中,连接AM。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1767.jpg" }, { "index": 1768, "ocr_en": "In triangle ABC, point D lies on segment BC. Extend segment AD beyond D to point E. Connect points C and E.", "ocr_zh": "在三角形ABC中，D为BC上一点，连接AD并延长至点E，连接CE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1768.jpg" }, { "index": 1769, "ocr_en": "In triangles OAB and OMN, line segment MN intersects OB at point J.", "ocr_zh": "在三角形OAB和三角形OMN中，MN交OB于点J。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1769.jpg" }, { "index": 1770, "ocr_en": "In triangle ABC, points D and E lie on segments BC and AC respectively. Connect BE and AD; they intersect at point F. Extend AD beyond D to point A'. Connect points A' and B.", "ocr_zh": "在三角形ABC中，D、E分别为BC、AC上一点，连接BE、AD相交于点F，延长AD至点A'，连接A'B。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1770.jpg" }, { "index": 1771, "ocr_en": "In rectangle ABCD, points F and G lie on diagonal BD. From point B, draw BE perpendicular to EF, with foot at E. Connect points C and G. Extend segment CG to point H. Connect points E and H, H and F, and C and E.", "ocr_zh": "在矩形ABCD中，点F、G分别为对角线BD上两点，过点B作BE垂直于EF，垂足为E，连接CG。延长CG至点H，连接EH、HF、CE。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1771.jpg" }, { "index": 1772, "ocr_en": "In rectangle ABCD, points E, F, and G lie on segments AB, BC, and AD respectively. Connect EF, EG, and FG. Extend segment GE to intersect the extension of FB at point M.", "ocr_zh": "在矩形ABCD中，点E、F、G分别在AB、BC、AD上，连接EF、EG、FG，延长GE交FB的延长线于点M。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1772.jpg" }, { "index": 1773, "ocr_en": "AD is perpendicular to AB, with the foot of the perpendicular at point A. AB is perpendicular to BC, with the foot of the perpendicular at point B. Connect points C and D. Point E lies on segment CD. Connect AE and extend it to intersect BC at point F.", "ocr_zh": "AD垂直于AB，垂足为A，AB垂直于BC，垂足为B，连接CD，点E在CD上，连接AE并延长，交BC于点F。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1773.jpg" }, { "index": 1774, "ocr_en": "In trapezoid ABDC, point O lies on segment BD. Connect AO and extend it to intersect the extension of CD at point E. Connect OC.", "ocr_zh": "在梯形ABDC中，点O为BD上一点，连接AO并延长交CD延长线于点E，连接OC。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1774.jpg" }, { "index": 1775, "ocr_en": "In parallelogram ABCD, points E and M lie on segments BC and AB respectively. Connect segment DM and extend it to intersect the extension of CB at point H. Connect segments DE, ME, and AE.", "ocr_zh": "在平行四边形ABCD中，点E、M分别在BC、AB上，连接DM并延长交CB的延长线于点H，连接DE、ME、AE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1775.jpg" }, { "index": 1776, "ocr_en": "In triangle ABC, point D lies on BC, and point F lies on AB. Construct parallelogram BDEF with vertices B, D, and F. Connect points A and E, and points C and E. Extend CE to intersect AB at point G.", "ocr_zh": "在三角形ABC中，点D在BC上，点F在AB上，以B、D、F为顶点作平行四边形BDEF，连接AE、CE，延长CE交AB于点G。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1776.jpg" }, { "index": 1777, "ocr_en": "In triangle ABC, from point A, draw line AP perpendicular to BP, with foot P. Connect points C and P. Extend line AP to intersect BC at point E.", "ocr_zh": "在三角形ABC中，过点A作AP垂直于BP，垂足为P，连接CP，延长AP交BC于点E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1777.jpg" }, { "index": 1778, "ocr_en": "In triangle ABC, from point B, draw BD perpendicular to CD, with foot at D. Extend BD to intersect AC at point E.", "ocr_zh": "在三角形ABC中，过点B作BD垂直于CD，垂足为D，延长BD交AC于点E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1778.jpg" }, { "index": 1779, "ocr_en": "In triangle ABC, points D and E lie on segment BC. From point B, draw line BF perpendicular to BC, with foot at point F. Connect points A and F, and points E and F.\nAngle BAE is angle 1, angle CAD is angle 2, and angle BAF is angle 3.", "ocr_zh": "在三角形ABC中，点D、E在BC上，过点B作BF垂直于BC，垂足为F，连接AF、EF。角BAE为角1，角CAD为角2，角BAF为角3.", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1779.jpg" }, { "index": 1780, "ocr_en": "In triangle ABC, from point A, draw line AP perpendicular to BP, with foot at point P. Connect points C and P. Extend line AP to intersect BC at point E.", "ocr_zh": "在三角形ABC中，过点A作AP垂直于BP，垂足为P，连接CP，延长AP交BC于点E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1780.jpg" }, { "index": 1781, "ocr_en": "In square ABCD, points E and F lie on segments BC and CD respectively. Connect BD; it intersects AF at point N and AE at point M. Extend segment CB to point G. Connect AG, and take point H on AG. Connect BH and HM.\nAngle DAF is angle 1, angle BAG is angle 2, angle BAE is angle 3, angle BDC is angle 4, angle ABD is angle 5, angle HBG is angle 6, and angle AFD is angle 7.", "ocr_zh": "在正方形ABCD中，点E、F分别在BC、CD上，连接BD交AF于点N，交AE于点M。延长CB至点G，连接AG，在AG上取一点H，连接BH、HM。角DAF为角1，角BAG为角2，角BAE为角3，角BDC为角4，角ABD为角5，角HBG为角6，角AFD为角7.", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_1781.jpg" }, { "index": 1782, "ocr_en": "In square ABCD, points E and F lie on segments BC and CD respectively. Connect BD; it intersects AF at point N and AE at point M. Connect segments EF and EN.", "ocr_zh": "在正方形ABCD中，点E、F分别在BC、CD上，连接BD交AF于点N，交AE于点M。连接EF、EN。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1782.jpg" }, { "index": 1783, "ocr_en": "In angle ABC, ray BN bisects angle ABC. From point A, draw line AP parallel to BC, intersecting BN at point P. From point P, draw PD perpendicular to BC, with foot at point D. From point P, draw PK perpendicular to AB, with foot at point K.", "ocr_zh": "在角ABC中，BN平分角ABC，过点A作AP平行于BC角BN于点P，过点P作PD垂直于BC，垂足为D，过点P作PK垂直于AB，垂足为K。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1783.jpg" }, { "index": 1784, "ocr_en": "In angle AOB, line OM bisects angle AOB. Point D lies on segment AO, and point E lies on segment OB. Connect points C and D, and points C and E. From point C, draw CF perpendicular to AO, with foot at point F. From point C, draw CG perpendicular to OB, with foot at point G.", "ocr_zh": "在角AOB中，OM平分角AOB，点D在AO上，点E在OB上，连接CD、CE，过点C作CF垂直于AO，垂足为F，过点C作CG垂直于BO，垂足为G。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1784.jpg" }, { "index": 1785, "ocr_en": "In angle MON, line OC bisects angle MON. Points A and B lie on segments OM and ON respectively. Connect points A and C, and points B and C. From point C, draw CF perpendicular to OM, with foot at point F. From point C, draw CE perpendicular to ON, with foot at point E.", "ocr_zh": "在角MON中，OC平分角MON，点A、B分别在OM、ON上，连接AC、BC。过点C作CF垂直于OM，垂足为F，过点C作CE垂直于ON，垂足为E。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1785.jpg" }, { "index": 1786, "ocr_en": "In angle AOB, line OM bisects angle AOB. Point D lies on the extension of AO, and point E lies on OB. Connect points C and D, and points C and E. From point C, draw CF perpendicular to AO, with foot at point F. From point C, draw CG perpendicular to BO, with foot at point G.", "ocr_zh": "在角AOB中，OM平分角AOB，点D在AO延长线上，点E在OB上，连接CD、CE，过点C作CF垂直于AO，垂足为F，过点C作CG垂直于BO，垂足为G。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1786.jpg" }, { "index": 1787, "ocr_en": "In the coordinate plane \n𝑥\n𝑜\n𝑦\nxoy, point A lies on the positive x-axis, and point C lies on the positive y-axis. Rectangle OABC is constructed with vertices O, A, and C. Points D and E lie on segments OA and AB respectively. Connect segments DE and CD. Extend segment CE to intersect the x-axis at point H.", "ocr_zh": "坐标轴xoy中，点A在x轴正半轴，点C在y轴正半轴，以O、A、C为顶点作矩形OABC，点D、E分别为OA、AB上的点，连接DE、CD。连接CE并延长交x轴于点H。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1787.jpg" }, { "index": 1788, "ocr_en": "In angle AOB, there is a point P. Connect points O and P. Reflect point P about OA and OB to obtain points C and D, respectively. Connect points C and D; this line intersects OA and OB at points M and N, respectively. Connect points N and P, M and P, and M and N. From point O, draw OH perpendicular to MN, with foot at point H.", "ocr_zh": "在角AOB中有一点P，连接OP。分别以OA、OB为对称轴作P的对称点C、D，连接CD分别交OA、OB于M、N两点，连接NP、MP、MN，过点O作OH垂直于MN，垂足为H。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1788.jpg" }, { "index": 1789, "ocr_en": "In triangle ABC, point D lies on segment BC. Connect points A and D. Point P lies on segment AD. Connect points C and P, and points E and P.\nConnect points B and E; they intersect segment AD at point P'.", "ocr_zh": "在三角形ABC中，点D在BC上，连接AD，点P在AD上，连接CP、EP。连接BE交AD于点P'。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1789.jpg" }, { "index": 1790, "ocr_en": "In triangles ABC and CEF, points A and F lie on opposite sides of line CE, and point B lies on segment CE. Connect points A and F. Point M lies on segment AF. Connect points E and M. Extend line BM to intersect segment EF at point D.", "ocr_zh": "在三角形ABC和三角形CEF中，AF两点在CE两侧，点B在CE上，连接AF，点M在AF上，连接EM。连接BM并延长，交EF于点D。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1790.jpg" }, { "index": 1791, "ocr_en": "In triangles ABC and CEF, points A and F lie on opposite sides of line CE, point B lies on segment CE. Connect points A and F. Point M lies on segment AF. Connect points E and M. Extend segment AB to intersect CF at point D.", "ocr_zh": "在三角形ABC和三角形CEF中，AF两点在CE两侧，点B在CE上，连接AF，点M在AF上，连接EM。延长AB交CF于点D。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1791.jpg" }, { "index": 1792, "ocr_en": "In rectangle ABCD, points E and P lie on segments AB and BC respectively. Connect points E and P. Point Q lies on diagonal AC. Connect points P and Q. Extend segment AB beyond point E to point E'. Connect points E' and P and extend it to intersect diagonal AC at point Q'.", "ocr_zh": "在矩形ABCD中，点E、P分别在AB、BC上，连接EP，点Q在对角线AC上，连接PQ。延长AB至点E'，连接E'P并延长，交AC于Q'。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1792.jpg" }, { "index": 1793, "ocr_en": "In triangles ABC and CEF, points A and F lie on opposite sides of line CE, connect points A and F. Take a point M on segment AF. Connect points B and M, and points E and M. Extend segment AB to intersect CE at point D. Connect points D and F. Extend segments FE and CB to intersect at point G. Connect points A and G.", "ocr_zh": "在三角形ABC和三角形CEF中，AF两点在CE两侧，连接AF，在AF上取一点M，连接BM、EM。延长AB交CE于点D，连接DF。延长FE、CB相交于点G，连接AG。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1793.jpg" }, { "index": 1794, "ocr_en": "In triangles ABC and CEF, points A and F lie on opposite sides of line CE, point B lies on segment CE. Connect points A and F. Point M lies on segment AF. Connect points B and M, and points E and M. Extend segment AB to intersect CF at point D. Extend segment CA to intersect the extension of FE at point G.", "ocr_zh": "在三角形ABC和三角形CEF中，AF两点在CE两侧，点B在CE上，连接AF，点M在AF上，连接BM、EM。延长AB交CF于点D。延长CA交FE延长线于点G。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1794.jpg" }, { "index": 1795, "ocr_en": "In triangles ABC and CEF, points A and F lie on opposite sides of line CE, point B lies on segment CE. Connect points A and F. Point M lies on segment AF. Connect points B and M, and points E and M. Extend segment BM to intersect EF at point D.", "ocr_zh": "在三角形ABC和三角形CEF中，AF两点在CE两侧，点B在CE上，连接AF，点M在AF上，连接BM、EM。延长BM交EF于点D。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1795.jpg" }, { "index": 1796, "ocr_en": "In triangles ABC and CEF, points A and F lie on opposite sides of line CE, connect points A and F. Take a point M on segment AF. Connect points B and M, and points E and M. Extend segment BM to intersect CF at point D. Connect points B and E, and points D and E.", "ocr_zh": "在三角形ABC和三角形CEF中，AF两点在CE两侧，连接AF，在AF上取一点M，连接BM、EM。延长BM交CF于点D，连接BE、DE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1796.jpg" }, { "index": 1797, "ocr_en": "In triangle ABC, point E lies inside the triangle. Connect points B and E, C and E, and A and E. Extend segment CE to point D. Connect points B and D. Point F lies on segment AE. Connect points D and F. Extend segment AC to point M. Connect points B and M. Extend segment CE to point N. Connect points A and N, and B and N. Connect segments M and E, and extend it to intersect AN at point H.", "ocr_zh": "在三角形ABC中有一点E，连接BE、CE、AE，延长CE至点D，连接BD，点F在AE上，连接DF。延长AC至点M，连接BM。延长CE至点N，连接AN、BN，连接ME并延长，交AN于点H。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_1797.jpg" }, { "index": 1798, "ocr_en": "In triangles ABC and BDE, point E lies on the extension of segment AB. Connect points C and E. Point F lies on segment AB. Connect points C and F.", "ocr_zh": "在三角形ABC中和三角形BDE中，点E在AB的延长线上，连接CE。点F在AB上，连接CF、CF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1798.jpg" }, { "index": 1799, "ocr_en": "In triangle ABC, there is a point E. Connect points B and E, C and E, and A and E. Point F lies on segment AE. Quadrilateral BCDF is formed. Connect points C and D. Extend segment CF to point M. Connect points A and M, and D and M. Connect segment M and E, and extend it to intersect BC at point H.", "ocr_zh": "在三角形ABC中有一点E，连接BE、CE、AE，点F在AE上，BCDF为四边形，连接CD。延长CF至点M，连接AM、DM，连接ME并延长，交BC于点H。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_1799.jpg" }, { "index": 1800, "ocr_en": "In triangle ABC, points E and F lie on segment AB. From point B, draw line BD perpendicular to DE, with foot at point D. Connect points D and F", "ocr_zh": "在三角形ABC中，点E、F在AB上，过点B作BD垂直于DE，垂足为D，连接DF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1800.jpg" }, { "index": 1801, "ocr_en": "In rectangle ABCD, point E lies outside the rectangle. Connect points A and E, and points C and E. From point E, draw EO perpendicular to AB, with foot at point O. Point P lies on segment CE, and point Q lies on segment AB. Connect points P and B, and points P and Q. Extend segment OP to intersect BC at point F.", "ocr_zh": "在矩形ABCD中，点E为矩形外一点，连接AE、CE，过点E作EO垂直于AB，垂足为O。点P在CE上，点Q在AB上，连接PB、PQ，连接OP并延长，交BC于点F。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1801.jpg" }, { "index": 1802, "ocr_en": "In triangles ABC and AEF, point D lies on segment BC. Connect points A and D. Extend segment AD to point M. Connect points C and M. Extend segment DA to intersect segment EF at point N.", "ocr_zh": "在三角形ABC和AEF中，点D在BC上，连接AD，延长AD至点M，连接CM，延长DA交EF于点N。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1802.jpg" }, { "index": 1803, "ocr_en": "In the quadrilateral ABCD, angle ADC is obtuse, DC is perpendicular to CB, the foot of the pendant is C. The extension of AD and BC intersects at point M, connecting DM, CM as an auxiliary line. Point E is on the line AD, connect BE as an auxiliary line and BE is perpendicular to AD, the vertical foot is E. Point N is the midpoint of CD, point F is the midpoint of BC, there is a point P on the line BE, connect PA, PC, PD, PF, PN, DF as an auxiliary line, the line DF and PC intersect at point O.", "ocr_zh": "在四边形ABCD中，角ADC是钝角，DC垂直于CB，垂足为C。AD与BC的延长线交于点M，连接DM、CM作辅助线。点E在线段AD上，连接BE作辅助线且BE垂直于AD，垂足是E。点N是CD的中点，点F是BC的中点，在线段BE上有一点P，连接PA、PC、PD、PF、PN、DF作辅助线，线段DF与PC交于点O。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1803.jpg" }, { "index": 1804, "ocr_en": "There is a triangle ABC, line AB around point A clockwise rotation α degrees to AB ′, line AC around point A counterclockwise rotation β degrees to AC ′, connect B ′ C ′, forming a triangle AB ′ C ′, where α and β are greater than 0 and less than 180. point D is the mid-point of the B ′ C ′, connected to the AD, the extension of the line in the segment AD take a point M, so that the MD is equal to the AD, connected to MD, MB ′, MC ′ as auxiliary lines. ′ as an auxiliary line.", "ocr_zh": "有三角形ABC，线段AB绕点A顺时针旋转α度至 AB′，线段AC绕点A逆时针旋转β度至AC′，连接B′C′，组成三角形AB′C′，其中α和β均大于0且小于180。点D是B′C′的中点，连接AD，在线段AD的延长线上取一点M，使MD等于AD，连接MD、MB′、MC′作辅助线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1804.jpg" }, { "index": 1805, "ocr_en": "In the quadrilateral ABCD there is a point E, connect AE, BE, CE, DE, point F is a point on the line segment AD, connect EF, point G is the intersection of the extension line of EF along the direction of E and BC, connect EG, point M is on the line segment EG, connect BM as an auxiliary line, take a point N on the extension line of MG along the direction of G, connect NG, NC as an auxiliary line.", "ocr_zh": "在四边形ABCD内有一点E，连接AE、BE、CE、DE，点F是线段AD上一点，连接EF，点G是EF沿E方向延长线与BC的交点，连接EG，点M在线段EG上，连接BM作辅助线，在MG沿G方向延长线上取一点N，连接NG、NC作辅助线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1805.jpg" }, { "index": 1806, "ocr_en": "There is a trapezium ABCD, BC perpendicular to CD, the pendant is C, AD perpendicular to CD, the pendant is D, point E in the line CD, connect BE, AE, BE perpendicular to AE, the pendant is E, in the AD along the extension of the line in the direction of the D to take a point F, connect EF, DF. in the trapezium ABCD outside the one point G, connect BG, GE, GF, GE is perpendicular to the EF, the pendant is E, BG and the CD intersect at point H, point M in the line CD, connect GM as an auxiliary line, GM perpendicular to the CD, pendant is M. BG and CD intersect at point H, point M is on line CD, connect GM as an auxiliary line, GM is perpendicular to CD, the foot is M.", "ocr_zh": "有梯形ABCD，BC垂直于CD，垂足是C，AD垂直于CD，垂足是D，点E在线段CD上，连接BE、AE，BE垂直于AE，垂足是E，在AD沿D方向延长线上取一点F，连接EF、DF。在梯形ABCD外有一点G，连接BG、GE、GF，GE垂直于EF，垂足是E，BG与CD交于点H，点M在线段CD上，连接GM作辅助线，GM垂直于CD，垂足是M。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1806.jpg" }, { "index": 1807, "ocr_en": "In the triangle ABC outside there are two non-crossing square ABDE and square ACFG, connect EG, point M in the line segment BC, over the point A for AM perpendicular BC at point M, extend MA to EG at point N, connect AN, point Q in the line segment AN, connect QG as an auxiliary line, QG perpendicular to AN at point Q, in the direction of the extension of the line along the N direction of the QN to take a point P, connect PE, PN as an Take a point P on the extension line of QN along the direction of N, connect PE and PN as auxiliary line, PE is perpendicular to PN at point P.\n", "ocr_zh": "在三角形ABC外部有两个不交叉的正方形ABDE和正方形ACFG，连接EG，点M在线段BC上，过点A作AM垂直BC于点M，延长MA交EG于点N，连接AN，点Q在线段AN上，连接QG作辅助线，QG垂直AN于点Q，在QN沿N方向延长线上取一点P，连接PE、PN作辅助线，PE垂直PN于点P。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1807.jpg" }, { "index": 1808, "ocr_en": "There are two non-crossing squares ABDE and square ACFG in the exterior of right triangle ABC, BA perpendicular to AC at point A, connect EG, point M is on the line segment BC, make AM perpendicular to BC at point M over point A, extend MA to intersect EG at point N, connect AN, point Q is on line segment AN, connect QG as an auxiliary line, QG perpendicular to AN at point Q, and take a point in the extension of the line along the direction of N in QN P, connect PE, PN as an auxiliary line, PE perpendicular PN at point P.\n", "ocr_zh": "在直角三角形ABC外部有两个不交叉的正方形ABDE和正方形ACFG，BA垂直AC于点A，连接EG，点M在线段BC上，过点A作AM垂直BC于点M，延长MA交EG于点N，连接AN，点Q在线段AN上，连接QG作辅助线，QG垂直AN于点Q，在QN沿N方向延长线上取一点P，连接PE、PN作辅助线，PE垂直PN于点P。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1808.jpg" }, { "index": 1809, "ocr_en": "In quadrilateral ABCD, CD is perpendicular to DA at point D, BA is perpendicular to AD at point A, angle BCD is obtuse, point E is a point on line segment AB, connect CE as an auxiliary line, CE is perpendicular to AB at point E.", "ocr_zh": "在四边形ABCD中，CD垂直DA于点D，BA垂直AD于点A，角BCD是钝角，点E是线段AB上一点，连接CE作辅助线，CE垂直AB于点E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1809.jpg" }, { "index": 1810, "ocr_en": "There is a triangle ABC, point A is on the horizontal plane, AC is perpendicular to CB at point C, BC is parallel to the horizontal plane, line segment BC is 2 metres long, point D is a point on the extension line of AC along the direction of C, connecting CD, line segment CD is 1 metre long.", "ocr_zh": "有三角形ABC，点A在水平面上，AC垂直CB于点C，BC平行于水平面，线段BC长2米，点D是AC沿C方向延长线上一点，连接CD，线段CD长1米。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1810.jpg" }, { "index": 1811, "ocr_en": "In triangle ABC, angle ABC is obtuse, circle I is the incircle of triangle ABC, and the tangent points to AC, BC, and AB are point E, point D, and point F. Connect AI, BI, CI, DI, EI, and FI as auxiliary lines.", "ocr_zh": "在三角形ABC中，角ABC是钝角，圆I是三角形ABC的内切圆，与AC、BC、AB的切点分别是点E、点D和点F，连接AI、BI、CI、DI、EI、FI作辅助线。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1811.jpg" }, { "index": 1812, "ocr_en": "In rectangle A′ACC′, point B′ is the midpoint of A′C′ and point B is the midpoint of AC, connecting BB′ and AC′.", "ocr_zh": "在矩形A′ACC′中，点B′是A′C′的中点，点B是AC的中点，连接BB′和AC′。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1812.jpg" }, { "index": 1813, "ocr_en": "In rectangle ABC′D′, connect AC′, point A′ is a point on line AD′, point B′ is a point on line BC′, connect A′B′ and A′B′ is parallel to AB.", "ocr_zh": "在矩形ABC′D′中，连接AC′，点A′是线段AD′上一点，点B′是线段BC′上一点，连接A′B′且A′B′平行于AB。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1813.jpg" }, { "index": 1814, "ocr_en": "In quadrilateral ABCD, connect AC and BD as auxiliary lines intersecting at point O. Points M, N, P, and Q are the midpoints on line segments AB, BC, CD, and DA, respectively; connect MQ, QP, NP, and MN.", "ocr_zh": "在四边形ABCD中，连接AC、BD作辅助线交于点O，点M、N、P、Q分别是线段AB、BC、CD、DA上的中点，连接MQ、QP、NP、MN。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1814.jpg" }, { "index": 1815, "ocr_en": "In quadrilateral ABCD, points E, F, G and H are points on line segments AB, BC, CD and AD respectively, connecting EH, HG, GF and EF to form rhombus EHGF.", "ocr_zh": "在四边形ABCD中，点E、F、G和H分别是线段AB、BC、CD和AD上的点，连接EH、HG、GF、EF组成菱形EHGF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1815.jpg" }, { "index": 1816, "ocr_en": "In quadrilateral ABCD, angle ABC and angle BCD are obtuse angles, points E, F, G and H are points on line segments AB, BC, CD and AD respectively, connect EH, HG, GF and EF to form parallelogram EHGF.", "ocr_zh": "在四边形ABCD中，角ABC和角BCD是钝角，点E、F、G和H分别是线段AB、BC、CD和AD上的点，连接EH、HG、GF、EF组成平行四边形EHGF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1816.jpg" }, { "index": 1817, "ocr_en": "In the plane right-angle coordinate system in xOy, there is a triangle ABC in the second quadrant, point A is on the negative half-axis of the x-axis, point B and point D are on the positive half-axis of the y-axis, AC is perpendicular to CB at point C, and connecting CD as an auxiliary line, CD is perpendicular to the y-axis at point D.", "ocr_zh": "在平面直角坐标系中xOy中，有三角形ABC在第二象限，点A在x轴负半轴上，点B和点D在y轴正半轴上，AC垂直CB于点C，连接CD作辅助线，CD垂直y轴于点D。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1817.jpg" }, { "index": 1818, "ocr_en": "In the plane right-angle coordinate system in xOy, there is a triangle ABC in the first quadrant, the point A is on the y-axis positive semiaxis, point C is on the x-axis positive semiaxis, the point M is the midpoint of AC, connect OM, BM, and OB for the auxiliary line.", "ocr_zh": "在平面直角坐标系中xOy中，有三角形ABC在第一象限，点A在y轴正半轴上，点C在x轴正半轴上，点M是AC的中点，连接OM、BM、OB作辅助线。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1818.jpg" }, { "index": 1819, "ocr_en": "There are rays OM, ON and angle MON is a right angle, rectangle ABCD in the interior of the angle MON, vertices A, B are on the rays OM, ON, the point E is the midpoint of AB, connect OD, DE, EO as auxiliary lines.", "ocr_zh": "有射线OM、ON且角MON是直角，矩形ABCD在角MON的内部，顶点A、B分别在射线OM、ON上，点E是AB的中点，连接OD、DE、EO作辅助线。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1819.jpg" }, { "index": 1820, "ocr_en": "In the plane right-angle coordinate system in xOy, triangle ABC and A′B′C′ are congruent and in the second quadrant, point A is on the negative half-axis of the x-axis, point B is on the positive half-axis of the y-axis, AC is perpendicular to CB at point C, point A′ is on the line segment OA, point B′ is on the extension of the line along the direction of B in OB, and A′C′ is perpendicular to B′C′ at point C′.", "ocr_zh": "在平面直角坐标系中xOy中，三角形ABC和A′B′C′全等且在第二象限，点A在x轴负半轴上，点B在y轴正半轴上，AC垂直CB于点C，点A′在线段OA上，点B′在OB沿B方向的延长线上，A′C′垂直B′C′于点C′。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1820.jpg" }, { "index": 1821, "ocr_en": "There is a quadrilateral ABCD, angles ADC and DCB are obtuse angles, connect AC, point E is CB along the B direction of the extension line on the point, connect AE, BE for the auxiliary line.", "ocr_zh": "有四边形ABCD，角ADC和DCB是钝角,连接AC，点E是CB沿B方向延长线上的一点，连接AE、BE作辅助线。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1821.jpg" }, { "index": 1822, "ocr_en": "There is a quadrilateral ABCD, angle BAD is a right angle, DC perpendicular to CB at point C, connect AC, point E is a point on the extension line of CB along the direction of B, connect AE, BE for auxiliary line.", "ocr_zh": "有四边形ABCD，角BAD是直角，DC垂直CB于点C，连接AC，点E是CB沿B方向延长线上的一点，连接AE、BE作辅助线。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1822.jpg" }, { "index": 1823, "ocr_en": "Inside the equilateral triangle ABC there is a point O, connect AO, BO, CO. rotate the line segment OB around the point B counterclockwise by a certain angle to O′B, the points O and O′ are on both sides of AB, connect AO′, where the angle of rotation is greater than O and less than 90. connect OO′ as an auxiliary line.", "ocr_zh": "在等边三角形ABC内部有一点O，连接AO、BO、CO。将线段OB绕点B逆时针旋转一定的角度至O′B，点O和O′分别在AB的两侧,连接AO′，其中旋转角度大于O小于90。连接OO′作辅助线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1823.jpg" }, { "index": 1824, "ocr_en": "Inside the equilateral triangle ABC there is a point O, connect AO, BO, CO. Rotate the line segment OB around the point B counterclockwise by a certain angle to O′B, the points O and O′ are on both sides of AB, and connect AO′, where the angle of rotation is greater than O and less than 90. Rotate the triangle ABO around the point A counterclockwise by 60 degrees, and get the triangle ACO′′, and connect CO′′, AO′′, and OO′′. ACO′′, connect CO′′, AO′′ and OO′′ as auxiliary lines.", "ocr_zh": "在等边三角形ABC内部有一点O，连接AO、BO、CO。将线段OB绕点B逆时针旋转一定的角度至O′B，点O和O′分别在AB的两侧，连接AO′，其中旋转角度大于O小于90。将三角形ABO绕点A逆时针旋转60度，得到三角形ACO′′，连接CO′′、AO′′、OO′′作辅助线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1824.jpg" }, { "index": 1825, "ocr_en": "Inside the equilateral triangle ABC there is a point P, connect AP, BP, CP. rotate the triangle ABP counterclockwise by 60 degrees to get the triangle ACP′, connect AP′, PP′, CP′ to make the auxiliary lines.", "ocr_zh": "在等边三角形ABC内部有一点P，连接AP、BP、CP。将三角形ABP逆时针旋转60度，得到三角形ACP′，连接AP′、PP′、CP′做辅助线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1825.jpg" }, { "index": 1826, "ocr_en": "Inside the right triangle ABC there is a point O, connect AO, BO, CO, AC perpendicular to CB at point C. Rotate the triangle ABO clockwise around the point B by a certain angle to get the triangle A′O′B, the points O and O′ are on both sides of AB, AB is equal to A′B, where the angle of rotation is greater than O is less than 90, connect AA′, BA′, BO′, OO′, O′A′ for the Auxiliary lines.", "ocr_zh": "在直角三角形ABC内部有一点O，连接AO、BO、CO，AC垂直CB于点C。将三角形ABO绕点B顺时针旋转一定的角度，得到三角形A′O′B，点O和O′分别在AB的两侧，AB等于A′B，其中旋转角度大于O小于90，连接AA′、BA′、BO′、OO′、O′A′作辅助线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1826.jpg" }, { "index": 1827, "ocr_en": "Inside the equilateral triangle ABC there is a point P, connect AP, BP, CP. rotate the triangle ABP around the point A counterclockwise by 60 degrees to get the triangle ACD, connect AD, PD, CD as auxiliary lines.", "ocr_zh": "在等边三角形ABC内部有一点P，连接AP、BP、CP。将三角形ABP绕点A逆时针旋转60度，得到三角形ACD，连接AD、PD、CD作辅助线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1827.jpg" }, { "index": 1828, "ocr_en": "Inside the equilateral triangle ABC there is a point P connecting AP, BP, CP. Rotate triangle CBP counterclockwise by 60 degrees to obtain triangle ABP′ connecting AP′, PP′, BP′ as auxiliary lines.", "ocr_zh": "在等边三角形ABC内部有一点P，连接AP、BP、CP。将三角形CBP逆时针旋转60度，得到三角形ABP′，连接AP′、PP′、BP′做辅助线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1828.jpg" }, { "index": 1829, "ocr_en": "In the isosceles triangle ABC, AB is equal to AC, point E and F are points on the line segment BC, and point E is closer to point B, connect AE, AF, rotate the triangle ABE around the point A counterclockwise to get the triangle ACE ′, the angle of rotation that is the degree of angle BAC, connect AE ′, FE ′, CE ′ as auxiliary lines.", "ocr_zh": "在等腰三角形ABC中，AB等于AC，点E和F是线段BC上的点，且点E更靠近点B，连接AE、AF，将三角形ABE绕点A逆时针旋转得到三角形ACE′，旋转的角度即为角BAC的度数，连接AE′、FE′、CE′作辅助线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1829.jpg" }, { "index": 1830, "ocr_en": "Inside triangle ABC there is a point P, connect AP, BP, CP. rotate APC clockwise around point C by a certain angle to obtain triangle EDC, where the angle of rotation is greater than 0 and less than 90, AC is equal to CE, and points P and D are on each side of AC. Connect BE, CE, DE, DP, DC as auxiliary lines.", "ocr_zh": "在三角形ABC内部有一点P，连接AP、BP、CP。将APC绕点C顺时针旋转一定的角度，得到三角形EDC，其中旋转角度大于0小于90，AC等于CE，点P和D分别在AC的两侧。连接BE、CE、DE、DP、DC作辅助线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1830.jpg" }, { "index": 1831, "ocr_en": "Inside the equilateral triangle ABC there is a point P, connect AP, BP, CP. rotate the triangle ABP around the point A counterclockwise by 60 degrees to get the triangle ACP′, connect AP′, CP′ and connect PP′ as an auxiliary line.", "ocr_zh": "在等边三角形ABC内部有一点P，连接AP、BP、CP。将三角形ABP绕点A逆时针旋转60度，得到三角形ACP′，连接AP′、CP′，连接PP′做辅助线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1831.jpg" }, { "index": 1832, "ocr_en": "Inside right triangle ABC there is a point P, connect AP, BP, CP, AC perpendicular CB at point C. Rotate triangle ABP clockwise around point B by a certain angle to get triangle A′P′B, AB and AB′ are equal in length, point P and P′ are on both sides of AB, where the angle of rotation is greater than O less than 90, and connect PP′ as an auxiliary line.", "ocr_zh": "在直角三角形ABC内部有一点P，连接AP、BP、CP，AC垂直CB于点C。将三角形ABP绕点B顺时针旋转一定的角度，得到三角形A′P′B，AB与AB′长度相等，点P和P′分别在AB的两侧，其中旋转角度大于O小于90，连接PP′作辅助线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1832.jpg" }, { "index": 1833, "ocr_en": "There is a point O inside the acute triangle MNG, connect NO and GO as auxiliary lines. There is a point D on the outside of triangle MNG, with point O on each side of MG, connect DM, DG as auxiliary lines to get triangle MDG, connect ND as auxiliary line. Point E is a point inside triangle MDG, connect OE, OM, EM as auxiliary line to form triangle EOM, connect ED as auxiliary line. Pass point D to make DF auxiliary line perpendicular to NM along the M direction to extend the line auxiliary line, the foot of the pendant is F.", "ocr_zh": "在锐角三角形MNG内有一点O，连接NO、GO作辅助线。在三角形MNG外部有一点D，与点O分别在MG的两侧，连接DM、DG作辅助线，得到三角形MDG，连接ND作辅助线。点E是三角形MDG内一点，连接OE、OM、EM作辅助线，形成三角形EOM，连接ED作辅助线。过点D作DF辅助线垂直于NM沿M方向延长线辅助线，垂足为F。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1833.jpg" }, { "index": 1834, "ocr_en": "In the circle O there are two strings AB and CD, AB perpendicular to the CD at point E, point H and G are in AB and CD, connect OH, OA, OG for auxiliary line, OH perpendicular to AB at point H, OG perpendicular to the CD at point G.", "ocr_zh": "在圆O中有两条弦AB和CD,AB垂直CD于点E，点H和G分别在AB和CD上，连接OH、OA、OG作辅助线，OH垂直AB于点H，OG垂直CD于点G。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1834.jpg" }, { "index": 1835, "ocr_en": "In the circle O there is a string AB, M is a point on AB, connect OA, OM, extend MO along the direction of M to cross the circle O at point P, extend OM along the direction of O to cross the circle 0 at point Q, connect OP, MQ for auxiliary line.", "ocr_zh": "在圆O中有弦AB，M是AB上一点，连接OA、OM，延长MO沿M方向交圆O于点P，延长OM沿O方向交圆0于点Q，连接OP、MQ作辅助线。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1835.jpg" }, { "index": 1836, "ocr_en": "In circle O, P is a point on the chord AB, connect OP, over the point P make a chord intersecting circle O at points C and D, connect CP, connect DP, OD, OC as auxiliary line, OP perpendicular to CD at point P.", "ocr_zh": "在圆O中，P为弦AB上一点，连接OP，过点P作弦交圆O于点C和D，连接CP，连接DP、OD、OC作辅助线，OP垂直CD于点P。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1836.jpg" }, { "index": 1837, "ocr_en": "In the quadrilateral ABCD, DC is parallel to AB, connect AC, BD, point B, C, D in the circle with A as the centre and the length of AB as the radius, along the direction of A to extend BA to intersect the circle A at the point F, connect DF, AF for auxiliary line.", "ocr_zh": "在四边形ABCD中，DC平行于AB，连接AC、BD，点B、C、D在以A为圆心，AB长度为半径的圆上，沿A方向延长BA交圆A于点F，连接DF、AF作辅助线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1837.jpg" }, { "index": 1838, "ocr_en": "Points C and D are two points on a semicircle with AB as the diameter and O as the centre of the circle, point P is on OB, connect AB, CO, CP, PD and OD, and connect CD as an auxiliary line.", "ocr_zh": "点C、D是以AB为直径的半圆上的两点，圆心为O，点P在OB上，连接AB、CO、CP、PD、OD，连接CD作辅助线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1838.jpg" }, { "index": 1839, "ocr_en": "Inside the rectangle ABCD there is a centre O, point P is a non-centre point inside the rectangle, connect AP, BP, OP, connect OA, OP as auxiliary lines.", "ocr_zh": "在矩形ABCD内部有中心O，点P为矩形内部的非中心的一点，连接AP、BP、OP，连接OA、OP作辅助线。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1839.jpg" }, { "index": 1840, "ocr_en": "In the right triangle ABC, AC perpendicular CB at point C, point D is a point on BC, connect AD, over point B for BE perpendicular AD to cross the extension of AD in E, connect DE, EB. point O is the mid-point of AB, connect OE as an auxiliary line to cross the BC at point F, with O as the centre of the circle, the length of the OE as the radius of the circle for the dotted circle, points A, C, E, B common circle O.", "ocr_zh": "在直角三角形ABC中，AC垂直CB于点C，点D是BC上一点，连接AD，过点B作BE垂直AD交AD的延长线于E，连接DE、EB。点O是AB的中点，连接OE作辅助线交BC于点F，以O为圆心，OE长度为半径作虚线圆，点A、C、E、B共圆O。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1840.jpg" }, { "index": 1841, "ocr_en": "In the plane right-angle coordinate system in xOy, the circle G passes through the first, second and fourth quadrant and the origin, and intersects the y-axis positive semiaxis at the point B. The point A is on the circle and located in the first quadrant, and connects OA, OB, and AB to form the triangle ABO, and connects BG, and AG for the auxiliary line, and connects OG and extends it along the direction of the point G and intersects with AB for the auxiliary line.", "ocr_zh": "在平面直角坐标系中xOy中，圆G经过第一、二、四象限和原点，交y轴正半轴于点B，点A在圆上且位于第一象限，连接OA、OB、AB组成三角形ABO，连接BG、AG作辅助线，连接OG并沿G点方向延长与AB相交作辅助线。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1841.jpg" }, { "index": 1842, "ocr_en": "In rhombus ABCD, connect BD, point P is a point on BC, connect AP, connect auxiliary line AC to intersect BD at point O, point E is on AP, point G is on BD, connect EG, GP, EG is the perpendicular bisector of AP, connect OE, AG to make the auxiliary line.", "ocr_zh": "在菱形ABCD中，连接BD，点P是BC上一点，连接AP，连接辅助线AC交BD于点O，点E在AP上，点G在BD上，连接EG、GP，EG是AP的垂直平分线，连接OE、AG做辅助线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1842.jpg" }, { "index": 1843, "ocr_en": "There is a point P in the isosceles right triangle ABC, AC is equal to BC, AC is perpendicular to BC at point C, connect AP, BP, CP, the point P is on the dashed circle O with AB as the chord, connect OP, OC, OA, OB for the auxiliary line.", "ocr_zh": "在等腰直角三角形ABC内有一点P，AC等于BC，AC垂直BC于点C，连接AP、BP、CP，点P在以AB为弦的虚线圆O上，连接OP、OC、OA、OB作辅助线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1843.jpg" }, { "index": 1844, "ocr_en": "Circle O is the outer dashed circle of triangle ABC, connect OA, OB, OC as an auxiliary line, point M is a point on the chord AB, connect CM as an auxiliary line perpendicular to AB at point M, CM passes through the centre of the circle O.", "ocr_zh": "圆O是三角形ABC的外接虚线圆，连接OA、OB、OC作辅助线，点M是弦AB上一点，连接CM作辅助线垂直AB于点M，CM经过圆心O。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1844.jpg" }, { "index": 1845, "ocr_en": "In isosceles triangle ABC, AB is equal to AC, point H is on BC, connect AH to make an auxiliary line perpendicular to BC at point H. Take A as the centre of the circle and the length of AB as the radius to make a dashed circle, point D is the intersection of the extension of the line of AH along the direction of A and the circle A, connect BD and CD, and connect AD to make an auxiliary line.", "ocr_zh": "在等腰三角形ABC中，AB等于AC，点H在BC上，连接AH作辅助线垂直BC于点H，以A为圆心，AB的长为半径作虚线圆，点D是AH沿A方向延长线与圆A的交点，连接BD、CD，连接AD作辅助线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1845.jpg" }, { "index": 1846, "ocr_en": "Points A, B, C and D are on circle O, connect CD, AD, OA and BC, OA is perpendicular to BC at point E. Points A and D are on both sides of BC, connect CO as an auxiliary line.", "ocr_zh": "点A、B、C、D在圆O上，连接CD、AD、OA、BC，OA垂直BC于点E，点A和点D分别在BC两侧，连接CO作辅助线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1846.jpg" }, { "index": 1847, "ocr_en": "In the plane right-angle coordinate system in xOy, the circle M passes through the first and second quadrants, is tangent to the x-axis positive semiaxis at point A, and intersects with the y-axis at points B and C, respectively, with point B below point C. Connect BM, ON, and AM to make an auxiliary line, and point H is on BC; connect MH to make an auxiliary line perpendicular to BC at point H.", "ocr_zh": "在平面直角坐标系中xOy中，圆M经过第一和第二象限，与x轴正半轴相切于点A，与y轴分别交于B、C点，B点在C点下方，连接BM、ON、AM作辅助线，点H在BC上，连接MH作辅助线垂直BC于点H。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1847.jpg" }, { "index": 1848, "ocr_en": "In the isosceles triangle ABC, AB is equal to AC, with point A as the centre of the circle, a certain length for the radius of the arc, respectively, intersect AB, AC at point E, F, and then respectively with point E, F as the centre of the circle, with more than half the length of EF as the radius of the arc to intersect at point H, point H in the point of point E and F below, make the ray AH. respectively, with point A, B as the centre of the circle, with more than half the length of the arc to intersect at point M, N, and then a straight line MN intersect AH at point 0, point M is above AB, point N is below AB. N, make a straight line MN to intersect ray AH at point 0, point M is above AB, point N is below AB. Point M is above AB, and point N is below AB. Take point 0 as the centre of the circle, and the length of OA as the radius of the circle. Points A, B and C are on circle O.", "ocr_zh": "在等腰三角形ABC中，AB等于AC，以点A为圆心，一定的长度为半径作弧，分别交AB, AC于点E，F，再分别以点E，F为圆心，以大于一半EF的长度为半径作弧相交于点H，点H在点E和F的下方，作射线AH。分别以点A、B为圆心，大于一半AB的长度为半径作弧相交于点M、N，作直线MN交射线AH于点0，点M在AB的上方，点N在AB的下方。以0点为圆心，OA的长度为半径作圆。点A、B、C均在圆O上。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1848.jpg" }, { "index": 1849, "ocr_en": "In circle O, line CD is the diameter, line AB is the chord, AB is perpendicular to CD at point M , connect OA as an auxiliary line.", "ocr_zh": "在圆O中，线段CD是直径，线段AB是弦，AB垂直CD于点M ，连接OA作辅助线。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1849.jpg" }, { "index": 1850, "ocr_en": "There is an isosceles triangle ABC connected to the circle O, AB is equal to AC, point D is a point on the arc AC, connect AD, BD, connect CD as an auxiliary line, point E is the extension of AD along the direction of D and BC along the direction of the extension of the line of the intersection of the C direction, connected to DE, CE. point H in the BD, connected to the AH, AH perpendicular to the BD in the point H. point M is a point on the BC, connected to the AM as an auxiliary line, AM passes through the centre of O and AM perpendicular to BC in point M. point N is a point on the BD, and point B respectively on both sides of AM, and AN as an auxiliary line. Point N is a point on BD, and point B are on both sides of AM, connect AN as an auxiliary line. Angle ACD is angle 1 and angle ABD is angle 2.", "ocr_zh": "有等腰三角形ABC内接于圆O，AB等于AC，点D是弧AC上一点，连接AD、BD，连接CD作辅助线，点E是AD沿D方向延长线与BC沿C方向延长线的交点，连接DE、CE。点H在BD上，连接AH，AH垂直BD于点H。点M是BC上一点，连接AM作辅助线，AM经过圆心O且AM垂直BC于点M。点N是BD上一点，与点B分别在AM的两侧，连接AN作辅助线。角ACD为角1，角ABD为角2。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1850.jpg" }, { "index": 1851, "ocr_en": "There are two parallel line segments AB and EF of lengths 1 and 2 respectively and their perpendicular distance is 1, endpoints A and B are the vertices of a small square of side length 1 and A is to the left of point B, endpoints E and F are the vertices of a large square of side length 2 and E is to the left of point F. The two squares are side by side, point B is the midpoint of one side of the large square, AB and BE are both of length 1, and the connection AF intersects the line segment BE at point C,. The shaded triangle ABC is obtained.", "ocr_zh": "有两条平行的线段AB和EF长度分别为1和2，且两者的垂直距离为1，端点A、B是边长为1的小正方形顶点且A在B点左侧，端点E、F是边长为2的大正方形顶点且E在F点左侧，两个正方形并排，点B是大正方形一条边的中点，AB和BE长度均为1，连接AF交线段BE于点C，得到阴影三角形ABC。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1851.jpg" }, { "index": 1852, "ocr_en": "In the plane right-angle coordinate system in xOy, point A is located in the first quadrant of the inverse function image of a point, point B is located in the second quadrant of the inverse function image of a point, connecting AB, A0, B0 to get the triangle ABO, AO perpendicular to the BO at the point O, through the point B as BC auxiliary line perpendicular to the x-axis at the point C, through the point A as AD auxiliary line perpendicular to the x-axis at the point D.", "ocr_zh": "在平面直角坐标系中xOy中，点A是位于第一象限的反比例函数图像上一点，点B是位于第二象限的反比例函数图像上一点，连接AB、A0、B0得到三角形ABO，AO垂直BO于点O，过点B作BC辅助线垂直x轴于点C，过点A作AD辅助线垂直x轴于点D。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1852.jpg" }, { "index": 1853, "ocr_en": "There is a right triangle ABC, AB perpendicular BC at point B, point P is a point on AC, point M and point P are on both sides of the line segment AB, point N and point P are on both sides of the line segment BC, connect MP, NP, MN to get the triangle MNP, MP perpendicular to NP at point P. MP intersects AB at point E, PN intersects BC at point F. Point Q and point R are on AB and BC respectively, connect PQ and PR. PQ is perpendicular to AB and PR is perpendicular to BC.", "ocr_zh": "有直角三角形ABC，AB垂直BC于点B，点P是AC上一点，点M与点P分别在线段AB的两侧，点N与点P分别在线段BC的两侧，连接MP、NP、MN得到三角形MNP，MP垂直NP于点P。MP交AB于点E，PN交BC于点F。点Q和点R分别在AB、BC上，连接PQ、PR作辅助线，PQ垂直AB于点Q，PR垂直BC于点R。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1853.jpg" }, { "index": 1854, "ocr_en": "Inside the rhombus ABCD there is a point G, point H is a point on the side AD, point E is a point on the side AB, connect GH, GE to get the rhombus AHGE, connect CG, extend HG to cross BC at point M, extend EG to cross DC at point N, connect MN to cross GC at point O, connect AG, GN, GM, GM for the auxiliary line.", "ocr_zh": "在菱形ABCD内部有一点G，点H是边AD上一点，点E是边AB上一点，连接GH、GE得到菱形AHGE，连接CG，延长HG交BC于点M，延长EG交DC于点N，连接MN交GC于点O，连接AG、GN、GM、GM作辅助线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1854.jpg" }, { "index": 1855, "ocr_en": "In the rhombus ABCD interior point H and point G, point E in the rhombus ABCD exterior, and point G are on both sides of the side AB, connect AH, GH, GE, AE to get rhombus AHGE, connect AC, AG for auxiliary line, connect DH, CG, BE.", "ocr_zh": "在菱形ABCD内部有点H和点G，点E在菱形ABCD外部，与点G分别在边AB的两侧，连接AH、GH、GE、AE得到菱形AHGE，连接AC、AG作辅助线，连接DH、CG、BE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1855.jpg" }, { "index": 1856, "ocr_en": "Inside the rectangle ABCD point H and point G, point E in the rectangle ABCD outside, and point G are on both sides of the side AB, connect AH, GH, GE, AE to get the rectangle AHGE, connect AC, AG for auxiliary line, connect DH, CG, BE.", "ocr_zh": "在矩形ABCD内部有点H和点G，点E在矩形ABCD外部，与点G分别在边AB的两侧，连接AH、GH、GE、AE得到矩形AHGE，连接AC、AG作辅助线，连接DH、CG、BE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1856.jpg" }, { "index": 1857, "ocr_en": "In the plane right-angle coordinate system xOy, point A is on the positive half-axis of the y-axis, point B is on the positive half-axis of the x-axis, connect AB, with point A as the centre of the circle, the length of AB as the radius of the dotted circle, intersecting the x-axis at the point P1 and the point B, the point P1 in the negative half-axis of the x-axis, intersection of positive half-axis of the y-axis at the point P2, intersection of the negative half-axis of the y-axis at the point P3. with the point B as the centre of the circle, the length of AB as the radius of the dotted circle, intersecting the x-axis at the points P6 and P5, point P5 is on the positive half-axis of the x-axis, intersect the y-axis at points A and P4, point P4 is on the negative half-axis of the y-axis, make the perpendicular bisector auxiliary line of the line segment AB intersecting the positive half-axis of the y-axis at point P7, intersecting the x-axis at point P8, the point P1 coincides with the point P6 and the point P8, and labelled on the left side of the point P1 (P6, P8).", "ocr_zh": "在平面直角坐标系xOy中，点A在y轴正半轴上，点B在x轴正半轴上，连接AB，以点A为圆心，AB的长度为半径作虚线圆，交x轴于点P1和点B，点P1在x轴负半轴上，交y轴正半轴于点P2，交y轴负半轴于点P3。以点B为圆心，AB的长度为半径作虚线圆，交x轴于点P6和P5，点P5在x轴正半轴上，交y轴于点A和P4，点P4在y轴负半轴上，作线段AB的垂直平分线辅助线交y轴正半轴于点P7，交x轴于点P8，点P1和点P6以及点P8重合，在点P1左侧标注（P6，P8）。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1857.jpg" }, { "index": 1858, "ocr_en": "In the plane right-angle coordinate system xOy, there is a primary function passing through the first, second and fourth quadrants, intersecting the y-axis proper semiaxis at point A, and intersecting the x-axis proper semiaxis at point B. There is a parabolic function opening downward, with the vertex in the first quadrant, intersecting the y-axis at point A. With point B as the centre of the circle, and the length of AB as the radius make a dashed circle, intersecting the parabola at point C, point M, and point N. Point C is in the first quadrant, and the auxiliary lines connecting AC and BC are made. Point M and point N are on the x-axis and to the left of point M and point N. Make a vertical auxiliary line to the x-axis through point B, with the foot of the line at B.", "ocr_zh": "在平面直角坐标系xOy中，有一次函数经过第一、二和四象限，交y轴正半轴于点A，交x轴正半轴于点B。有开口向下的抛物线函数，顶点在第一象限，交y轴于点A。以点B为圆心，AB的长度为半径作虚线圆，交抛物线于点C、点M和点N。点C在第一象限，连接AC和BC作辅助线，点M和点N在x轴上且点M和N的左侧。过点B作x轴的垂线辅助线，垂足为B。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1858.jpg" }, { "index": 1859, "ocr_en": "In the plane right-angle coordinate system xOy, there is a primary function passing through the first, second and third quadrant, the inverse function is located in the first and third quadrant, the intersection of the images of the primary function and the inverse function are the point A (3,4) is located in the first quadrant, and the point B (n, -1) is located in the third quadrant. Connect OA, the vertical bisector of OA for the auxiliary line intersecting the x-axis half-axis at point C, point C in the point A right below, connect AC for the auxiliary line. Make a vertical auxiliary line through point A to intersect the x-axis at point D. AD is perpendicular to the x-axis at point D.", "ocr_zh": "在平面直角坐标系xOy中，有一次函数经过第一、二和三象限，反比例函数位于第一和三象限，一次函数和反比例函数图像的交点分别是点A（3,4）位于第一象限，和点B（n，-1）位于第三象限。连接OA，作OA的垂直平分线辅助线交x轴正半轴于点C，点C在点A右下方，连接AC作辅助线。过点A作垂线辅助线交x轴于点D，AD垂直x轴于点D。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1859.jpg" }, { "index": 1860, "ocr_en": "In the plane right-angle coordinate system xOy, point A is in the first quadrant, point B is in the fourth quadrant, connect AB. through the point A for AB's plumb line auxiliary line intersecting the negative half-axis of the x-axis at the point C1, the positive half-axis of the y-axis at the point C2, the foot of the plumb line is A, connected to the auxiliary line BC1, BC2. Point B as AB for the vertical line of the auxiliary line intersecting the positive semi-axis of the x-axis at point C3, the negative semi-axis of the y-axis at point C4, connected to AC3, AC4 for the auxiliary line. Take the line segment AB as the diameter to draw the dashed circle, tangent to the y-axis, the tangent point is C5, and the positive half-axis of the x-axis intersects at C6 and C7, the point C6 is closer to the point O, connect C5A, C5B, C6A, C6B, C7A, C7B as the auxiliary line.", "ocr_zh": "在平面直角坐标系xOy中，点A在第一象限，点B在第四象限，连接AB。过点A作AB的垂线辅助线交x轴负半轴于点C1，y轴正半轴于点C2，垂足是A，连接BC1、BC2作辅助线。过点B作AB的垂线辅助线交x轴正半轴于点C3，y轴负半轴于点C4，连接AC3、AC4作辅助线。以线段AB为直径画虚线圆，与y轴相切，切点为C5，与x轴正半轴交于C6和C7，点C6更靠近点O，连接C5A、C5B、C6A、C6B、C7A、C7B作辅助线。", "class": "Analytic Geometry", "difficulty": 3, "image_path": "fig_1860.jpg" }, { "index": 1861, "ocr_en": "In the plane right-angle coordinate system xOy, there is an opening up the quadratic function graph through the four quadrants, the vertex in the fourth quadrant, the parabola and the x-axis intersected at the point A and B, the point A in the x-axis on the positive semiaxis, the point B in the x-axis on the negative semiaxis, and the negative semiaxis of the y-axis intersected in the point C, connecting the AC, over the point C for CP perpendicular to the AC intersecting the quadratic image of the function for the point P, the foot of the pituitary is C, connecting the PC, AP.", "ocr_zh": "在平面直角坐标系xOy中，有开口向上的二次函数图象经过四个象限，顶点在第四象限，抛物线与x轴交于点A和B，点A在x轴正半轴上，点B在x轴负半轴上，与y轴负半轴交于C点，连接AC，过点C作CP垂直AC交二次函数图像为点P，垂足是C，连接PC、AP。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1861.jpg" }, { "index": 1862, "ocr_en": "In the plane right-angle coordinate system xOy, point B is on the negative half-axis of the x-axis, point C is on the positive half-axis of the x-axis, OC is equal to OB, and point A is on the positive half-axis of the y-axis, connecting AB and AC. Make an auxiliary line BH through point B to intersect AC at point H, and to intersect the positive half-axis of the y-axis at point D. BH is perpendicular to AC at point H. Connect CD.", "ocr_zh": "在平面直角坐标系xOy中，点B在x轴负半轴上，点C在x轴正半轴上，OC等于OB，点A在y轴正半轴上，连接AB、AC。过点B作辅助线BH交AC于点H，交y轴正半轴于点D，BH垂直AC于点H。连接CD。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1862.jpg" }, { "index": 1863, "ocr_en": "In right triangle ABC, BA is perpendicular to AC at point A, point A is above BC, point B is to the left of C, and the length of AC is greater than that of AB. point D is a point on the line segment BC, connect to AD. make the symmetry point A′ of point A about BC, and connect to A′A, A′D as an auxiliary line. aA′ intersects BC at point H. make the auxiliary line DE perpendicular to AC at point E, and point E is on AC.", "ocr_zh": "在直角三角形ABC中，BA垂直AC于点A，点A在BC的上方，点B在C的左侧，AC长度大于AB。点D是线段BC上一点，连接AD。作点A关于BC的对称点A′，连接A′A、A′D作辅助线。AA′交BC于点H。过点D作DE辅助线垂直AC于点E，点E在AC上。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1863.jpg" }, { "index": 1864, "ocr_en": "In the plane right-angle coordinate system xOy, there is a quadratic function image with the opening down through four quadrants, the vertex C is located in the first quadrant, intersecting the x-axis negative semiaxis at point A, the x-axis positive semiaxis at point B, connecting BC. parabola axis of symmetry intersects the x-axis positive semiaxis at point D. Point P is a point on the line segment CD, through the point P for the x-axis of the parallel line intersecting the line segment BC at point E, through the point E for the perpendicular PE for the parabola at point F. The pendant is E. Connect CF and BF. The foot is E. Connect CF and BF. mark triangle BCF in black. connect AC to make an auxiliary line. Make a PG auxiliary line perpendicular to segment AC at point G. Point G is on segment AC. Make a BH auxiliary line perpendicular to segment AC at point H. Point H is on segment AC. Points G and H are in the first quadrant and point G is to the left and below H.", "ocr_zh": "在平面直角坐标系xOy中，有开口向下的二次函数图像经过四个象限，顶点C位于第一象限，交x轴负半轴于点A，x轴正半轴于点B，连接BC。抛物线对称轴交x轴正半轴于点D。点P是线段CD上一点，过点P作x轴的平行线交线段BC于点E，过点E作EF垂直PE交抛物线于点F，垂足是E，连接CF、BF。用黑色标记三角形BCF。连接AC作辅助线。过点P作PG辅助线垂直线段AC于点G，点G在线段AC上。过点B作BH辅助线垂直线段AC于点H，点H在线段AC上。点G和H在第一象限且点G在H左下方。", "class": "Analytic Geometry", "difficulty": 3, "image_path": "fig_1864.jpg" }, { "index": 1865, "ocr_en": "In a right-angled triangle ABC, AC is perpendicular to CB at point C. Point A is above CB and point C is to the left of B. Make a circle with point C as the centre and not exceeding the lengths of AC and CB as the radius. Point P is on the circle and also inside the triangle ABC, connect AP and BP. connect CP as an auxiliary line, and point M is inside the circle and on AC, connect PM and BM as auxiliary lines.", "ocr_zh": "在直角三角形ABC中，AC垂直CB于点C，点A在CB上方，点C在B的左侧。以点C为圆心，不超过AC和CB的长度为半径作圆。点P在圆上也在三角形ABC内，连接AP、BP。连接CP作辅助线，点M在圆的内部且在AC上，连接PM、BM作辅助线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1865.jpg" }, { "index": 1866, "ocr_en": "In the plane rectangular coordinate system xOy, a circle is made with O as the centre and a certain length as the radius. Point P is a point on the circle and lies in the first quadrant, point A is on the positive semiaxis of the x-axis and outside the circle, and point B is in the first quadrant and to the right of point P. Connect PB and PA. Connect PB and PA. Point K is on the half-axis of the x-axis and inside the circle, connect OP, PK and BK as auxiliary lines.", "ocr_zh": "在平面直角坐标系xOy中，以O为圆心，一定的长度为半径作圆。点P是圆上一点，位于第一象限，点A在x轴正半轴上且在圆外，点B在第一象限且位于点P右上方。连接PB、PA。点K在x轴正半轴上且在圆内，连接OP、PK、BK作辅助线。 ", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1866.jpg" }, { "index": 1867, "ocr_en": "There is a square ABCD, with vertex B as the centre of the circle, less than the length of AB as the radius of the circle, point P is a point on the circle and in the square, connect DP, CP. point G is a point on BC and in the circle. Connect BP, DG, PG as an auxiliary line, extend DG along the direction of G and cross the circle at point P1, connect GP1 as an auxiliary line.", "ocr_zh": "有正方形ABCD，以顶点B为圆心，小于AB的长度为半径作圆，点P是圆上一点且在正方形内，连接DP、CP。点G是BC上一点且在圆内。连接BP、DG、PG作辅助线，沿G方向延长DG交圆于点P1，连接GP1作辅助线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1867.jpg" }, { "index": 1868, "ocr_en": "In the plane right-angle coordinate system xOy, there is a primary function passing through the first, second and fourth quadrants, intersecting the y-axis semiaxis at point C and the x-axis semiaxis at point A. There is a parabola with an upward point, the vertex is in the fourth quadrant and passes through points A and C. Another intersection with the x-axis semiaxis is at point B, which is on the right side of A, and connects to BC. The point M is on the parabola and below the x-axis between points A and B. Connect AM to MH, and make an auxiliary line MH perpendicular to the x-axis. Connect AM, and make an auxiliary line MH through point M to intersect the x-axis at point H. MH is perpendicular to the x-axis at point H.", "ocr_zh": "在平面直角坐标系xOy中，有一次函数经过第一、二、四象限，交y轴正半轴于点C，x轴正半轴于点A。有开口向上的抛物线，顶点在第四象限且经过点A和C，与x轴正半轴的另一个交点为点B，点B在A右侧，连接BC。点M在抛物线上且在x轴下方，位于点A和B之间。连接AM，过点M作MH辅助线交x轴于点H，MH垂直x轴于点H。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1868.jpg" }, { "index": 1869, "ocr_en": "In the plane right-angle coordinate system xOy, point B is on the negative half-axis of the x-axis, point C is on the positive half-axis of the x-axis, and point A is in the first quadrant, connecting AB and AC. point F is the mid-point of the line segment BC and is located on the positive half-axis of the x-axis, so connect AF to make the straight line L.", "ocr_zh": "在平面直角坐标系xOy中，点B在x轴负半轴上，点C在x轴正半轴上，点A在第一象限，连接AB、AC。点F是线段BC的中点且位于x轴正半轴，连接AF作直线L。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_1869.jpg" }, { "index": 1870, "ocr_en": "In the plane rectangular coordinate system xOy, the image of a primary function passing through the origin passes through the first and third quadrants. A point C on the image of this function lies in the first quadrant. There is a primary function whose image passes through point C and passes through the first, second and fourth quadrants. Points A and B are located in the first quadrant, to the left and right of OC respectively, and both are to the lower left of point C. Connect OA, OB, AC, and BC. Connect AB and intersect OC at point G, which is the midpoint of the line segments AB and OC.", "ocr_zh": "在平面直角坐标系xOy中，经过原点的一次函数图像经过第一和三象限。该函数图像上有一点C位于第一象限。有一次函数图像经过点C且经过第一、二和四象限。点A和B位于第一象限，分别位于OC的左侧和右侧且均在点C的左下方，连接OA、OB、AC、BC。连接AB交OC于点G，点G是线段AB和OC的中点。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1870.jpg" }, { "index": 1871, "ocr_en": "In triangle ABC, angles B and C are acute angles, point B is to the left of C, point A is above BC, point D is on BC and is the midpoint of BC, connect AD, point E is the point on the line segment CD, connect AE, AE is perpendicular to BC at point E.", "ocr_zh": "在三角形ABC中，角B和角C是锐角，点B在C的左侧，点A在BC上方，点D在BC上且是BC的中点，连接AD，点E是线段CD上的点，连接AE，AE垂直BC于点E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1871.jpg" }, { "index": 1872, "ocr_en": "In the plane right-angle coordinate system xOy, there is a primary function that passes through the first, second and fourth quadrants and intersects the y-axis semiaxis at point C and the x-axis semiaxis at point A. There is a parabola with an upward pointing vertex in the fourth quadrant that passes through points A and C. The other intersection with the x-axis semiaxis is point B, which is on the right side of A. The parabola has two points of intersection with AB. A circle is made with point B as the centre and a length less than AB as the radius, which has two points of intersection with the parabola. Point P is a point on the circle and is inside the parabola, located in the first quadrant, connecting PC and PA. point D is on the positive semiaxis of the x-axis, located inside the circle, to the left of point B. Point D is on the positive semiaxis of the x-axis, located inside the circle, to the right of point B. Connect DC, PD and PB as auxiliary lines.", "ocr_zh": "在平面直角坐标系xOy中，有一次函数经过第一、二、四象限，交y轴正半轴于点C，x轴正半轴于点A。有开口向上的抛物线顶点在第四象限且经过点A和C，与x轴正半轴的另一个交点为点B，点B在A右侧。以点B为圆心，小于AB的长度为半径作圆，与抛物线有两个交点。点P是圆上一点且在抛物线内部，位于第一象限，连接PC和PA。点D在x轴正半轴上，位于圆内，在点B的左侧。连接DC、PD、PB作辅助线。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1872.jpg" }, { "index": 1873, "ocr_en": "In trapezium ABCD, AB is parallel to DC and above DC, point A is on the left side of B, point D is on the left side of C. The length of AB is less than CD. point M is the midpoint of AD, connect BM as an auxiliary line, extend MB along M to intersect the extension line of line CD in the direction of D at point N, connect MN as an auxiliary line, connect DN as a ray auxiliary line, point G is the midpoint of line CN between point C and D, connect BG as a ray auxiliary line. G is the midpoint of the line segment CN, located between the points C and D, connect BG as the ray auxiliary line.", "ocr_zh": "在梯形ABCD中，AB平行DC且在DC上方，点A在B左侧，点D在C左侧。AB长度小于CD。点M是AD的中点，连接BM作辅助线，沿M方向延长MB交线段CD沿D方向延长线于点N，连接MN作辅助线，连接DN作射线辅助线，点G是线段CN的中点，位于点C和D之间，连接BG作射线辅助线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1873.jpg" }, { "index": 1874, "ocr_en": "In the plane right-angle coordinate system xOy, there is a primary function y = 4/3x + m image intersecting the negative half-axis of the x-axis at point A, the positive half-axis of the y-axis at point B. The point D is on the line segment OB, the point E is on the line segment AB, and the line is made by connecting DE. Point F is on line BD, connect EF as an auxiliary line, EF perpendicular to BD in point F.", "ocr_zh": "在平面直角坐标系xOy中，有一次函数y=4/3x+m图像交x轴负半轴于点A，y轴正半轴于点B。点D在线段OB上，点E在线段AB上，连接DE作直线。点F在线段BD上，连接EF作辅助线，EF垂直BD于点F。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1874.jpg" }, { "index": 1875, "ocr_en": "In the plane right-angle coordinate system xOy, point B is on the y-axis of the positive semi-axis, point C is on the x-axis of the positive semi-axis, point A and D are the first quadrant of the point, are located in the left side of the point C, and the point A is in the upper left side of D. Connect AB, AD and DC to form a pentagon ABOCD. Connect AB, AD, DC to form a pentagon ABOCD. connect OA as a ray auxiliary line, point B as a line BM auxiliary line parallel to OA to the x-axis at point M, connect the line AC as an auxiliary line, point D as a line DN auxiliary line parallel to AC to the x-axis at point N. Point F is the mid-point of the line MN, which is located in the positive half-axis of the x-axis, and connect AF as a straight line auxiliary line. Point H is on the line FC, connect AH as an auxiliary line, AH is perpendicular to the x-axis at point H, and point A is above H.", "ocr_zh": "在平面直角坐标系xOy中，点B在y轴正半轴上，点C在x轴正半轴上，点A和D是第一象限的点，均位于点C左侧，且点A在D的左上方。连接AB、AD、DC组成五边形ABOCD。连接OA作射线辅助线，过点B作线段BM辅助线平行于OA交x轴于点M，连接线段AC作辅助线，过点D作线段DN辅助线平行于AC交x轴于点N。点F是线段MN的中点，位于x轴正半轴，连接AF作直线辅助线。点H在线段FC上，连接AH作辅助线，AH垂直x轴于点H，点A在H上方。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_1875.jpg" }, { "index": 1876, "ocr_en": "In two parallel line segments AB and CD, AB is above CD, point A is to the left of B, and point C is to the right of D. Point B is directly above C, point A is directly above D. Point E is between AB and CD and lies to the lower left of B and to the upper left of C. Connect BE and EC to form a section of the polyline convex to the left. Make an auxiliary line EF parallel to AB through point E. Point F is to the right of point E. Angle BEF is angle 1 and angle CEF is angle 2. Label angles ABE, DCE, 1 and 2 with arcs.", "ocr_zh": "在平行的两条线段AB与CD中，AB在CD的上方，点A在B的左侧，点C在D的右侧。点B在C的正上方，点A在D的正上方，点E在AB和CD之间，且位于B的左下方、C的左上方。连接BE和EC，构成一段向左凸的折线。过点E作辅助线EF平行于AB，点F在点E的右侧。角BEF为角1，角CEF为角2。用弧线标记角ABE、角DCE、角1和角2。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1876.jpg" }, { "index": 1877, "ocr_en": "In two parallel red line segments AB and CD, AB is above CD, point A is to the left of B, point C is to the right of D. Points E and F are between AB and CD, with point E below the left of B and above the left of F, and point F above the right of D. Connect the line segments BE, EF, and DF to form a section of the polyline from top to bottom that is convex to the left and then to the right. Mark angles ABE, BEF, EFD, and FDC with red arcs. mark segments AB and CD in red.", "ocr_zh": "在平行的两条红色线段AB与CD中，AB在CD的上方，点A在B的左侧，点C在D的右侧，点E和F在AB和CD之间，点E在B的左下方、F的左上方，点F在D的右上方。连接线段BE、EF和DF，从上至下构成一段先向左凸再向右凸的折线。用红色弧线标记角ABE、角BEF、角EFD和角FDC。用红色标记线段AB和CD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1877.jpg" }, { "index": 1878, "ocr_en": "In two parallel red line segments AB and CD, AB is above CD, point A is to the left of B, and point C is to the right of D. Points E1, E2, E3, En, F1, F2, and Fn-1 are points between the parallel lines AB and CD, and points E1, E2, E3, and En are arranged in order from top to bottom along the vertical, and points F1, F2, and Fn-1 are arranged in order from top to bottom along the vertical, and points E1, E2, E3, En are to the left of points B, F1, F2, Fn-1 and C. Point E1 is above F1 and points F1, F2, Fn-1 are above points E2, E3, En respectively. Connect BE1, E1F1, E2F1, E2F2, E3F2, EnFn-1, EnC, and connect E3Fn-1 as an auxiliary line, and these segments form a section of a fold. Mark with red arcs the angles ABE1, BE1F1, E1F1E2, F1E2F2, E2F2E3, Fn-1EnC and EnCD, which are all acute angles. Mark the line segments AB and CD in red.", "ocr_zh": "在平行的两条红色线段AB与CD中，AB在CD的上方，点A在B的左侧，点C在D的右侧，点E1、E2、E3、En、F1、F2、Fn-1是平行线AB和CD间的点，点E1、E2、E3、En沿竖直方向由上至下依次排列，点F1、F2、Fn-1沿竖直方向由上至下依次排列，点E1、E2、E3、En均在点B、F1、F2、Fn-1和C的左侧，点E1在F1上方，且点F1、F2、Fn-1分别在点E2、E3、En的上方，连接BE1、E1F1、E2F1、E2F2、E3F2、EnFn-1、EnC，连接E3Fn-1作辅助线，这些线段构成一段折线。用红色弧线标记角ABE1、角BE1F1、角E1F1E2，角F1E2F2、角E2F2E3、角Fn-1EnC和角EnCD，这些角均为锐角。用红色标记线段AB和CD。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1878.jpg" }, { "index": 1879, "ocr_en": "There is a point C between parallel lines DB and EA, DB is above EA, point D is to the left of B, point B is to the upper right of A, and point E is to the left of A. Connect AC, BC, and AB to form triangle ABC, with AC perpendicular to BC at point C.", "ocr_zh": "在平行的直线DB和EA之间有一点C，DB在EA的上方，点D在B的左侧，点B在A的右上方，点E在A的左侧，连接AC、BC、AB组成三角形ABC，AC垂直BC于点C。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1879.jpg" }, { "index": 1880, "ocr_en": "There are parallel lines a and b, with line a above b. Points D and A are on line a, and point D is to the left of A. Points E and C are on line b, and point E is to the left of C. There is a point B between the lines a and b. Connect AC, BC, and AB to form an acute triangle ABC. Mark the angle DAB as angle 1 and the angle BCE as angle 2 with an arc, and both angles 1 and 2 are acute.", "ocr_zh": "有平行线a和b，直线a在b的上方，点D和A在直线a上，且点D在A的左侧，点E和C在直线b上，且点E在C的左侧。在直线a和b之间有一点B，连接AC、BC、AB组成锐角三角形ABC，用弧线标记角DAB为角1、角BCE为角2，角1和2均为锐角。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1880.jpg" }, { "index": 1881, "ocr_en": "There are parallel lines a and b, the line a is above b. Points D and A are on the line a, and point D is to the left of A. Points E and C are on the line b, and point E is to the left of C. Between the lines a and b there is a point B. Connect AC, BC and AB to form an acute triangle ABC, and mark the angle DAB with an arc as angle 1 and the angle BCE as angle 2. Make an auxiliary line c parallel to a and b through the point B. Divide the angle ABC into angles 3 and 4 and mark them with an arc, with the angle 3 being nearer to the line a. Mark the line segments AD, AB, BC, and EC in red and the line c in yellow.", "ocr_zh": "有平行线a和b，直线a在b的上方，点D和A在直线a上，且点D在A的左侧，点E和C在直线b上，且点E在C的左侧。在直线a和b之间有一点B，连接AC、BC、AB组成锐角三角形ABC，用弧线标记角DAB为角1、角BCE为角2。过点B作辅助线直线c平行于a和b，将角ABC分割为角3和角4并用弧线标记，角3更靠近直线a。标记线段AD、AB、BC、EC为红色，直线c为黄色。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1881.jpg" }, { "index": 1882, "ocr_en": "There are parallel lines L1 and L2, L1 above L2, between L1 and L2 in the vertical direction from top to bottom there are two points of inflection, respectively, connected with a point on L1 and a point on L2, and then connect the two points of inflection, by three line segments connected in turn to form the folding line, first to the left and then to the right convex. Angle 1 is the angle between the line L1 and the beginning of the fold line, angle α and angle β are the interior angles of the fold line, angle α is closer to L1, and angle 2 is the angle between the line L2 and the end of the fold line. Mark angles 1, 2, α, and β with arcs, where angle 1 is acute and angle 2 is obtuse.", "ocr_zh": "有平行线L1和L2，L1在L2上方，在L1和L2之间在竖直方向从上至下有两个拐点，分别与L1上某点和L2上某点连接，再连接两个拐点，由三条线段依次相连形成折线，先向左凸再向右凸。角1是直线L1与折线起始段的夹角，角α、角β是是折线转折处的内角，角α更靠近L1，角2是直线L2与折线末端段的夹角。用弧线标记角1、角2、角α、角β，其中角1是锐角，角2是钝角。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1882.jpg" }, { "index": 1883, "ocr_en": "There are parallel segments AB and CD, AB above CD, point A to the left of B and directly above C, and point D to the right of C and directly below B. Connect BD to point BD. Connect BD, AB perpendicular to BD at point B and CD perpendicular to DB at point D. Between AB and CD points E, G and F are arranged vertically from top to bottom, with point E at the lower right of point A, point G at the lower left of point E and point F at the lower right of point G. Connect AE, EG, GF and FC to form a section of folds, convex to the right, then to the left and finally to the right. Use the arcs to mark angle BAE as angle 1, angle AEG as angle 2, angle EGF as angle 3, angle GFC as angle 4, and angle FCD as angle 5, all of which are acute angles.", "ocr_zh": "有平行线段AB和CD，AB在CD上方，点A在B的左侧、C的正上方，点D在C的右侧、B的正下方。连接BD，AB垂直BD于点B，CD垂直DB于点D。在AB和CD之间有点E、G、F在竖直方向从上至下排列，点E在点A的右下方，点G在点E的左下方，点F在点G的右下方，连接AE、EG、GF、FC形成一段折线，先向右凸再向左凸最后向右凸。用弧线标记角BAE为角1、角AEG为角2，角EGF为角3，角GFC为角4，角FCD为角5，这些角均是锐角。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1883.jpg" }, { "index": 1884, "ocr_en": "There are parallel segments AB and CD, AB above CD, point A to the left of B and directly above C, and point D to the right of C and directly below B. Connect BD with AB perpendicular to BD and CD perpendicular to DB. Connect BD, AB perpendicular to BD at point B, CD perpendicular to DB at point D. There is a point E between AB and CD, point E is to the lower right of point A. Connect AE and EC to form a section of the fold that is convex to the right, and use the arcs to mark the angle BAE as Angle 1, Angle AEC as Angle 2, and Angle ECD as Angle 3, all of which are acute angles.", "ocr_zh": "有平行线段AB和CD，AB在CD上方，点A在B的左侧、C的正上方，点D在C的右侧、B的正下方。连接BD，AB垂直BD于点B，CD垂直DB于点D。在AB和CD之间有点E，点E在点A的右下方，连接AE、EC形成一段向右凸的折线，用弧线标记角BAE为角1、角AEC为角2、角ECD为角3，这些角均是锐角。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1884.jpg" }, { "index": 1885, "ocr_en": "There are parallel lines AB and CD, AB is above CD, point A is to the left of B and directly above C, and point D is to the right of C and directly below B. There are five points of inflection between AB and CD in the vertical direction from top to bottom. Connect BD, AB perpendicular to BD at point B, CD perpendicular to DB at point D. Between AB and CD there are five points of inflection in the vertical direction from top to bottom, the first point of inflection is connected to point A, the points of inflection are connected to each other, the last point of inflection is connected to point C, where the fourth point of inflection and the last point of inflection are connected to each other by dotted lines, and the six lines are connected to each other to form a section of the folded line. The angle between line AB and the beginning of the line, and the angle between line CD and the end of the line are both acute angles.", "ocr_zh": "有平行线段AB和CD，AB在CD上方，点A在B的左侧、C的正上方，点D在C的右侧、B的正下方。连接BD，AB垂直BD于点B，CD垂直DB于点D。在AB和CD之间在竖直方向从上至下有五个拐点，首个拐点与点A连接，拐点之间相互连接，末个拐点与点C连接，其中第四个拐点与末个拐点之间用虚线连接，由六条线段依次相连形成一段折线。线段AB与折线起始段的夹角、线段CD与折线末端段的夹角均为锐角。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1885.jpg" }, { "index": 1886, "ocr_en": "There are parallel segments AB and DC, AB is above DC, point A is to the left of B and point D is to the left of C. A point E is between AB and DC. Between AB and DC there is a point E, which is to the lower right of B and to the upper right of C. Connect BE and CE. Make an auxiliary line EF parallel to AB through point E. Point F is to the left of E. Mark the angle BEF with an arc as angle 1 and the angle CEF as angle 2, both of which are acute angles.", "ocr_zh": "有平行线段AB和DC，AB在DC上方，点A在B的左侧，点D在C的左侧。在AB和DC之间有一点E，点E在B的右下方、在C的右上方，连接BE、CE。过点E作辅助线直线EF平行于AB，点F在E的左侧，用弧线标记角BEF为角1，角CEF为角2，均为锐角。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1886.jpg" }, { "index": 1887, "ocr_en": "There are parallel lines AD and CE, AD is above CE, point A is to the left of D, and point C is to the left of E and to the right of point A. Point E is directly below point D. Between AD and CE there is a point B, to the left of points D and E, connected to a point on the line AD, joining BC, and the two lines form a section of a bisecting line convex to the right, with the point B to the lower right of point A and to the upper right of point C. Mark with an arc the angle between AD and the start of the line as angle 1, which is acute. The angle between the arc marker CE and the end segment of the fold line is angle 2, which is obtuse, and angle BCE, which is acute.", "ocr_zh": "有平行线AD和CE，AD在CE上方，点A在D的左侧，点C在E的左侧且在点A的右下方。点E在点D的正下方。在AD和CE之间有一点B，在点D和E的左侧，与线段AD上一点相连接，连接BC，两条线段构成一段向右凸的折线，点B在点A的右下方且在点C的右上方。用弧线标记AD与折线起始段的夹角为角1，角1为锐角。用弧线标记CE与折线末端段的夹角为角2，角2为钝角，角BCE为锐角。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1887.jpg" }, { "index": 1888, "ocr_en": "There are parallel segments BA and DE, BA is above DE, point B is to the left of A, and point D is to the left of E. Point C is a point between BA and DE and to the left of point B and point D. Connect BC and CD to form a leftward convex polyline. Take a point F on the extension of BC along C and connect it to CF. Mark angle ABC with an arc as angle 1 and angle CDE as angle 2, both obtuse angles. Mark angle FCD with an arc as angle α, which is acute.", "ocr_zh": "有平行线段BA和DE，BA在DE上方，点B在A的左侧，点D在E的左侧。点C是BA和DE之间一点，且在点B和点D的左侧，连接BC和CD，构成一段向左凸的折线。在BC沿C方向延长线上取一点F，连接CF。用弧线标记角ABC为角1、角CDE为角2，两个均是钝角。用弧线标记角FCD为角α，为锐角。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1888.jpg" }, { "index": 1889, "ocr_en": "There are parallel segments MA1 and NA4 and MA1 is above NA4, point M is to the left of A1, point N is to the left of A4, point M is directly above N and point A1 is directly above A4. Points A2 and A3 are between the parallel lines, point A2 is below the right of A1 and above A3, and point A3 is above the right of A4, connecting A1A2, A2A3, and A3A4 to form a section of a folded line convex to the right. Mark the line segments MA1 and NA4 in red, and the corners MA1A2, A1A2A3, A2A3A4, and A3A4N with red arcs.", "ocr_zh": "有平行线段MA1和NA4且MA1在NA4上方，点M在A1左侧，点N在A4左侧，点M在N正上方，点A1在A4正上方。点A2和A3在平行线之间，点A2位于A1右下方、A3上方，点A3位于A4右上方，连接A1A2、A2A3、A3A4，构成一段向右凸的折线。用红色标记线段MA1、NA4，用红色弧线标记角MA1A2、A1A2A3、A2A3A4、A3A4N。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1889.jpg" }, { "index": 1890, "ocr_en": "There are parallel segments MA1 and NAn and MA1 is above NAn, point M is to the left of A1 and point N is to the left of An. Point M is directly above N and point A1 is directly above An. Points A2, A3, A4, and A5 are between the parallel lines, point A2 is at the lower right of A1 and above A3, and point A5 is at the upper right of An. Points A2, A3, A4, and A5 are arranged vertically from top to bottom, and are connected to A1A2, A2A3, A3A4, and A4A5, and A5An is connected to the auxiliary line, and together they form a section of a bisecting line that is convex to the right. Mark the line segments MA1, NAn in red and the corners MA1A2, A1A2A3, A2A3A4, A3A4A5 with red arcs.", "ocr_zh": "有平行线段MA1和NAn且MA1在NAn上方，点M在A1左侧，点N在An左侧。点M在N正上方，点A1在An正上方。点A2、A3、A4、A5在平行线之间，点A2位于A1右下方、A3上方，点A5位于An右上方，点A2、A3、A4、A5在竖直方向从上到下排列，连接A1A2、A2A3、A3A4、A4A5，连接A5An作辅助线，共同构成一段向右凸的折线。用红色标记线段MA1、NAn，用红色弧线标记角MA1A2、A1A2A3、A2A3A4、A3A4A5。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1890.jpg" }, { "index": 1891, "ocr_en": "There are parallel line segments MA1 and NA5 and MA1 is above NA5, point M is to the left of A1, point N is to the left of A5, point M is directly above N and point A1 is directly above A5. Points A2, A3, A4 are between parallel lines, point A2 is below right of A1 and above A3, point A4 is above right of A5, points A2, A3, A4 are arranged vertically in the vertical direction from top to bottom, connecting A1A2, A2A3, A3A4, A4A5. mark the line segments MA1, NA5 in red, and mark the corners MA1A2, A1A2A3, A2A3A4 in red arcs, A3A4A5, A4A5N.", "ocr_zh": "有平行线段MA1和NA5且MA1在NA5上方，点M在A1左侧，点N在A5左侧，点M在N正上方，点A1在A5正上方。点A2、A3、A4在平行线之间，点A2位于A1右下方、A3上方，点A4位于A5右上方，点A2、A3、A4在竖直方向从上到下排列，连接A1A2、A2A3、A3A4、A4A5。用红色标记线段MA1、NA5，用红色弧线标记角MA1A2、A1A2A3、A2A3A4、A3A4A5、A4A5N。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1891.jpg" }, { "index": 1892, "ocr_en": "There are parallel segments BA and DE, BA is above DE, point B is to the left of A and point D is to the left of E. Point C is a point between BA and DE and to the left of points B and D. Connect BC and CD to form a leftward convex polyline. Take a point F on the line extending BC along C and connect it to CF. Mark the angle FCD as angle α, acute, with an arc. Make a ray CG auxiliary parallel to BA through point C. Point G is to the right of C. Label the line segments AB, BC, CD, and DE in red, and the ray CG in yellow. Label the angle ABC as angle 1, the angle CDE as angle 2, the angle BCG as angle 3, and the angle DCG as angle 4 using the arc. angles 1 and 2 are obtuse angles, and angles 3 and 4 are acute angles.", "ocr_zh": "有平行线段BA和DE，BA在DE上方，点B在A的左侧，点D在E的左侧。点C是BA和DE之间一点，且在点B和点D的左侧，连接BC和CD构成一段向左凸的折线。在BC沿C方向延长线上取一点F，连接CF。用弧线标记角FCD为角α，为锐角。过点C作射线CG辅助线平行于BA，点G在C的右侧。用红色标记线段AB、BC、CD、DE，用黄色标记射线CG，用弧线标记角ABC为角1，角CDE为角2，角BCG为角3，角DCG为角4，角1和角2是钝角，角3和角4是锐角。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1892.jpg" }, { "index": 1893, "ocr_en": "There are parallel line segments AB and CD, AB is above CD, point A is to the left of B, point C is to the left of D, point C is directly below A, and point D is directly below B. Between AB and CD there are four points of inflection in the vertical direction from top to bottom, the first point of inflection is connected to a point on the line segment AB, the points of inflection are connected to each other, and the last point of inflection is connected to a point on the line segment CD, which is connected to the line segment A and B in order to form a convex fold line to the right. The angle between line AB and the start of the line, the four angles of inflection, and the angle between line CD and the end of the line are labelled from top to bottom as Angle 1, Angle 2, Angle 3, Angle 4, Angle 5, and Angle 6, which are all obtuse angles.", "ocr_zh": "有平行线段AB和CD，AB在CD上方，点A在B的左侧，点C在D的左侧，点C在A的正下方，点D在B的正下方。在AB和CD之间在竖直方向从上至下有4个拐点，首个拐点与线段AB上一点连接，拐点之间相互连接，末个拐点与线段CD上某点连接，由五条线段依次相连形成凸向右侧的折线。用弧线标记线段AB与折线起始段的夹角、4个拐角、线段CD与折线末端段的夹角从上至下依次为角1、角2、角3、角4、角5和角6，这些角均为钝角。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1893.jpg" }, { "index": 1894, "ocr_en": "Points A and E are on a horizontal plane and A is to the left of E. Point B and point F are above AE. Point B and point F are above point AE, BA is perpendicular to AE at point A and EF is perpendicular to AE at point E. Make congruent rectangles with BA and EF as long sides, and the two rectangles do not intersect. Point C is to the upper right of point B, point D is to the upper left of point F, and point C is to the left of D. Connect the lines BC and CD. CD is parallel to AE.", "ocr_zh": "点A和E在水平面上，且A在E左侧。点B和点F在AE上方，BA垂直AE于点A，EF垂直AE于点E。以BA、EF为长边分别作全等的矩形，两矩形不相交。点C位于点B右上方，点D位于点F左上方，点C在D左侧，连接线段BC、CD。CD平行于AE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1894.jpg" }, { "index": 1895, "ocr_en": "There are parallel lines DF and CE, DF is above CE, point D is to the right of F, and point E is to the right of C. There is a point B between the parallel lines, and a point A between the lines DF, connecting AB and BC, the two lines form a bisecting line convex to the left, with angles FAB and BCE acute, and angle ABC obtuse.", "ocr_zh": "有平行线DF与CE，DF在CE上方，点D在F右侧，点E在C右侧。在平行线之间有点B，在线段DF之间有一点A，连接AB、BC，两条线段构成一条凸向左侧的折线，角FAB和角BCE为锐角，角ABC为钝角。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1895.jpg" }, { "index": 1896, "ocr_en": "There are parallel segments AB and CD, AB above CD, point A to the left of B, and point C to the left of D. Between AB and CD there are points E and F. Point E is to the lower right of point A, point F is to the lower right of point E and to the upper right of point C. Points E and F are to the left of points B and D. Connecting AE, EF, and FC, the three line segments form a bisecting line that is convex to the right, and use arcs to mark angles BAE, AEF, EFC, and FCD as angles 1, 2, 3, and 4, respectively. angles 1, 3, and 4 are acute angles. Angle 2 is obtuse.", "ocr_zh": "有平行线段AB和CD，AB在CD上方，点A在B的左侧，点C在D的左侧。在AB和CD之间有点E和F，点E在点A的右下方，点F在点E右下方且在点C右上方，点E和F均在点B和D的左侧，连接AE、EF、FC，三条线段构成一条凸向右侧的折线，用弧线标记角BAE、角AEF、角EFC、角FCD分别为角1、角2、角3和角4。角1、3和4是锐角，角2是钝角。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1896.jpg" }, { "index": 1897, "ocr_en": "There are parallel segments AB and CD, AB is above CD, point A is to the left of B, and point C is to the left of D. Point E is a point to the right of B. Connect BE and DE. Point E is a point to the upper right of point B. Connect BE and DE. Make the auxiliary line EF parallel to AB through point E. Point E is to the left of F. Angle BED is acute. Angle CDE is obtuse.", "ocr_zh": "有平行线段AB和CD，AB在CD上方，点A在B的左侧，点C在D的左侧。点E是点B右上方一点，连接BE、DE。过点E作辅助线EF平行于AB，点E在F左侧。角BED是锐角。角CDE是钝角。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1897.jpg" }, { "index": 1898, "ocr_en": "There are parallel line segments AB and CD, AB is above CD, point A is to the left of B, and point C is to the left of D. The angle BED is acute. Point E is a point to the upper right of point B. Connect BE and DE. mark line segments AB, CD and point E in red. angle BED is acute.", "ocr_zh": "有平行线段AB和CD，AB在CD上方，点A在B的左侧，点C在D的左侧。点E是点B右上方一点，连接BE、DE。用红色标记线段AB、CD和点E。角BED是锐角。角CDE是钝角。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1898.jpg" }, { "index": 1899, "ocr_en": "There are parallel line segments AB and CD, AB is above CD, point A is to the left of B, and point C is to the left of D. The angle BED is acute. Point E is a point to the upper right of point B. Connect BE and DE. mark line segments AB, CD and point E in red. angle BED is acute. Angle CDE is obtuse.", "ocr_zh": "有平行线段AB和CD，AB在CD上方，点A在B的左侧，点C在D的左侧。点E是点B右上方一点，连接BE、DE。用红色弧线标记角ABE、角BED和角CDE。角BED是锐角。角CDE是钝角。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1899.jpg" }, { "index": 1900, "ocr_en": "There are parallel lines a and b, with the line a above b. There is acute triangle ABC with vertex A above line a, vertex B between a and b, and vertex C on line c. Side AB intersects line a at point D, and side AC intersects line a at point E. Point D is to the left of E, point B is to the left of C, and point A is to the upper right of point E. Mark with an arc the angle between line segment AD and line a as angle 1, angle BCA as angle 3, and the angle between BC and line b as angle 2. Angle 1 is obtuse, and angles 2 and 3 are acute.", "ocr_zh": "有平行线a和b，直线a在b的上方。有锐角三角形ABC，顶点A在直线a的上方，顶点B在a和b之间，顶点C在直线c上。边AB交直线a于点D，边AC交直线a于点E。点D在E左侧，点B在C左侧，点A在点E的右上方。用弧线标记线段AD与直线a的夹角为角1，角BCA为角3，BC与直线b的夹角为角2。角1是钝角，角2和角3是锐角。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1900.jpg" }, { "index": 1901, "ocr_en": "There are parallel lines m and n, and the line m is above n. There is right triangle ABC with BA perpendicular to AC at point A. Vertex A is on line m, vertex C is between m and n, and vertex B is below n. Point B is to the left of point A, and point C is to the right of point n. Point B is below the left of point A and point C is below the right of point A. BC intersects the line n at point D. Mark with an arc the angle between side AB and the line m as angle 2, and the angle between line segment BD and the line n as angle 1. Angles 1 and 2 are both acute angles.", "ocr_zh": "有平行线m和n，直线m在n上方。有直角三角形ABC，BA垂直AC于点A。顶点A在直线m上，顶点C在m和n之间，顶点B在n下方。点B在A的左下方，点C在点A的右下方。BC交直线n于点D。用弧线标记边AB与直线m的夹角为角2，线段BD与直线n的夹角为角1。角1和角2均为锐角。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1901.jpg" }, { "index": 1902, "ocr_en": "There are parallel lines m and n, and the line m is above n. There is right triangle ABC with BA perpendicular to AC at point A. Vertex A is on line m, vertex C is between m and n, and vertex B is below n. Point B is to the left of point A, and point C is to the right of point n. Point B is below the left of point A and point C is below the right of point A. BC intersects the line n at point D. Mark with an arc the angle between the side AB and the line m as angle 2, and the angle between the line segment BD and the line n as angle 1. Angles 1 and 2 are acute angles. Make the BE auxiliary line parallel to m and n through point B. Point E is to the right of B. Mark in red the left part of point A in line m, the left part of point D in line n, AB and BD.", "ocr_zh": "有平行线m和n，直线m在n上方。有直角三角形ABC，BA垂直AC于点A。顶点A在直线m上，顶点C在m和n之间，顶点B在n下方。点B在A的左下方，点C在点A的右下方。BC交直线n于点D。用弧线标记边AB与直线m的夹角为角2，线段BD与直线n的夹角为角1。角1和角2均为锐角。过点B作BE辅助线平行于m和n，点E在B的右侧。用红色标记直线m中点A的左侧部分、直线n中点D的左侧部分、AB和BD。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1902.jpg" }, { "index": 1903, "ocr_en": "There are parallel lines a and b, with the line a above b. There is acute triangle ABC with vertex A above line a, vertex B between a and b, and vertex C on line c. Side AB intersects line a at point D, and side AC intersects line a at point E. Point D is to the left of E, point B is to the left of C, and point A is to the upper right of point E. Mark with an arc the angle between line segment AD and line a as angle 1, angle BCA as angle 3, and the angle between line segment BC and line b as angle 2. Angle 1 is obtuse, and angles 2 and 3 are acute. An auxiliary line AF is drawn parallel to a and b through point A. Point F is to the right of A. Mark in red the line segments AF, AC, AD, the left-hand part of point D in line a, and the left-hand part of point C in line b.", "ocr_zh": "有平行线a和b，直线a在b的上方。有锐角三角形ABC，顶点A在直线a的上方，顶点B在a和b之间，顶点C在直线c上。边AB交直线a于点D，边AC交直线a于点E。点D在E左侧，点B在C左侧，点A在点E的右上方。用弧线标记线段AD与直线a的夹角为角1，角BCA为角3，线段BC与直线b的夹角为角2。角1是钝角，角2和角3是锐角。过点A作AF辅助线平行于a和b，点F在A右侧。用红色标记线段AF、AC、AD、直线a中点D的左侧部分，直线b中点C的左侧部分。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1903.jpg" }, { "index": 1904, "ocr_en": "There are parallel line segments AB and EF, AB is above EF, point A is to the left of B, and point E is to the left of F. Point E is to the lower right of A and to the lower left of B. There is a point D on line segment EF, which lies to the lower right of point B. There is a point C on the line EF below the point E, which lies to the lower right of the point E and to the lower left of the point D. Connect BC and CD. mark with an arc that angle ABC is 75 degrees and angle CDF is 135 degrees.", "ocr_zh": "有平行线段AB和EF，AB在EF上方，点A在B的左侧，点E在F的左侧。点E在A的右下方、B的左下方。线段EF上有一点D，位于点B的右下方。线段EF下方有一点C，位于点E的右下方，点D的左下方。连接BC和CD。用弧线标记角ABC为75度，角CDF为135度。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1904.jpg" }, { "index": 1905, "ocr_en": "There are right-angled triangles ABC and DEF arranged right and left, AB perpendicular to BC at point B, point A is above side BC, point D is on the extension of BC in the direction of C, point E is on AC, ED is perpendicular to DF at point D, point F is on the upper right side of point D, and DE equals DF.", "ocr_zh": "有直角三角形ABC和DEF左右排列，AB垂直BC于点B，点A在边BC的上方，点D在BC沿C方向的延长线上，点E在AC上，ED垂直DF于点D，点F在点D的右上方，DE等于DF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1905.jpg" }, { "index": 1906, "ocr_en": "There are parallel lines a and b, with a above b. Two intersecting rays c and d, with c to the left of d and the point of intersection above a. The angle is labelled with an arc as angle 2. The angle between ray c and line b is labelled with an arc as angle 1, and the angle between ray d and line a is labelled with an arc as angle 3. Angles 1 and 3 are obtuse angles, and angle 2 is acute.", "ocr_zh": "有平行线a和b，a在b的上方。两条相交的射线c和d，c在d左侧，交点位于a的上方，夹角用弧线标记为角2。射线c与直线b的夹角用弧线标记为角1，射线d与直线a的夹角用弧线标记为角3。角1和角3是钝角，角2是锐角。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1906.jpg" }, { "index": 1907, "ocr_en": "There are parallel line segments AB and CD, AB is above CD, point A is to the left of B, point C is to the left of D, and point C is to the lower right of point B. Point E is a point to the upper right of point B. Connect BE and EC, with EC perpendicular to CD at point C.", "ocr_zh": "有平行线段AB和CD，AB在CD上方，点A在B左侧，点C在D左侧，点C在点B的右下方。点E是点B右上方一点，连接BE、EC，EC垂直CD于点C。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1907.jpg" }, { "index": 1908, "ocr_en": "There are parallel line segments AB and CD, AB is above CD, point A is to the left of B, point C is to the left of D, point C is directly below point A, and point D is directly below point B. Point F is a point between AB and CD, connect AF and CF, Angle BAF and Angle FCD are acute angles, Point B is above AB, to the right of point A and to the left of point B. Connect AE and CE. AE is perpendicular to AF at point A.", "ocr_zh": "有平行线段AB和CD，AB在CD上方，点A在B左侧，点C在D左侧，点C在点A的正下方，点D在点B的正下方。点F是AB和CD之间的一点，连接AF和CF，角BAF和角FCD为锐角，点B位于AB上方，点A的右侧，点B的左侧，连接AE、CE。AE垂直AF于点A。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1908.jpg" }, { "index": 1909, "ocr_en": "There are parallel line segments AB and CD, AB is above CD, point A is to the left of B, point C is to the left of D, point C is directly below point A, point D is directly below point B. Point E is above AB, to the right of point A and to the left of point B. Connect AE and CE.", "ocr_zh": "有平行线段AB和CD，AB在CD上方，点A在B左侧，点C在D左侧，点C在点A的正下方，点D在点B的正下方。点E位于AB上方，点A的右侧，点B的左侧，连接AE、CE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1909.jpg" }, { "index": 1910, "ocr_en": "There is a quadrilateral ABCD with point A above B, C and D. Point B is to the left of A, D and C. Point D is to the upper right of point B and to the upper left of point C. Angles BAC, ABD and ACD are all acute and angle BDC is obtuse. Extend BD along D to intersect line AC at point E. Connect DE as an auxiliary line and mark it in red.", "ocr_zh": "点C没截全。（有四边形ABCD，点A在B、C和D的上方，点B在A、D、C左侧，点D在点B的右上方、点C的左上方。角BAC、ABD、ACD均为锐角，角BDC为钝角。沿D方向延长BD交线段AC于点E，连接DE作辅助线并用红色标记。）", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1910.jpg" }, { "index": 1911, "ocr_en": "There is an acute triangle ABC with point A above BC and point B to the left of C. Points F and E are points on sides AB and AC respectively, connecting CF and BE to intersect at point H. CF is perpendicular to AB at point F and BE is perpendicular to AC at point E.", "ocr_zh": "有锐角三角形ABC，点A在BC上方，点B在C的左侧。点F和E分别是边AB和AC上的点，连接CF、BE交于点H。CF垂直AB于点F，BE垂直AC于点E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1911.jpg" }, { "index": 1912, "ocr_en": "There is an acute triangle ABC with point A above BC and point B to the left of C. There is a right-angled triangular ruler with a circle inside. The short right-angled side and the long right-angled side of the right-angled triangular ruler pass through points B and C. The right-angled vertex D is inside the triangle ABC, BD is perpendicular to DC at point D, and the circle of the triangular ruler intersects the side BC. Extend BD along D to intersect segment AC at point E. Connect DE as an auxiliary line. Mark the line segments AB, BD, DC, and AC in red.", "ocr_zh": "有锐角三角形ABC，点A在BC上方，点B在C的左侧。有一把内部有一个圆的直角三角尺。直角三角尺的短边直角边和长边直角边分别经过点B和C，直角顶点D在三角形ABC内部，BD垂直DC于点D，三角尺的圆与边BC相交。沿D方向延长BD交线段AC于点E，连接DE作辅助线。用红色标记线段AB、BD、DC、AC。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1912.jpg" }, { "index": 1913, "ocr_en": "There is an acute triangle ABC with point A above BC and point B to the left of C. There is a right-angled triangular ruler with a circle inside. The short right-angled side and the long right-angled side of the right-angled triangular ruler pass through points B and C. The right-angled vertex D is inside the triangle ABC, BD is perpendicular to DC at point D. The circle of the triangular ruler intersects the side BC.", "ocr_zh": "有锐角三角形ABC，点A在BC上方，点B在C的左侧。有一把内部有一个圆的直角三角尺。直角三角尺的短边直角边和长边直角边分别经过点B和C，直角顶点D在三角形ABC内部，BD垂直DC于点D，三角尺的圆与边BC相交。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1913.jpg" }, { "index": 1914, "ocr_en": "In triangle ABC, point A is above BC, point B is to the left of C, angles B and C are acute angles, point O is a point inside triangle ABC, connect AO, BO and CO.", "ocr_zh": "在三角形ABC中，点A在BC上方，点B在C左侧，角B和C是锐角，点O是三角形ABC内一点，连接AO、BO、CO。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1914.jpg" }, { "index": 1915, "ocr_en": "There is an acute angle BAC with point A to the lower left of B and the upper left of C. Inside the acute angle BAC there are two points D and E, both of which lie to the lower left of B and to the upper left of C. Point D is to the upper left of E. Connecting DE, BE intersects CD at point P. Triangle DEP is an acute triangle and angle BPC is obtuse.", "ocr_zh": "有锐角BAC，点A在B的左下方，C的左上方。在锐角BAC内部有两点D和E，均位于B的左下方、C的左上方。点D在E的左上方。连接DE，BE与CD交于点P，三角形DEP是锐角三角形，角BPC是钝角。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1915.jpg" }, { "index": 1916, "ocr_en": "In triangle ABC, angle C is obtuse, vertex A is above BC, and point B is to the left of C. Point O is to the right of AC, below the right of A, and above the right of C. Connect BO to bisect angle ABC. Point P is a point on AC, connect OP, which is perpendicular to BO at point O. Point P is closer to A.", "ocr_zh": "在三角形ABC中，角C是钝角，顶点A在BC上方，点B在C的左侧。点O在AC的右侧、A的右下方、C的右上方，连接BO平分角ABC，点P是AC上一点，连接OP，OP垂直BO于点O，点P更靠近A。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1916.jpg" }, { "index": 1917, "ocr_en": "In an acute triangle ABC, vertex A is above BC and point B is to the left of C. Point C coincides with point P, labelled (P) to the right of C. Point O is a point inside triangle ABC, connecting OB and OC, with OB perpendicular to OC at point O.", "ocr_zh": "在锐角三角形ABC中，顶点A在BC上方，点B在C的左侧。点C与点P重合，在C右边标注（P）。点O是三角形ABC内部一点，连接OB和OC，OB垂直OC于点O。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1917.jpg" }, { "index": 1918, "ocr_en": "There are lines AD and BC intersecting at point O, connecting AB and CD. Triangles ABO and CDO are both acute triangles, AB is to the left of CD, point B is to the upper left of A, and point D is to the upper right of C.", "ocr_zh": "有线段AD和BC相交于点O，连接AB和CD，三角形ABO和CDO均为锐角三角形，AB在CD左侧，点B在A左上方，点D在C右上方。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1918.jpg" }, { "index": 1919, "ocr_en": "There are lines AD and BC intersecting at point O, connecting AB and CD. Triangles ABO and CDO are both acute triangles, AB is to the left of CD, point B is to the upper left of A, and point D is to the upper right of C. Label the segments AD and BC in red, the segments AB and CD in yellow, the angles OBA and BAO in red, and the angles ODC and OCD in yellow.", "ocr_zh": "有线段AD和BC相交于点O，连接AB和CD，三角形ABO和CDO均为锐角三角形，AB在CD左侧，点B在A左上方，点D在C右上方。用红色标记线段AD和BC，黄色标记线段AB和CD，红色弧线标记角OBA和角BAO，黄色弧线标记角ODC和角OCD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1919.jpg" }, { "index": 1920, "ocr_en": "There are lines AD and BC intersecting at point E, connecting AB and CD. Triangles ABE and CDE are both right-angled triangles, with AB to the right of CD, point B to the lower right of A, and point D to the lower left of C. DC is perpendicular to CE at point C and BA is perpendicular to AE at point A.", "ocr_zh": "有线段AD和BC相交于点E，连接AB和CD，三角形ABE和CDE均为直角三角形，AB在CD右侧，点B在A右下方，点D在C左下方。DC垂直CE于点C，BA垂直AE于点A。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1920.jpg" }, { "index": 1921, "ocr_en": "In the horizontal direction there is line segment CD and point C is to the left of D. In the vertical direction there is line segment AB intersecting CD at point G. Point G is between the segments. point A is to the lower right of C and lower left of D. point B is to the upper right of C and to the upper left of D. connect AC. point F is to the left of AB and above CD. connect FB. connect FD intersecting line segment BG at point E. triangle FBE is an acute triangle.", "ocr_zh": "在水平方向上有线段CD，点C在D左侧。在竖直方向上有线段AB交CD于点G，点G在线段之间，点A在C的右下方、D的左下方，点B在C的右上方、D的左上方，连接AC。点F位于AB左侧、CD的上方，连接FB，连接FD交线段BG于点E。三角形FBE是锐角三角形。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1921.jpg" }, { "index": 1922, "ocr_en": "There are two right triangles ABC and DEF, point A is above CB, point C is to the left of B, and AC is perpendicular to CB at point C. Point D is on line segment AB, side DE intersects AC at point G, and ED is perpendicular to DF at point D. ED is equal to DF. point E is on the prolongation of CB along direction C, and point F is on the prolongation of CB along direction B. Mark the line segments AD, AC, DE and CE in red.", "ocr_zh": "有两个直角三角形ABC和DEF，点A在CB上方，点C在B的左侧，AC垂直CB于点C。点D在线段AB上，边DE交AC于点G，ED垂直DF于点D，ED等于DF。点E在CB沿C方向的延长线上，点F在CB沿B方向的延长线上。用红色标记线段AD、AC、DE和CE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1922.jpg" }, { "index": 1923, "ocr_en": "There is a pentagram ABCDE with point A above BE, point B to the left of E, point C to the lower right of B and lower left of E, and point D to the right of C and lower left of E. BD intersects CE at point O.", "ocr_zh": "有一个五角星ABCDE，点A在BE上方，点B在E左侧，点C在B的右下方、E的左下方，点D在C的右侧，E的左下方。BD交CE于点O。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1923.jpg" }, { "index": 1924, "ocr_en": "There is a quadrilateral ABCD with AD above BC and AB to the left of CD, connecting AC and BD intersecting at point O.", "ocr_zh": "有四边形ABCD，AD在BC上方，AB在CD左侧，连接AC和BD交于点O。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1924.jpg" }, { "index": 1925, "ocr_en": "There are lines AB and CD, AB is above and longer than CD, point C is to the right of A and to the left of B, and point D is to the right of C and to the left of B. The lines AD and CD are joined at point O. Take a point E on the line extending from CD along C. The angle BCE is obtuse and the bisector of angle BAD is obtuse. Connect the line segments AD and BC to intersect at point O. Take a point E on the extension of CD along the direction of C, connect CE, angle BCE is obtuse, and the bisector of angle BCE intersects the bisector of angle BAD at a point P. Point P is inside triangle ABO, connect AP, and connect PC to AD at point F.", "ocr_zh": "有线段AB和CD，AB在上方，且长度大于CD，点C在A的右下方、B的左下方，点D在C的右侧、B的左下方。连接线段AD和BC交于点O。在CD沿C方向延长线上取一点E，连接CE，角BCE是钝角，角BCE的平分线与角BAD的平分线交于一点P，点P在三角形ABO内部，连接AP，连接PC交AD于点F。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1925.jpg" }, { "index": 1926, "ocr_en": "There are lines AB and CD, AB is above and less long than CD, point C is to the lower left of A, and point D is to the lower right of B. Connect the line segments AD and BC to intersect at point O.", "ocr_zh": "有线段AB和CD，AB在上方，且长度小于CD，点C在A的左下方，点D在B的右下方。连接线段AD和BC交于点O。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1926.jpg" }, { "index": 1927, "ocr_en": "In triangle ABC, angles B and C are acute angles, point A is above BC, point B is to the left of C. AE is the bisector of angle BAC and point E is on BC. Point D is on the line segment CE, connecting AD, which is perpendicular to BC at point D. Label angles ABC and ACB with yellow arcs, and angle EAD with a red arc.", "ocr_zh": "在三角形ABC中，角B和角C是锐角，点A在BC上方，点B在C左侧，AE是角BAC的平分线，且点E在BC上。点D在线段CE上，连接AD，AD垂直BC于点D。用黄色弧线标记角ABC、ACB，用红色弧线标记角EAD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1927.jpg" }, { "index": 1928, "ocr_en": "In triangle ABC, angles B and C are acute angles, point A is above BC, point B is to the left of C. AE is the bisector of angle BAC and point E is on BC. Point D is on line segment CE, connect AD, which is perpendicular to BC at point D. Mark line segment AD in red and line segment AE in yellow.", "ocr_zh": "在三角形ABC中，角B和角C是锐角，点A在BC上方，点B在C左侧，AE是角BAC的平分线，且点E在BC上。点D在线段CE上，连接AD，AD垂直BC于点D。用红色标记线段AD，黄色标记线段AE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1928.jpg" }, { "index": 1929, "ocr_en": "In triangle ABC, angles B and C are acute angles, point A is above BC, point B is to the left of C, AE is the bisector of angle BAC, and point E is on BC. Point D is on the line BE, connect AD, AD is perpendicular to BC at point D.", "ocr_zh": "在三角形ABC中，角B和角C是锐角，点A在BC上方，点B在C左侧，AE是角BAC的平分线，且点E在BC上。点D在线段BE上，连接AD，AD垂直BC于点D。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1929.jpg" }, { "index": 1930, "ocr_en": "In triangle ABC, angles B and C are acute angles, point A is above BC, point B is to the left of C, AE is the bisector of angle BAC, and point E is on BC. Point D is on line BE, connect AD as an auxiliary line, AD is perpendicular to BC at point D. Take a point M on the extension line of AE along the direction of E, connect EM. point N is on line CE, connect MN, MN is perpendicular to BC at point N. Mark the line segment AM in yellow, and the yellow arcs to mark the angles ABC and ACB, and mark the line segment MN in red, and the red arcs to mark the angles AMN and ANC.", "ocr_zh": "在三角形ABC中，角B和角C是锐角，点A在BC上方，点B在C左侧，AE是角BAC的平分线，且点E在BC上。点D在线段BE上，连接AD作辅助线，AD垂直BC于点D。在AE沿E方向延长线上取一点M，连接EM。点N在线段CE上，连接MN，MN垂直BC于点N。用黄色标记线段AM，黄色弧线标记角ABC和角ACB，红色标记线段MN、红色弧线标记角AMN和角ANC。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1930.jpg" }, { "index": 1931, "ocr_en": "In triangle ABC, angles B and A are acute angles, point C is above BA, point A is to the left of B, CE is the bisector of angle ACB, and point E is on AB. Point D is on line segment BE, connect CD, CD is perpendicular to BA at point D. Point F is on line segment CE, connect DF, DF is perpendicular to CE at point F. Mark line segment CD in red and line segment CE in yellow.", "ocr_zh": "在三角形ABC中，角B和角A是锐角，点C在BA上方，点A在B左侧，CE是角ACB的平分线，且点E在AB上。点D在线段BE上，连接CD，CD垂直BA于点D。点F在线段CE上，连接DF，DF垂直CE于点F。用红色标记线段CD，黄色标记线段CE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1931.jpg" }, { "index": 1932, "ocr_en": "In triangle ABC, angles B and C are acute angles, point A is above BC, point B is to the left of C. AE is the bisector of angle BAC and point E is on BC. Point D is on line CE, connect AD as an auxiliary line, AD is perpendicular to BC at point D. Point M is on line AE, point N is on line DE, connect MN, MN is perpendicular to BC at point N. Label the line segment AE in yellow, the yellow arcs marking the angles ABC and ACB, and label the line segment MN in red, the red arcs marking the angles EMN and MND.", "ocr_zh": "在三角形ABC中，角B和角C是锐角，点A在BC上方，点B在C左侧，AE是角BAC的平分线，且点E在BC上。点D在线段CE上，连接AD作辅助线，AD垂直BC于点D。点M在线段AE上，点N在线段DE上，连接MN，MN垂直BC于点N。用黄色标记线段AE，黄色弧线标记角ABC和角ACB，红色标记线段MN，红色弧线标记角EMN和角MND。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1932.jpg" }, { "index": 1933, "ocr_en": "In triangle ABC, angles B and C are acute angles, point A is above BC, point B is to the left of C, AD is the bisector of angle BAC and point D is on BC. Point E is on the line segment CD, connect AE, AE is perpendicular to BC at point E.", "ocr_zh": "在三角形ABC中，角B和角C是锐角，点A在BC上方，点B在C左侧，AD是角BAC的平分线，且点D在BC上。点E在线段CD上，连接AE，AE垂直BC于点E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1933.jpg" }, { "index": 1934, "ocr_en": "In triangle ABC, angles B and C are acute angles, point A is above BC, point B is to the left of C, AD is the bisector of angle BAC, and point D is on BC. Point F is on line segment CD, connect AF, AF is perpendicular to BC at point F.", "ocr_zh": "在三角形ABC中，角B和角C是锐角，点A在BC上方，点B在C左侧，AD是角BAC的平分线，且点D在BC上。点F在线段CD上，连接AF，AF垂直BC于点F。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1934.jpg" }, { "index": 1935, "ocr_en": "In triangle ABC, angles B and C are acute angles, point A is above BC, point B is to the left of C. AD is the bisector of angle BAC and point D is on BC. Take a point P on the extension line of AD along D and connect it to DP. Point E is on the line segment CD and connect it to PE, which is perpendicular to BC at point E.", "ocr_zh": "在三角形ABC中，角B和角C是锐角，点A在BC上方，点B在C左侧，AD是角BAC的平分线，且点D在BC上。在AD沿D方向延长线上取一点P，连接DP。点E在线段CD上，连接PE，PE垂直BC于点E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1935.jpg" }, { "index": 1936, "ocr_en": "In triangle ABC, angles B and C are acute angles, point A is above BC, point B is to the left of C. AD is the bisector of angle BAC and point D is on BC. Take a point F on the extension line of AD along the direction of A and connect it to AF. Point E is on the line segment CD and connect it to FE, which is perpendicular to BC at point E.", "ocr_zh": "在三角形ABC中，角B和角C是锐角，点A在BC上方，点B在C左侧，AD是角BAC的平分线，且点D在BC上。在AD沿A方向延长线上取一点F，连接AF。点E在线段CD上，连接FE，FE垂直BC于点E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1936.jpg" }, { "index": 1937, "ocr_en": "There are parallel line segments AB and CD, AB is above CD, point A is to the left of B, point C is to the right of D, point C is directly below point B, and point D is directly below point A. Points Q and P are points between AB and CD. Connecting BQ and QC, the two line segments form a polyline convex to the left, and connecting BP and PC, the two line segments form a polyline convex to the right. Points Q and P are to the left and right of point B. BQ is the bisector of angle ABP and CQ is the bisector of angle PCD. Mark angles ABQ and QBP as angle α and angles QCD and QCP as angle β with arcs.", "ocr_zh": "有平行线段AB和CD，AB在CD上方，点A在B左侧，点C在D右侧，点C在点B的正下方，点D在点A的正下方。点Q和P是AB和CD之间的点，连接BQ和QC，两线段构成向左凸的折线，连接BP和PC，两线段构成向右凸的折线。点Q与P分别在点B的左侧和右侧。BQ是角ABP的平分线，CQ是角PCD的平分线。用弧线标记角ABQ和角QBP为角α，角QCD和角QCP为角β。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1937.jpg" }, { "index": 1938, "ocr_en": "There are parallel line segments AB and CD, AB is above CD, point A is to the left of B, point C is to the right of D, point C is directly below point B, and point D is directly below point A. Points Q and P are points between AB and CD. Connecting BQ and QC, the two line segments form a polyline convex to the left, and connecting BP and PC, the two line segments form a polyline convex to the left. Point Q is to the left of P and both points are to the lower left of point B and to the upper left of point C. BQ is the bisector of angle ABP and CQ is the bisector of angle PCD. Mark angles ABQ and QBP as angle α and angles QCD and QCP as angle β with arcs.", "ocr_zh": "有平行线段AB和CD，AB在CD上方，点A在B左侧，点C在D右侧，点C在点B的正下方，点D在点A的正下方。点Q和P是AB和CD之间的点，连接BQ和QC，两线段构成向左凸的折线，连接BP和PC，两线段构成向左凸的折线。点Q在P的左侧，且两点均在点B的左下方、C的左上方。BQ是角ABP的平分线，CQ是角PCD的平分线。用弧线标记角ABQ和角QBP为角α，角QCD和角QCP为角β。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1938.jpg" }, { "index": 1939, "ocr_en": "There are parallel lines AB and CD, AB above CD, point A to the left of B and point C to the left of D. The line EF intersects AB and CD. The straight line EF intersects AB at point E and CD at point F. Point E is to the upper right of F. Points M1 and M are between the lines AB and CD and to the left of EF. Point M1 is to the left of M. Connect EM1 and FM1, and the two line segments form a polyline that is convex to the left. connect EM and FM, and the two line segments form a polyline that is convex to the left. em is the bisector of angle AEF, and MF is the bisector of angle EFC. em1 is the bisector of angle AEM, and M1F is the bisector of angle MFC.", "ocr_zh": "有平行线段AB和CD，AB在CD上方，点A在B左侧，点C在D左侧。直线EF分别交AB于点E、交CD于点F。点E在F右上方。点M1和M在直线AB和CD之间，且在EF的左侧。点M1在M的左侧，连接EM1、FM1，两线段构成向左凸的折线，连接EM和FM，两线段构成向左凸的折线。EM是角AEF的平分线，MF是角EFC的平分线。EM1是角AEM的平分线，M1F是角MFC的平分线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1939.jpg" }, { "index": 1940, "ocr_en": "There are parallel lines AB and CD, AB above CD, point A to the left of B and point C to the left of D. The line EF intersects AB and CD. The line EF intersects AB at point E and CD at point F. Point E is to the upper right of F. Point M is between the lines AB and CD and to the left of EF. Connect EM and FM, the two line segments form a bisector convex to the left. EM is the bisector of angle AEF and MF is the bisector of angle EFC.", "ocr_zh": "有平行线段AB和CD，AB在CD上方，点A在B左侧，点C在D左侧.直线EF分别交AB于点E、交CD于点F。点E在F右上方。点M在直线AB和CD之间，且在EF的左侧。连接EM和FM，两线段构成向左凸的折线。EM是角AEF的平分线，MF是角EFC的平分线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1940.jpg" }, { "index": 1941, "ocr_en": "There are parallel lines AB and CD, AB above CD, point A to the left of B and point C to the left of D. The line EF intersects AB and CD. The straight line EF intersects AB at point E and CD at point F. Point E is to the upper right of F. Points M1 and M are between the lines AB and CD and to the left of EF. Point M1 is to the left of M. Connect EM1 and FM1, and the two line segments form a polyline convex to the left. EM is the bisector of angle AEF, and MF is the bisector of angle EFC. EM1 is the bisector of angle AEM, and M1F is the bisector of angle MFC. Make the MJ auxiliary line parallel to AB through point M. Point J is to the right of M. Mark angles AEM1 and M1EM with arcs for angle α and angles MFM1 and M1FC for angle β. Mark the dashed line MJ in red, and the yellow lines ME, MF, M1E, M1F, the portion to the left of point E in the line AB, and the portion to the left of point F in the line CD.", "ocr_zh": "有平行线段AB和CD，AB在CD上方，点A在B左侧，点C在D左侧。直线EF分别交AB于点E、交CD于点F。点E在F右上方。点M1和M在直线AB和CD之间，且在EF的左侧。点M1在M的左侧，连接EM1、FM1，两线段构成向左凸的折线，连接EM和FM，两线段构成向左凸的折线。EM是角AEF的平分线，MF是角EFC的平分线。EM1是角AEM的平分线，M1F是角MFC的平分线。过点M作MJ辅助线平行于AB，点J在M的右侧。用弧线标记角AEM1和角M1EM为角α，角MFM1和角M1FC为角β。用红色标记虚线MJ，黄线标记ME、MF、M1E、M1F、直线AB中点E左侧的部分、直线CD中点F左侧的部分。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1941.jpg" }, { "index": 1942, "ocr_en": "There are parallel line segments AB and CD, AB is above CD, point A is to the left of B, point C is to the left of D, point C is directly below point A, and point D is to the right of point B. Connect BD. point E is the intersection of the bisector of angle ABD with the bisector of angle BDC. connect EB and ED.", "ocr_zh": "有平行线段AB和CD，AB在CD上方，点A在B左侧，点C在D左侧，点C在点A的正下方，点D在点B的右下方。连接BD。点E是角ABD平分线与角BDC平分线的交点，连接EB、ED。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1942.jpg" }, { "index": 1943, "ocr_en": "There are parallel segments AB and CD, AB is above CD, point A is to the left of B, point C is to the left of D, and point C is directly below point B. Point E is a point between AB and CD, connecting BE and CE, and the two segments form a fold convex to the right. Angle ABE is obtuse and angle ECD is acute.CG is the bisector of angle DCE and point G is to the upper right of C. Extend CG in the direction of C to intersect angle ECD. Extend CG along C to intersect the bisector of angle ABE at point F, connecting CF and BF.", "ocr_zh": "有平行线段AB和CD，AB在CD上方，点A在B左侧，点C在D左侧，点C在点B的正下方，点E是AB和CD之间一点，连接BE和CE，两线段构成向右凸的折线。角ABE为钝角，角ECD为锐角。CG是角DCE的平分线，点G在C的右上方。沿C方向延长CG交角ABE的平分线于点F，连接CF和BF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1943.jpg" }, { "index": 1944, "ocr_en": "There are parallel line segments AB and CD, AB is above CD, point A is to the left of B, point C is to the left of D, point C is directly below point A, and point D is to the right below point B. Point E is the intersection of the angle ABD bisector and the angle BDC bisector. Point E is the intersection of the bisector of angle ABD with the bisector of angle BDC, connect EB and ED. point F is the point between AB and CD and to the left of points E, A and C, connect BF and DF.", "ocr_zh": "有平行线段AB和CD，AB在CD上方，点A在B左侧，点C在D左侧，点C在点A的正下方，点D在点B的右下方。点E是角ABD平分线与角BDC平分线的交点，连接EB、ED。点F是AB和CD之间的点且位于点E、A、C左侧，连接BF和DF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1944.jpg" }, { "index": 1945, "ocr_en": "There are parallel lines AB and CD, AB is below CD, point A is to the left of B, point C is to the left of D, point C is directly above point A, and point D is directly above point B. Point E is below the right of point D and above the right of point B. Connect BE and DE, the bisectors of angles CDE and ABE intersect at point F. Connect DF and BF.", "ocr_zh": "有平行线AB和CD，AB在CD下方，点A在B左侧，点C在D左侧，点C在点A的正上方，点D在点B的正上方。点E位于点D的右下方，点B的右上方。连接BE和DE，角CDE和角ABE的平分线相交于点F，连接DF和BF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1945.jpg" }, { "index": 1946, "ocr_en": "There are parallel lines AB and CD, AB is above CD, point A is to the right of B, point C is to the right of D. Point C is below the right side of point A and point D is below the left side of point B. Point P is a point below CD to the lower left of point C and to the lower right of point D. Connect PC and PA. The bisector of angle BAP and the bisector of angle DCP intersect at point K. Connect AK and CK.", "ocr_zh": "有平行线AB和CD，AB在CD上方，点A在B右侧，点C在D右侧，点C在点A的右下方方，点D在点B的左下方。点P是CD下方一点，位于C点左下方，D点右下方，连接PC、PA。角BAP的平分线和角DCP的平分线相交于点K，连接AK、CK。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1946.jpg" }, { "index": 1947, "ocr_en": "There are parallel lines AB and CD, AB is above CD, point A is to the right of B, point C is to the right of D, point C is to the lower right of point A, and point D is to the lower left of point B. Connect AC with point P lying to the left of AC, between AB and CD, and to the right of points B and D. Connect PA and PC.", "ocr_zh": "有平行线AB和CD，AB在CD上方，点A在B右侧，点C在D右侧，点C在点A的右下方，点D在点B的左下方。连接AC，点P位于AC左侧、AB和CD之间、点B和D的右侧。连接PA、PC。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1947.jpg" }, { "index": 1948, "ocr_en": "There are parallel lines AB and CD, AB is above CD, point A is to the right of B, point C is to the right of D, point C is to the lower right of point A, and point D is to the lower left of point B. Connect AC. Point P lies to the left of AC, between AB and CD, and to the right of points B and D. Connect PA and PC. the bisector of angle BAP and the bisector of angle DCP intersect at point K. Connect AK and CK. point K is to the left of point P.", "ocr_zh": "有平行线AB和CD，AB在CD上方，点A在B右侧，点C在D右侧，点C在点A的右下方，点D在点B的左下方。连接AC，点P位于AC左侧、AB和CD之间、点B和D的右侧。连接PA、PC。角BAP的平分线和角DCP的平分线相交于点K，连接AK、CK。点K在P左侧。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1948.jpg" }, { "index": 1949, "ocr_en": "There are parallel line segments AB and CD, AB is above CD, point A is to the left of B, point C is to the left of D, point A is directly above C, and point B is directly above D. Point E is on AB, point F is on CD, and point P lies to the lower left of point E and to the upper left of point F. Connecting PE and PF, the two line segments form a bisector convex to the left.", "ocr_zh": "有平行线段AB和CD，AB在CD上方，点A在B的左侧，点C在D的左侧，点A在C的正上方，点B在D的正上方。点E在AB上，点F在CD上，点P位于点E左下方、点F左上方。连接PE和PF，两线段构成向左凸的折线。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1949.jpg" }, { "index": 1950, "ocr_en": "In the acute triangle ABC, point A is above BC, point B is to the left of C. Take a point D on the extension line of AB along the direction of B and connect it to BD, take a point E on the extension line of AC along the direction of C and connect it to CE, the bisectors of angle DBC and angle BCE intersect at point P, connect them to BP and CP, and use the red arcs to mark the angle BAC and the yellow arcs to mark the angle BPC.", "ocr_zh": "在锐角三角形ABC中，点A在BC上方，点B在C左侧，在AB沿B方向延长线上取一点D，连接BD，在AC沿C方向延长线上取一点E，连接CE，角DBC和角BCE的平分线相交于点P，连接BP、CP，用红色弧线标记角BAC，黄色弧线标记角BPC。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1950.jpg" }, { "index": 1951, "ocr_en": "In an acute triangle ABC, point A is above BC and point B is to the left of C. Take a point D on the extension of BC along the direction of C and connect it to CD. The bisector of angle ABC and angle ACD intersects at a point P, which lies to the right and above the point C. Connect BP and CP, and use red arcs to mark the angle BAC and yellow arcs to mark the angle BPC.", "ocr_zh": "在锐角三角形ABC中，点A在BC上方，点B在C左侧，在BC沿C方向延长线上取一点D，连接CD，角ABC和角ACD的平分线相交于点P，位于点C右上方，连接BP和CP，用红色弧线标记角BAC，黄色弧线标记角BPC。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1951.jpg" }, { "index": 1952, "ocr_en": "In acute triangle ABC, point A is above BC and point B is to the left of C. The bisectors of angles ABC and ACB intersect at point P, which lies inside triangle ABC. Connect BP and CP and mark angle BAC with a red arc and angle BPC with a yellow arc.", "ocr_zh": "在锐角三角形ABC中，点A在BC上方，点B在C左侧，角ABC和角ACB的平分线相交于点P，位于三角形ABC内部，连接BP和CP，用红色弧线标记角BAC，黄色弧线标记角BPC。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1952.jpg" }, { "index": 1953, "ocr_en": "In the acute triangle ABC, point A is above BC and point B is to the left of C. Take a point D on the extension line of AB along the direction of B and connect it to BD, take a point E on the extension line of AC along the direction of C and connect it to CE, and the bisectors of the angle DBC and the angle BCE meet at the point P. Connect BP and CP and mark them in red.", "ocr_zh": "在锐角三角形ABC中，点A在BC上方，点B在C左侧，在AB沿B方向延长线上取一点D，连接BD，在AC沿C方向延长线上取一点E，连接CE，角DBC和角BCE的平分线相交于点P，连接BP、CP并以红色标记。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1953.jpg" }, { "index": 1954, "ocr_en": "In acute triangle ABC, point A is above BC and point B is to the left of C. The bisectors of angles ABC and ACB intersect at point P, inside triangle ABC, connecting BP and CP and labelled in red.", "ocr_zh": "在锐角三角形ABC中，点A在BC上方，点B在C左侧，角ABC和角ACB的平分线相交于点P，位于三角形ABC内部，连接BP和CP并以红色标记。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1954.jpg" }, { "index": 1955, "ocr_en": "In an acute triangle ABC, point A is above BC and point B is to the left of C. Take a point D on the extension of BC along the direction of C and connect it to CD. The bisectors of angle ABC and angle ACD intersect at point P, which lies to the upper right of point C. Connect BP and CP. Label the angle PBC with an arc to show that the angle is α, and the angle PCD is β.", "ocr_zh": "在锐角三角形ABC中，点A在BC上方，点B在C左侧，在BC沿C方向延长线上取一点D，连接CD，角ABC和角ACD的平分线相交于点P，位于点C右上方，连接BP和CP。用弧线标记角PBC为角α，角PCD为角β。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1955.jpg" }, { "index": 1956, "ocr_en": "In an acute triangle ABC, point A is above BC and point B is to the left of C. The bisectors of angles ABC and ACB intersect at point P, which lies inside triangle ABC, connecting BP and CP, and labelling the angle PBC with an arc for angle α and the angle PCB for angle β.", "ocr_zh": "在锐角三角形ABC中，点A在BC上方，点B在C左侧，角ABC和角ACB的平分线相交于点P，位于三角形ABC内部，连接BP和CP，用弧线标记角PBC为角α，角PCB为角β。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1956.jpg" }, { "index": 1957, "ocr_en": "In acute triangle ABC, point A is above BC and point B is to the left of C. Take a point D on the extension of AB along the B direction and connect it to BD, take a point E on the extension of AC along the C direction and connect it to CE, the bisectors of angles DBC and BCE intersect at point P and connect them to BP and CP. Label the angle PBC with an arc to show that angle α is angle α, and the angle PCB to show that it is angle β.", "ocr_zh": "在锐角三角形ABC中，点A在BC上方，点B在C左侧，在AB沿B方向延长线上取一点D，连接BD，在AC沿C方向延长线上取一点E，连接CE，角DBC和角BCE的平分线相交于点P，连接BP、CP。用弧线标记角PBC为角α，角PCB为角β。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1957.jpg" }, { "index": 1958, "ocr_en": "There is a horizontal straight line OM and vertical straight line ON, OM perpendicular ON in point O, point M in the lower right of N. Point A and B are on the ray OM and ON respectively, connect AB. Points A and B are on the rays OM and ON respectively, connect AB. the BE ray is the bisector of the angle NBA through point B, and point E is on the upper right of B. The line extending BE along the direction of B intersects the bisector of the angle BAO at point C, and connects BC and AC.", "ocr_zh": "有水平方向的直线OM和竖直方向的直线ON，OM垂直ON于点O，点M在N的右下方。点A和B分别在射线OM和ON上，连接AB。过点B作BE射线为角NBA的平分线，E点在B的右上方。BE沿B方向延长线与角BAO的平分线相交于点C，连接BC、AC。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1958.jpg" }, { "index": 1959, "ocr_en": "In an acute triangle ABC, point A is above BC and point B is to the left of C. The bisectors of angles ABC and ACB are BD and CE, respectively, which intersect at point O. Points D and E are on AC and AB, respectively.", "ocr_zh": "在锐角三角形ABC中，点A在BC上方，点B在C左侧，角ABC和角ACB的平分线分别是BD和CE，两者相交于点O。点D和E分别在AC和AB上。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1959.jpg" }, { "index": 1960, "ocr_en": "In acute triangle ABC, point A is to the left of BC, point B is below C. The bisector of angle ACB and angle ABC intersects at point M. M is inside triangle ABC, connect CM and BM. Extend AB in the direction of B, extend AC in the direction of C, and the bisectors of the outer angle of angle ACB and the outer angle of angle ABC intersect at point N. Connect BN and CN.", "ocr_zh": "在锐角三角形ABC中，点A在BC左侧，点B在C的下方，角ACB和角ABC的平分线交于点M，M在三角形ABC内，连接CM和BM。沿B方向延长AB，沿C方向延长AC，角ACB外角和角ABC外角的平分线相交于点N。连接BN和CN。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1960.jpg" }, { "index": 1961, "ocr_en": "In an acute triangle ABC, point A is above BC, point B is to the left of C. Points O1 and O2 are inside the triangle ABC and on the bisector of angle ACB, and O1 is to the lower right of O2. Connect BO2, CO2, and BO1, and use the arcs to mark angle BO2C as angle 1 and angle BO1C as angle 2.", "ocr_zh": "在锐角三角形ABC中，点A在BC上方，点B在C左侧，点O1和O2在三角形ABC内部，且在角ACB的平分线上，O1在O2的右下方。连接BO2、CO2和BO1，用弧线标记角BO2C为角1、角BO1C为角2。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1961.jpg" }, { "index": 1962, "ocr_en": "In an acute triangle ABC, point A is above BC, point B is to the left of C. The bisectors of angles ABC and ACB intersect at point O. Connect OB and OC, and point O lies inside triangle ABC.", "ocr_zh": "在锐角三角形ABC中，点A在BC上方，点B在C左侧，角ABC和角ACB的平分线相交于点O，连接OB和OC，点O位于三角形ABC内部。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1962.jpg" }, { "index": 1963, "ocr_en": "In the acute triangle ABC, point A is to the right of BC and is to the lower right of B and to the upper right of C. Point B is above C. Points D and E are points on sides AB and AC respectively, connect DE and fold triangle ADE to the left along DE to triangle DEA1, with point A1 on the outside of triangle ABC, to the lower left of point A and to the right of point C. Mark the line segments AD and AE with dotted lines, the angle BDA1 as angle 1 and A1EC as angle 2 with yellow arcs, and the angle BAC with red arcs.", "ocr_zh": "在锐角三角形ABC中，点A在BC右侧，且在B的右下方、C的右上方。点B在C上方。点D和E分别是边AB和AC上的点，连接DE，将三角形ADE沿DE向左翻折至三角形DEA1，点A1在三角形ABC外部，位于点A的左下方、点C的右侧。用虚线标记线段AD和AE，用黄色弧线标记角BDA1为角1、A1EC为角2，红色弧线标记角BAC。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1963.jpg" }, { "index": 1964, "ocr_en": "In the acute triangle ABC, point A is to the right of BC and is to the lower right of B and to the upper right of C. Point B is above C. Points D and E are points on sides AB and AC, respectively, connecting DE, and folding triangle ADE to the left along DE to triangle DEA1, with point A1 inside triangle ABC. Mark the line segments AD and AE with dotted lines, the angles BDA1 as angle 1 and A1EC as angle 2 with yellow arcs, and the angle BAC with red arcs.", "ocr_zh": "在锐角三角形ABC中，点A在BC右侧，且在B的右下方、C的右上方。点B在C上方。点D和E分别是边AB和AC上的点，连接DE，将三角形ADE沿DE向左翻折至三角形DEA1，点A1在三角形ABC内部。用虚线标记线段AD和AE，用黄色弧线标记角BDA1为角1、A1EC为角2，红色弧线标记角BAC。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1964.jpg" }, { "index": 1965, "ocr_en": "In the acute triangle ABC, point A is to the right of BC and is to the lower right of B and to the upper right of C. Point B is above C. Points D and E are points on sides AB and AC, respectively, connect DE, and fold triangle ADE to the left along DE to triangle DEA1, where point A1 is on the outside of triangle ABC, to the lower left of point A and to the right of point C. Mark the line segments AD and AE with yellow dashed lines and the line segments DE, DA1, and A1E in red. mark the line segment where side AC intersects triangle DEA1 with a dashed line.", "ocr_zh": "在锐角三角形ABC中，点A在BC右侧，且在B的右下方、C的右上方。点B在C上方。点D和E分别是边AB和AC上的点，连接DE，将三角形ADE沿DE向左翻折至三角形DEA1，点A1在三角形ABC外部，位于点A的左下方、点C的右侧。用黄色虚线标记线段AD和AE，红色标记线段DE、DA1、A1E。用虚线标记边AC与三角形DEA1相交的线段。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1965.jpg" }, { "index": 1966, "ocr_en": "In an acute triangle ABC, point A is to the right of BC and to the lower right of B and upper right of C. Point B is above C. Points D and E are points on sides AB and AC, respectively, connecting DE, and folding triangle ADE to the left along DE to triangle DEA1, with point A1 inside triangle ABC. Mark line segments AD and AE with yellow dashed lines and line segments DE, DA1, and A1E in red.", "ocr_zh": "在锐角三角形ABC中，点A在BC右侧，且在B的右下方、C的右上方。点B在C上方。点D和E分别是边AB和AC上的点，连接DE，将三角形ADE沿DE向左翻折至三角形DEA1，点A1在三角形ABC内部。用黄色虚线标记线段AD和AE，红色标记线段DE、DA1、A1E。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1966.jpg" }, { "index": 1967, "ocr_en": "In an acute triangle ABC, point A is to the right of BC and to the lower right of B and upper right of C. Point B is above C. Points D and E are points on sides AB and AC, respectively, connect DE, and fold triangle ADE to the left along DE to triangle DEA1, with point A1 on the outside of triangle ABC, to the lower left of point A and to the right of point C. Mark the line segments AD, AE, and the line segment where triangle DEA1 intersects AC with dotted lines, and mark angles BAC and BDA1 as angle 1 and A1EC as angle 2 with curved lines, and A1DE and ADE as angle α and A1ED and AED as angle β.", "ocr_zh": "点C没截全。（在锐角三角形ABC中，点A在BC右侧，且在B的右下方、C的右上方。点B在C上方。点D和E分别是边AB和AC上的点，连接DE，将三角形ADE沿DE向左翻折至三角形DEA1，点A1在三角形ABC外部，位于点A的左下方、点C的右侧。用虚线标记线段AD、AE、三角形DEA1与AC相交的线段，用弧线标记角BAC、角BDA1为角1、A1EC为角2，A1DE和ADE为角α，A1ED和AED为角β。）", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1967.jpg" }, { "index": 1968, "ocr_en": "In the acute triangle ABC, point C is above AB and point A is to the left of B. Points D and E are points on sides AB and AC respectively. connect DE as a straight line of auxiliary lines. fold triangle ADE to the right along DE to triangle DEA′, with A′ inside triangle ABC. mark line segments AE and AD in dashed lines, and line segments DE, DA′, and A′E in red. mark line segments AD and BD in purple. mark angle CEA′ in arcs as Angle 1, and angle BDA′ as Angle 2.", "ocr_zh": "在锐角三角形ABC中，点C在AB的上方，点A在B的左侧。点D和E分别是边AB和AC上的点，连接DE作辅助线直线，将三角形ADE沿DE向右翻折至三角形DEA′，A′在三角形ABC内部，用虚线标记线段AE和AD，用红色标记线段DE、DA′、A′E。用紫色标记线段AD和BD。用弧线标记角CEA′为角1、角BDA′为角2。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1968.jpg" }, { "index": 1969, "ocr_en": "In an acute triangle ABC, point B is to the right of AC and is to the lower right of A and to the upper right of C. Point A is above C. Points D and E are points on sides BC and AB, respectively, connecting to DE.Fold triangle BDE to the left along DE to triangle DEB′, with point B′ to the left of AC.The line segments where BE, BD, and triangle DEB′ intersect side AC are labelled as dashed lines. Mark angles AEB′ and CDB′ with arcs for angles 1 and 2, respectively. dB′ intersects AC at point F.", "ocr_zh": "在锐角三角形ABC中，点B在AC右侧，且在A的右下方、C的右上方。点A在C上方。点D和E分别是边BC和AB上的点，连接DE。将三角形BDE沿DE向左翻折至三角形DEB′，点B′在AC的左侧。BE、BD和三角形DEB′与边AC相交的线段标记为虚线。用弧线标记角AEB′和角CDB′分别为角1和2。DB′交AC于点F。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1969.jpg" }, { "index": 1970, "ocr_en": "In the acute triangle ABC, point B is above AC and point C is to the right of A. Points D and E are points on sides AB and AC, respectively. Connect DE and fold triangle ADE to the right along DE to triangle DEA′, with A′ inside triangle ABC. Connect BA′ and CA′. BA′ bisects angle ABC and CA′ bisects angle ACB. mark the line segments AD and AE with dotted lines, and mark angles BDA′ and CEA′ with arcs for angles 1 and 2, respectively.", "ocr_zh": "在锐角三角形ABC中，点B在AC上方，点C在A的右侧。点D和E分别是边AB和AC上的点，连接DE，将三角形ADE沿DE向右翻折至三角形DEA′，A′在三角形ABC内部。连接BA′、CA′。BA′平分角ABC，CA′平分角ACB。用虚线标记线段AD和AE，用弧线分别标记角BDA′、角CEA′为角1和角2。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1970.jpg" }, { "index": 1971, "ocr_en": "In acute triangle ABC, point A is above BC and point B is to the left of C. Points D and E are points on sides AB and AC, respectively. Connect DE and fold triangle ADE down along DE to triangle DEA′, with A′ on side BC. Mark the line segments AD and AE with dashed lines, and the angles BDA′, BA′D, CEA′, and CA′E with arcs for angles 1, 2, 3, and 4, respectively.", "ocr_zh": "在锐角三角形ABC中，点A在BC上方，点B在C的左侧。点D和E分别是边AB和AC上的点，连接DE，将三角形ADE沿DE向下翻折至三角形DEA′，A′在边BC上。用虚线标记线段AD和AE，用弧线分别标记角BDA′、角BA′D、角CEA′、角CA′E分别为角1、2、3和4。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1971.jpg" }, { "index": 1972, "ocr_en": "In an obtuse triangle ABC, angle C is obtuse, point A is above BC, point B is to the left of C. Points M and N are points on sides AB and BC, respectively, connecting MN. Fold triangle BNM to the right along MN to triangle B′MN, with point B′ to the left of AB, to the upper right of B, and to the lower left of A. Label the line segments BM, BN, and side BA intersecting triangle B′MN as dashed lines.", "ocr_zh": "在钝角三角形ABC中，角C是钝角，点A在BC上方，点B在C左侧，点M、N分别是边AB和BC上的点，连接MN，将三角形BNM沿MN向右翻折至三角形B′MN，点B′在AB左侧，B的右上方、A的左下方。标记线段BM、BN、边BA与三角形B′MN相交的线段为虚线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1972.jpg" }, { "index": 1973, "ocr_en": "In triangle ABC, angles B and A are acute angles, and point A is to the right of BC and to the lower right of B and to the upper right of C. Point B is above C. Points D and E are points on sides AB and AC, respectively; connect DE and fold triangle ADE to the left along DE to triangle DEA′. Point A′ is on the outside of triangle ABC, to the lower left of point A and to the lower right of point C. Use dotted lines to mark the line segments AD, AE, and the line segment where side AC intersects triangle DEA′, and use arcs to mark angles BDA′ as angle 1 and A′EC as angle 2.", "ocr_zh": "在三角形ABC中，角B和A是锐角，点A在BC右侧，且在B的右下方、C的右上方。点B在C上方。点D和E分别是边AB和AC上的点，连接DE，将三角形ADE沿DE向左翻折至三角形DEA′。点A′在三角形ABC外部，位于点A的左下方、点C的右下方。用虚线标记线段AD、AE、边AC与三角形DEA′相交的线段，用弧线标记角BDA′为角1、A′EC为角2。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1973.jpg" }, { "index": 1974, "ocr_en": "In triangle ABC, angles B and A are acute angles, and point A is to the right of BC and to the lower right of B and to the upper right of C. Point B is above C. Points D and E are points on sides AB and AC, respectively; connect DE and fold triangle ADE to the left along DE to triangle DEA′. Point A′ is inside triangle ABC. Mark the line segments AD and AE with dotted lines and the angles BDA′ as angle 1 and A′EC as angle 2 with arcs.", "ocr_zh": "在三角形ABC中，角B和A是锐角，点A在BC右侧，且在B的右下方、C的右上方。点B在C上方。点D和E分别是边AB和AC上的点，连接DE，将三角形ADE沿DE向左翻折至三角形DEA′。点A′在三角形ABC内部。用虚线标记线段AD、AE，用弧线标记角BDA′为角1、A′EC为角2。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1974.jpg" }, { "index": 1975, "ocr_en": "There is a quadrilateral ABDC with point A above B, C and D, point B to the left of C, and point D to the upper right of point B and to the upper left of point C. Angles BAC, ABD and ACD are all acute and angle BDC is obtuse. Connect BC as an auxiliary line with points E and F on sides AC and AB, respectively. Connect CF and BE, the bisector of angle ACD and the bisector of angle ABD, respectively, which intersect at point G. Label the line segments AB, AC, BD, and DC in red, BG and GC in yellow, and use the arcs to label angles 1, 2, 3, and 4 for angle CBD, angle DCB, angle DBG, and angle DCG, respectively.", "ocr_zh": "有四边形ABDC，点A在B、C和D的上方，点B在C左侧，点D在点B的右上方、点C的左上方。角BAC、ABD、ACD均为锐角，角BDC为钝角。连接BC作辅助线，点E和F分别在边AC和AB上。连接CF和BE，分别是角ACD的平分线和角ABD的平分线，二者交于点G。用红色标记线段AB、AC、BD、DC，黄色标记BG、GC，用弧线标注角CBD、角DCB、角DBG、角DCG分别为角1、2、3、4。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1975.jpg" }, { "index": 1976, "ocr_en": "There is a quadrilateral ABDC with point A above B, C and D, point B to the left of C, and point D to the upper right of point B and the upper left of point C. Angles BAC, ABD and ACD are all acute and angle BDC is obtuse. Points E and F are on sides AC and AB, respectively. Connect CF and BE, the bisector of angle ACD and the bisector of angle ABD, respectively, which intersect at point G.", "ocr_zh": "有四边形ABDC，点A在B、C和D的上方，点B在C左侧，点D在点B的右上方、点C的左上方。角BAC、ABD、ACD均为锐角，角BDC为钝角。点E和F分别在边AC和AB上。连接CF和BE，分别是角ACD的平分线和角ABD的平分线，二者交于点G。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1976.jpg" }, { "index": 1977, "ocr_en": "In an acute triangle ABC, A is above BC, point B is to the left of C. The bisector of angle ABC and angle ACB intersects at point D 1, the bisector of angle ABD1 and angle ACD1 intersects at point D2, and points D1 and D2 are both in the interior of the triangle ABC, with D2 above D1, connecting BD1, CD1, BD2 and CD2.", "ocr_zh": "在锐角三角形ABC中，A在BC上方，点B在C的左侧，角ABC与角ACB的平分线交于点D 1，角ABD1与角ACD1的平分线交于点D2，点D1、D2均在三角形ABC内部，D2在D1上方，连接BD1、CD1、BD2、CD2。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1977.jpg" }, { "index": 1978, "ocr_en": "Lines AB and CD intersect at point O, connecting AC and BD, with point C to the right above A and to the left above O, and point B to the right above O and to the left above D. The bisector of angle CAB and angle BDC intersect at point P, above CD and AB. Connect AP to intersect CO at point M. Connect DP to intersect OB at point N.", "ocr_zh": "线段AB和CD相交于点O，连接AC和BD，点C在A的右上方、O的左上方，点B在O的右上方、D的左上方。角CAB和角BDC的平分线相交于点P，位于CD和AB的上方。连接AP交CO于点M，连接DP交OB于点N。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1978.jpg" }, { "index": 1979, "ocr_en": "Line segments AB and CD intersect at point O. Connect AC and BD, with point C to the upper right of A and to the upper left of O, and point B to the upper right of O and to the upper left of D.", "ocr_zh": "线段AB和CD相交于点O，连接AC和BD，点C在A的右上方、O的左上方，点B在O的右上方、D的左上方。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1979.jpg" }, { "index": 1980, "ocr_en": "In an acute triangle ABC, point A is above BC and point B is to the left of C. There is a right triangle XYZ with XY perpendicular to XZ at point X. The right angle sides XY and XZ pass through points B and C. Vertex X is inside triangle ABC, vertex Y is to the lower left of B, and vertex Z is to the lower right of C.", "ocr_zh": "在锐角三角形ABC中，点A在BC的上方，点B在C的左侧。有直角三角形XYZ，XY垂直XZ于点X，直角边XY和XZ分别经过点B和C，顶点X在三角形ABC内部，顶点Y在B的左下方，顶点Z在C的右下方。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1980.jpg" }, { "index": 1981, "ocr_en": "There is a quadrilateral ABDC with point A above B, C and D, point B to the left of C, and point D to the upper right of point B and to the upper left of point C. Angles BAC, ABD and ACD are all acute and angle BDC is obtuse. The twelve equidistant lines of angles ABD and ACD intersect at points G1, G2, G3.... ...G11. These points are arranged vertically from top to bottom, connecting G1B, G1C, G2B, G2C, G3B, G3C, G11B, G11C", "ocr_zh": "有噪音。（有四边形ABDC，点A在B、C和D的上方，点B在C左侧，点D在点B的右上方、点C的左上方。角BAC、ABD、ACD均为锐角，角BDC为钝角。角ABD和角ACD的十二等分线相交于点G1、G2、G3...G11。这些点由上至下在竖直方向上排列，连接G1B、G1C、G2B、G2C、G3B、G3C、G11B、G11C）", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_1981.jpg" }, { "index": 1982, "ocr_en": "There is a quadrilateral ADBE with point A above B, E and D, point D to the left of E, and point B to the upper right of point D and the upper left of point E. Angles BDA, BEA and DAE are all acute and angle DBE is obtuse. The bisectors of angles ADB and AEB intersect at point C, in the interior of the triangle, connecting CD and CE.", "ocr_zh": "有四边形ADBE，点A在B、E和D的上方，点D在E左侧，点B在点D的右上方、点E的左上方。角BDA、BEA、DAE均为锐角，角DBE为钝角。角ADB和角AEB的平分线交于点C，位于三角形内部，连接CD和CE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1982.jpg" }, { "index": 1983, "ocr_en": "There is a quadrilateral ABDC with point A above B, C and D, point B to the left of C, and point D to the upper right of point B and the upper left of point C. Angles BAC, ABD and ACD are all acute and angle BDC is obtuse.", "ocr_zh": "有四边形ABDC，点A在B、C和D的上方，点B在C左侧，点D在点B的右上方、点C的左上方。角BAC、ABD、ACD均为锐角，角BDC为钝角。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1983.jpg" }, { "index": 1984, "ocr_en": "In triangle ABC, angles B and C are acute angles, point B is to the left of C, point A is above BC, and the length of AB is greater than that of AC. point D is the midpoint of line segment BC, connecting it to AD, and marking line segment AD in red.", "ocr_zh": "在三角形ABC中，角B和角C是锐角，点B在C的左侧，点A在BC上方，AB长度大于AC。点D是线段BC的中点，连接AD，用红色标记线段AD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1984.jpg" }, { "index": 1985, "ocr_en": "In triangle ABC, angles B and C are acute angles, point B is to the left of C, point A is above BC, and the length of AB is greater than that of AC. point D is the midpoint of segment BC, connect AD, take a point E on the extension line of AD along the direction of D, and connect BE and DE to make an auxiliary line, with AD equalling to DE. mark the line segment DE in red, and mark the line segment BE in yellow.", "ocr_zh": "在三角形ABC中，角B和角C是锐角，点B在C的左侧，点A在BC上方，AB长度大于AC。点D是线段BC的中点，连接AD，在AD沿D方向延长线上取一点E，连接BE和DE作辅助线，AD等于DE。用红色标记线段DE，黄色标记线段BE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1985.jpg" }, { "index": 1986, "ocr_en": "In triangle ABC, angle B and angle C are acute angles, point B is on the left side of C, point A is above BC, and the length of AB is greater than that of AC. point D is the midpoint of the line segment BC, connect AD, take a point E on the extension of the line of AD along the direction of D, and connect BE and DE to make an auxiliary line, and AD is equal to DE.", "ocr_zh": "在三角形ABC中，角B和角C是锐角，点B在C的左侧，点A在BC上方，AB长度大于AC。点D是线段BC的中点，连接AD，在AD沿D方向延长线上取一点E，连接BE和DE作辅助线，AD等于DE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1986.jpg" }, { "index": 1987, "ocr_en": "In triangle ABC, angles A and B are acute and angle C is obtuse, point A is above BC and point B is to the left of C. A point D is on the prolongation of BC along C. Point E is on the line AC and connects DE. On the extension line of BC along C, there is a point D, connect CD, CD is equal to BC, point E is on the line AC, connect DE. On the extension line of AC along C, take a point F, connect BF, CF as an auxiliary line, CF is equal to CE. Mark the line CF in red, triangle ECD in blue, and triangle BCF in yellow.", "ocr_zh": "在三角形ABC中，角A和角B是锐角，角C是钝角，点A在BC上方，点B在C左侧。在BC沿C方向延长线上有一点D，连接CD，CD等于BC，点E在线段AC上，连接DE。在AC沿C方向延长线上取一点F，连接BF、CF作辅助线，CF等于CE。用红色标记线段CF，蓝色标记三角形ECD，黄色标记三角形BCF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1987.jpg" }, { "index": 1988, "ocr_en": "In triangle ABC, angles A and B are acute and angle C is obtuse, point A is above BC and point B is to the left of C. A point D is on the prolongation of BC along C. Point E is on the line AC and connects DE. A point D is on the extension line of BC along C. Connect CD, CD is equal to BC, point E is on the line AC, connect DE. Take a point F on the extension line of AC along C. Connect CF and DF as an auxiliary line, CF is equal to AC. Label the line CF in red, the triangle ABC in blue and the triangle DCF in yellow.", "ocr_zh": "在三角形ABC中，角A和角B是锐角，角C是钝角，点A在BC上方，点B在C左侧。在BC沿C方向延长线上有一点D，连接CD，CD等于BC，点E在线段AC上，连接DE。在AC沿C方向延长线上取一点F，连接CF和DF作辅助线，CF等于AC。用红色标记线段CF，用蓝色标记三角形ABC，黄色标记三角形DCF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1988.jpg" }, { "index": 1989, "ocr_en": "In triangle ABC, angles A and B are acute and angle C is obtuse, point A is above BC and point B is to the left of C. There is a point D on the line extending BC along C. Connect CD, which is equal to BC, and point E on the line AC, which is connected to DE.", "ocr_zh": "在三角形ABC中，角A和角B是锐角，角C是钝角，点A在BC上方，点B在C左侧。在BC沿C方向延长线上有一点D，连接CD，CD等于BC，点E在线段AC上，连接DE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1989.jpg" }, { "index": 1990, "ocr_en": "In quadrilateral ABCD, AD is parallel to and above BC, point A is to the left of D, point B is to the left of C. Point O is the midpoint of segment AC. Connect AC, point O is the midpoint of line AC, point E is a point on AD, connect OE, extend EO along the direction of O and intersect BC at point F, connect OF as an auxiliary line. Mark the line segment OF in red, the triangle AOE in blue and the triangle FOC in yellow.", "ocr_zh": "在四边形ABCD中，AD平行于BC且在BC上方，点A在D左侧，点B在C左侧。连接AC，点O是线段AC的中点，点E是AD上一点，连接OE，沿O方向延长EO交BC于点F，连接OF作辅助线。用红色标记线段OF，用蓝色标记三角形AOE，黄色标记三角形FOC。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1990.jpg" }, { "index": 1991, "ocr_en": "In triangle ABC, angles B and C are acute angles, point B is to the left of C, point A is above BC, and the length of AB is greater than that of AC. points D and E are on line segment BC, point D is to the left of E and nearer to B, point E is the midpoint of line segment CD, and point D is the midpoint of line segment BC. connect AD and AE. take a point F on the extension line of AE along the direction of E, and make an auxiliary line by connecting DF and EF so that FE is equal to AE. mark line segment EF in red and triangle DEF in yellow. Mark segment EF in red, triangle AEC in blue and triangle DEF in yellow.", "ocr_zh": "在三角形ABC中，角B和角C是锐角，点B在C的左侧，点A在BC上方，AB长度大于AC。点D和E在线段BC上，点D在E的左侧且更靠近B，点E是线段CD的中点，点D是线段BC的中点，连接AD、AE。在AE沿E方向延长线上取一点F，连接DF、EF作辅助线，使得FE等于AE。用红色标记线段EF，用蓝色标记三角形AEC，黄色标记三角形DEF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1991.jpg" }, { "index": 1992, "ocr_en": "In triangle ABC, angles B and C are acute angles, point B is to the left of C, point A is above BC, and the length of AB is greater than that of AC. points D and E are on line segment BC, point D is to the left of E and nearer to B, point E is the midpoint of line segment CD, and point D is the midpoint of line segment BC, connect AD and AE.", "ocr_zh": "在三角形ABC中，角B和角C是锐角，点B在C的左侧，点A在BC上方，AB长度大于AC。点D和E在线段BC上，点D在E的左侧且更靠近B，点E是线段CD的中点，点D是线段BC的中点，连接AD、AE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1992.jpg" }, { "index": 1993, "ocr_en": "In triangle ABC, angles B and C are acute angles, point B is to the left of C, point A is above BC, and the length of AB is greater than that of AC. points D and E are on line BC, point D is to the left of E and closer to B, and the length of DE is equal to that of EC. connect AE, and through point D, make DF parallel to AB and intersect with AE at point F. Take a point G on the extension of AE along the line of direction E, so that EG is equal to EF, and connect EG, CG as an auxiliary line. Mark the segment GE in red, triangle DEF in blue and triangle CEG in yellow.", "ocr_zh": "在三角形ABC中，角B和角C是锐角，点B在C的左侧，点A在BC上方，AB长度大于AC。点D和E在线段BC上，点D在E的左侧且更靠近B，DE长度等于EC。连接AE，过点D作DF平行于AB交AE于点F。在AE沿E方向延长线上取一点G，使得EG等于EF，连接EG、CG作辅助线。用红色标记线段GE，用蓝色标记三角形DEF、黄色标记三角形CEG。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1993.jpg" }, { "index": 1994, "ocr_en": "In triangle ABC, angles B and C are acute angles, point B is to the left of C, point A is above BC, and the length of AB is greater than that of AC. points D and E are on line BC, point D is to the left of E and closer to B, and the length of DE is equal to that of EC. connect AE, and cross point D and make DF parallel to AB and intersect point AE at point F.", "ocr_zh": "在三角形ABC中，角B和角C是锐角，点B在C的左侧，点A在BC上方，AB长度大于AC。点D和E在线段BC上，点D在E的左侧且更靠近B，DE长度等于EC。连接AE，过点D作DF平行于AB交AE于点F。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1994.jpg" }, { "index": 1995, "ocr_en": "In triangle ABC, angles B and C are acute angles, point B is to the left of C, point A is above BC, and the length of AB is greater than that of AC.Points D and E are on line segment BC, point D is to the left of E and nearer to B. Point E is the midpoint of line segment CD, and point D is the midpoint of line segment BC. Take a point F on the extension of the line AE along E. Connect EF and CF to make an auxiliary line such that AE equals EF. Label the line EF in red, the triangle ADE in blue, and the triangle CEF in yellow.", "ocr_zh": "在三角形ABC中，角B和角C是锐角，点B在C的左侧，点A在BC上方，AB长度大于AC。点D和E在线段BC上，点D在E的左侧且更靠近B，点E是线段CD的中点，点D是线段BC的中点。在线段AE沿E方向延长线上取一点F，连接EF和CF作辅助线，使得AE等于EF。用红色标记线段EF，用蓝色标记三角形ADE，黄色标记三角形CEF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1995.jpg" }, { "index": 1996, "ocr_en": "In quadrilateral ABCD, DC is parallel to AB, above AB and less than AB. point D is to the left of C, point A is to the left of B. point M is the midpoint of line BC, connect DM and AM, DM is the bisector of angle ADC. Extend the line DM along M to intersect the line AB along B at point P. Connect MP and BP as an auxiliary line. Label the line segment MP in red, the triangle DCM in blue and the triangle BMP in yellow.", "ocr_zh": "在四边形ABCD中，DC平行于AB，在AB上方，且长度小于AB。点D在C的左侧，点A在B的左侧，点M是线段BC的中点，连接DM和AM，DM是角ADC的平分线。沿M方向延长线段DM交AB沿B方向延长线于点P，连接MP、BP作辅助线。用红色标记线段MP，用蓝色标记三角形DCM，黄色标记三角形BMP。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1996.jpg" }, { "index": 1997, "ocr_en": "In a quadrilateral ABCD, DC is parallel to AB, above AB and of less length than AB. point D is to the left of C, point A is to the left of B, point M is the midpoint of the line segment BC, connect DM and AM, DM is the bisector of angle ADC.", "ocr_zh": "在四边形ABCD中，DC平行于AB，在AB上方，且长度小于AB。点D在C的左侧，点A在B的左侧，点M是线段BC的中点，连接DM和AM，DM是角ADC的平分线。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_1997.jpg" }, { "index": 1998, "ocr_en": "In triangle ABC, angles B and C are acute angles, point B is to the left of C, point A is above BC, and the length of AB is greater than that of AC. points D and E are on line BC, point D is to the left of E and closer to B, and the length of DE is equal to that of EC. connect AE, and make DF parallel to AB through point D to intersect with AE at point F. Take a point G on the extension of the line of AE along the direction of E, so that the length of AE is equal to that of EG, and connect DG and EG as auxiliary lines. Connect DG and EG as an auxiliary line. Mark the segment GE in red, triangle AEC in blue and triangle DEG in yellow.", "ocr_zh": "在三角形ABC中，角B和角C是锐角，点B在C的左侧，点A在BC上方，AB长度大于AC。点D和E在线段BC上，点D在E的左侧且更靠近B，DE长度等于EC。连接AE，过点D作DF平行于AB交AE于点F。在AE沿E方向延长线上取一点G，使得AE长度等于EG，连接DG、EG作辅助线。用红色标记线段GE，用蓝色标记三角形AEC、黄色标记三角形DEG。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1998.jpg" }, { "index": 1999, "ocr_en": "In isosceles triangle ABC, AB equals AC, angles B and C are acute angles, point C is to the right of B, and point A is above BC. Point D is a point on line segment CB, connect AD. rotate AD around point A counterclockwise by a certain angle to AE, point E is to the right of AC. Connect BE, point G is the midpoint of BE, inside triangle ABC, connect AG.", "ocr_zh": "在等腰三角形ABC中，AB等于AC，角B和角C是锐角，点C在B右侧，点A在BC上方。点D是线段CB上一点，连接AD。将AD绕点A逆时针旋转一定的角度至AE，点E在AC的右侧。连接BE，点G是BE的中点，在三角形ABC内部，连接AG。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_1999.jpg" }, { "index": 2000, "ocr_en": "In triangle ABC, angle ABC is acute, point C is to the right of B and point A is above BC. Point D is the midpoint of the line segment BC and is connected to AD. point E is on AC and is connected to BE to intersect AD at point F.", "ocr_zh": "在三角形ABC中，角ABC是锐角，点C在B右侧，点A在BC上方。点D是线段BC中点，连接AD。点E在AC上，连接BE交AD于点F。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2000.jpg" }, { "index": 2001, "ocr_en": "In triangle ABC, angles C and B are acute angles, point C is to the right of B, and point A is above BC. Point D is a point on the line segment CB, connected to AD, AD bisects the angle BAC. point E is the midpoint of the side BC, located between the line segments BD. Point E is the midpoint of side BC between the lines BD. Make a parallel line to AD through point E, intersect the line extending AC in the direction of A at point F, and intersect AB at point G, connecting EF and AF.", "ocr_zh": "在三角形ABC中，角C和角B是锐角，点C在B右侧，点A在BC上方。点D是线段CB上一点，连接AD，AD平分角BAC。点E是边BC的中点，位于线段BD之间。过点E作AD的平行线，交AC沿A方向延长线于点F，交AB于点G，连接EF、AF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2001.jpg" }, { "index": 2002, "ocr_en": "There is an acute triangle ABC, point B is to the left of point C, point A is above BC, point D is the midpoint of BC, connect AD. make isosceles triangles ABE and ACF with sides AB and AC respectively, point E is to the left of AB and point F is to the right of AC. Connect EF, BE and CF. EF intersects AB at point H and EH equals HF.", "ocr_zh": "有锐角三角形ABC，点B在点C左侧，点A在BC上方，点D是BC的中点，连接AD。分别以AB、AC为边作等腰三角形ABE和ACF，点E在AB的左侧，点F在AC的右侧。连接EF、BE和CF。EF交AB于点H，且EH等于HF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2002.jpg" }, { "index": 2003, "ocr_en": "In right triangle ABC, AC is perpendicular to BC at point C, point B is to the left of C, point A is above BC, the length of AC is greater than BC, point D is the midpoint of the line segment AB, point E is a point on the line segment CA and is closer to the point C, and DE is connected.", "ocr_zh": "在直角三角形ABC中，AC垂直BC于点C，点B在C的左侧，点A在BC上方，AC长度大于BC，点D是线段AB的中点，点E是线段CA上一点且更靠近点C，连接DE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2003.jpg" }, { "index": 2004, "ocr_en": "There is an acute triangle ABC, point B is to the left of point C, point A is above BC, point D is the midpoint of BC, connect AD. make isosceles triangles ABE and ACF with sides AB and AC respectively, with point E to the left of AB and point F to the right of AC. Connect EF, BE and CF.", "ocr_zh": "有锐角三角形ABC，点B在点C左侧，点A在BC上方，点D是BC的中点，连接AD。分别以AB、AC为边作等腰三角形ABE和ACF，点E在AB的左侧，点F在AC的右侧。连接EF、BE和CF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2004.jpg" }, { "index": 2005, "ocr_en": "In right triangle ABC, AC is perpendicular to BC at point C, point B is to the left of C, point A is above BC, the length of AC is greater than BC, point D is the midpoint of line segment AB, point E is a point on the extension line of line segment CA along the direction of A. Connect DE and AE.", "ocr_zh": "在直角三角形ABC中，AC垂直BC于点C，点B在C的左侧，点A在BC上方，AC长度大于BC，点D是线段AB的中点，点E是线段CA沿A方向延长线上一点，连接DE、AE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2005.jpg" }, { "index": 2006, "ocr_en": "In triangle ABC, angles C and B are acute angles, point C is above B, point A is to the left of BC, and point D is the midpoint of line segment CB. Point F is a point on line AC, and point E is a point on line AB, connecting DE, DF, and EF. DF is perpendicular to DE at point D.", "ocr_zh": "在三角形ABC中，角C和角B是锐角，点C在B上方，点A在BC左侧，点D是线段CB的中点。点F是线段AC上一点，点E是线段AB上一点，连接DE、DF、EF。DF垂直DE于点D。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2006.jpg" }, { "index": 2007, "ocr_en": "In triangle ABC, angles C and B are acute angles, point A is above BC, point C is to the right of B, point D is a point on line segment CB, connect AD, AD bisects angle BAC, and mark angle ACD as angle 2α and angle ABD as angle α with red arcs. mark line segment AB in red, line segments AC and CD in yellow, and line segment AD in blue.", "ocr_zh": "在三角形ABC中，角C和角B是锐角，点A在BC上方，点C在B右侧，点D是线段CB上一点，连接AD，AD平分角BAC，用红色弧线标记角ACD为角2α，角ABD为角α。用红色标记线段AB，黄色标记线段AC和CD，蓝色标记线段AD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2007.jpg" }, { "index": 2008, "ocr_en": "In triangle ABC, angles ACD and B are acute angles, point A is above BC, point C is to the right of B, point D is a point on segment CB, connect AD, AD bisecting angle BAC, point E is a point on the extension line of segment AC along direction C, connect DE and CE as auxiliary lines. Use arcs to mark angle ACD as angle 2α and angle ABD as angle α. Mark segments CD and CE in red and segment DE in yellow.", "ocr_zh": "在三角形ABC中，角ACD和角B是锐角，点A在BC上方，点C在B右侧，点D是线段CB上一点，连接AD，AD平分角BAC，点E是线段AC沿C方向延长线上一点，连接DE、CE作辅助线。用弧线标记角ACD为角2α，角ABD为角α。用红色标记线段CD、CE，黄色标记线段DE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2008.jpg" }, { "index": 2009, "ocr_en": "In triangle ABC, angles ACD and B are acute angles, point A is above BC, point C is to the right of B, point D is a point on line segment CB, connect AD, AD bisects angle BAC, point E is a point on the extension line of line segment AC along the direction of C, connect DE and CE as auxiliary lines. Use arcs to mark angle ACD as angle 2α and angle ABD as angle α. Mark segments AB, BC and CE in red and segment DE in yellow.", "ocr_zh": "在三角形ABC中，角ACD和角B是锐角，点A在BC上方，点C在B右侧，点D是线段CB上一点，连接AD，AD平分角BAC，点E是线段AC沿C方向延长线上一点，连接DE、CE作辅助线。用弧线标记角ACD为角2α，角ABD为角α。用红色标记线段AB、BC、CE，黄色标记线段DE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2009.jpg" }, { "index": 2010, "ocr_en": "In triangle ABC, angles ACD and B are acute angles, point A is above BC, point C is to the right of B, point D is a point on line segment CB, connect AD, AD bisecting angle BAC, point E is a point on the extension of line segment AC along the direction of C, connect DE and CE as auxiliary lines. Mark the angle ACD as angle 2α and the angle ABD as angle α with an arc. Mark the line segments CD and CE in red, the triangle ABD in blue and the triangle ADE in yellow.", "ocr_zh": "在三角形ABC中，角ACD和角B是锐角，点A在BC上方，点C在B右侧，点D是线段CB上一点，连接AD，AD平分角BAC，点E是线段AC沿C方向延长线上一点，连接DE、CE作辅助线。用弧线标记角ACD为角2α，角ABD为角α。用红色标记线段CD、CE，用蓝色标记三角形ABD，黄色标记三角形ADE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2010.jpg" }, { "index": 2011, "ocr_en": "In triangle ABC, angles C and B are acute angles, point A is above BC, point C is to the right of B, point D is a point on segment CB, connect AD, AD bisects angle BAC, point E is a point on segment AB and closer to point B, connect DE as an auxiliary line. Mark angle ACD as angle 2α and angle ABD as angle α. Mark segments AE and AC in red and DE in yellow.", "ocr_zh": "在三角形ABC中，角C和角B是锐角，点A在BC上方，点C在B右侧，点D是线段CB上一点，连接AD，AD平分角BAC，点E是线段AB上一点且更靠近点B，连接DE作辅助线。用弧线标记角ACD为角2α，角ABD为角α。标记线段AE、AC为红色，DE为黄色。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2011.jpg" }, { "index": 2012, "ocr_en": "In triangle ABC, angles C and B are acute angles, point A is above BC, point C is to the right of B, point D is a point on segment CB, connect AD, AD bisects angle BAC, point E is a point on segment AB and closer to point B, connect DE as an auxiliary line. Use arcs to mark angle ACD as angle 2α and angle ABD as angle α. Mark segments BE and ED in red, triangle AED in yellow and triangle ACD in blue.", "ocr_zh": "在三角形ABC中，角C和角B是锐角，点A在BC上方，点C在B右侧，点D是线段CB上一点，连接AD，AD平分角BAC，点E是线段AB上一点且更靠近点B，连接DE作辅助线。用弧线标记角ACD为角2α，角ABD为角α。标记线段BE、ED为红色，三角形AED为黄色，三角形ACD为蓝色。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2012.jpg" }, { "index": 2013, "ocr_en": "In triangle ABC, angles ACB and B are acute angles, point A is above BC, point C is to the right of B. Point D is a point on line segment CB, connect AD, which bisects angle BAC, and point E is a point on the extension of line segment AC along C. Connect DE and CE as auxiliary lines. Use the arcs to mark angle ACD as angle 2α and angle ABD as angle α.", "ocr_zh": "在三角形ABC中，角ACB和角B是锐角，点A在BC上方，点C在B右侧，点D是线段CB上一点，连接AD，AD平分角BAC，点E是线段AC沿C方向延长线上一点，连接DE、CE作辅助线。用弧线标记角ACD为角2α，角ABD为角α。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2013.jpg" }, { "index": 2014, "ocr_en": "In triangle ABC, angles C and B are acute angles, point A is above BC, point C is to the right of B, point D is a point on line segment CB, connect AD, AD bisects angle BAC, point E is a point on line segment AB and is closer to point B, connect DE as an auxiliary line. Use the arcs to mark angle ACD as angle 2α and angle ABD as angle α.", "ocr_zh": "在三角形ABC中，角C和角B是锐角，点A在BC上方，点C在B右侧，点D是线段CB上一点，连接AD，AD平分角BAC，点E是线段AB上一点且更靠近点B，连接DE作辅助线。用弧线标记角ACD为角2α，角ABD为角α。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2014.jpg" }, { "index": 2015, "ocr_en": "In triangle ABC, angles C and B are acute angles, point A is above BC, point C is to the left of B, point D is a point on line segment CB, connect AD, which is perpendicular to BC at point D. Point E is a point on line segment CD and the length of DE is equal to that of DB. connect AE as an auxiliary line. Mark line segments AE and AB in yellow and line segment BE in red.", "ocr_zh": "在三角形ABC中，角C和角B是锐角，点A在BC上方，点C在B左侧，点D是线段CB上一点，连接AD，AD垂直BC于点D。点E是线段CD上一点，且DE长度等于DB。连接AE作辅助线。标记线段AE、AB为黄色，线段BE为红色。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2015.jpg" }, { "index": 2016, "ocr_en": "In triangle ABC, angles C and B are acute angles, point A is above BC, point C is to the left of B, point D is a point on line segment CB, connect AD, AD is perpendicular to BC at point D.", "ocr_zh": "在三角形ABC中，角C和角B是锐角，点A在BC上方，点C在B左侧，点D是线段CB上一点，连接AD，AD垂直BC于点D。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2016.jpg" }, { "index": 2017, "ocr_en": "There are right triangles ABC and BDC, point A is above BC, point B is to the left of C. AB is equal to AC and AB is perpendicular to AC at point A. BD is perpendicular to CD at point D. Point D is to the right of AC. Point H is inside triangle ABC and on BD, connect AH, AH is perpendicular to BD at point H. Point E is a point on BH, connect AE, AD as auxiliary line. Mark line segments BE and CD in red, triangle ABE in yellow and triangle ADC in blue.", "ocr_zh": "有直角三角形ABC和BDC，点A在BC上方，点B在C左侧，AB等于AC且AB垂直AC于点A，BD垂直CD于点D，点D在AC的右侧。点H在三角形ABC内部且在BD上，连接AH，AH垂直BD于点H。点E是BH上一点，连接AE、AD作辅助线。标记线段BE、CD为红色，三角形ABE为黄色，三角形ADC为蓝色。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2017.jpg" }, { "index": 2018, "ocr_en": "There are right triangles ABC and BDC, point A is above BC, point B is to the left of C, AB is equal to AC and AB is perpendicular to AC at point A, BD is perpendicular to CD at point D, and point D is to the right of AC. Point H is inside triangle ABC and on BD, connect AH, AH is perpendicular to BD at point H.", "ocr_zh": "有直角三角形ABC和BDC，点A在BC上方，点B在C左侧，AB等于AC且AB垂直AC于点A，BD垂直CD于点D，点D在AC的右侧。点H在三角形ABC内部且在BD上，连接AH，AH垂直BD于点H。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2018.jpg" }, { "index": 2019, "ocr_en": "In triangle ABC, point D is the point on line segment BC, point A is above BC, point C is to the left of point B. Connect AD, which is perpendicular to CB at point D. Point E is a point on the extension line of CB along the direction of B. Connect BE and AE to make an auxiliary line. Mark lines BC and BE in red and lines AC and AE in yellow.", "ocr_zh": "在三角ABC中，点D是线段BC上的点，点A在BC上方，点C在点B的左侧，连接AD，AD垂直CB于点D。点E是CB沿B方向延长线上一点，连接BE、AE作辅助线。标记线段BC、BE为红色，线段AC、AE为黄色。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2019.jpg" }, { "index": 2020, "ocr_en": "In triangle ABC, point D is the point on line segment BC, point A is above BC, point C is to the left of point B. Connect AD, which is perpendicular to CB at point D. Point E is a point on the extension line of CB along the direction of B. Connect BE and AE to make an auxiliary line. Mark line segments AB and BE in red, triangle ACD in blue and triangle ADE in yellow.", "ocr_zh": "在三角ABC中，点D是线段BC上的点，点A在BC上方，点C在点B的左侧，连接AD，AD垂直CB于点D。点E是CB沿B方向延长线上一点，连接BE、AE作辅助线。标记线段AB、BE为红色，三角形ACD为蓝色，三角形ADE为黄色。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2020.jpg" }, { "index": 2021, "ocr_en": "There are right triangles ABC and BDC, point A is above BC, point B is to the left of C, AB is equal to AC and AB is perpendicular to AC at point A, BD is perpendicular to CD at point D, and point D is to the right of AC. Point H is inside triangle ABC and on BD, connect AH, AH is perpendicular to BD at point H. Point Q is a point on the extension line of CD along the direction of D, connect AQ, QD as an auxiliary line. Mark line segments BH, CD and QD in red, mark triangle ABH in blue and triangle ACQ in yellow.", "ocr_zh": "有直角三角形ABC和BDC，点A在BC上方，点B在C左侧，AB等于AC且AB垂直AC于点A，BD垂直CD于点D，点D在AC的右侧。点H在三角形ABC内部且在BD上，连接AH，AH垂直BD于点H。点Q是CD沿D方向延长线上一点，连接AQ、QD作辅助线。标记线段BH、CD、QD为红色，标记三角形ABH为蓝色，三角形ACQ为黄色。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2021.jpg" }, { "index": 2022, "ocr_en": "There are right triangles ABC and ABD, AB perpendicular to BC at point B, AD perpendicular to AB at point A and the length of AD is equal to AB. point C is on the left side of B, point E is a point on the line segment AB, connect DE, along the direction of E to extend DE to intersection with AC at point G, intersection with BC along the direction of the extension of the line in the direction of C at point F, connect EF. point H is a point on the line segment GE, connect AH, HB, HB perpendicular to the EF at point H. point H is a point on the line segment GE, connect AH, HB, HB perpendicular to the EF at point H. point H, point H, HB perpendicular to the EF.", "ocr_zh": "有直角三角形ABC和ABD，AB垂直BC于点B，AD垂直AB于点A且AD长度等于AB。点C在B的左侧，点E是线段AB上一点，连接DE，沿E方向延长DE交AC于点G，交BC沿C方向延长线于点F，连接EF。点H是线段GE上一点，连接AH、HB，HB垂直EF于点H。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2022.jpg" }, { "index": 2023, "ocr_en": "In an isosceles right triangle ABC, AB is perpendicular to BC at point B and AB is equal to BC, point B is above AC, point A is to the left of C, point D is on the line segment AB, connect CD, point E is on AC, connect BE to intersect CD at point F, BE is perpendicular to AC at point E.", "ocr_zh": "在等腰直角三角形ABC中，AB垂直BC于点B且AB等于BC，点B在AC上方，点A在C的左侧，点D在线段AB上，连接CD，点E在AC上，连接BE交CD于点F，BE垂直AC于点E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2023.jpg" }, { "index": 2024, "ocr_en": "There is an equilateral triangle ABC, point A is above BC, point B is on the left side of C, point M is the point where BC extends the line in the direction of C, make the ray CM. CN is a ray inside the angle ACM, and the symmetry point of point A about CN is D. Connect AD, BD, and CD, where AD and BD intersect the ray CN at the points E and P, respectively.", "ocr_zh": "有等边三角形ABC，点A在BC上方，点B在C的左侧，点M是BC沿C方向延长线的点，作射线CM。CN是角ACM内部的一条射线，点A关于CN的对称点为D，连接AD、BD、CD，其中AD、BD分别交射线CN于点E和P。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2024.jpg" }, { "index": 2025, "ocr_en": "There is an equilateral triangle ABC, point A is above BC, point B is to the left of C. Make a straight line a parallel to AB through point C. Point D is between the line segments BC, point E is on the straight line a and above point C. Connect AD and DE, angle ADE is acute.", "ocr_zh": "有等边三角形ABC，点A在BC上方，点B在C的左侧，过点C作直线a平行于AB。点D在线段BC之间，点E在直线a上且在点C上方，连接AD和DE，角ADE是锐角。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2025.jpg" }, { "index": 2026, "ocr_en": "There are acute isosceles triangles ABC and ADE, point A is above BC, point B is to the left of C, AB is equal to AC, and AD is equal to AE. where point D is inside triangle ABC, point E is to the right of AC, and the length of AD is less than that of AC. connect BD and CE to make an auxiliary line and mark it in red. Triangles ABD and ACE are labelled blue and yellow respectively.", "ocr_zh": "有锐角等腰三角形ABC和ADE，点A在BC上方，点B在C左侧，AB等于AC，AD等于AE。其中点D在三角形ABC内部，点E在AC右侧，且AD长度小于AC。连接BD、CE作辅助线并用红色标记。三角形ABD和ACE分别标记为蓝色和黄色。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2026.jpg" }, { "index": 2027, "ocr_en": "There are acute isosceles triangles ABC and ADE, point A is above BC, point B is to the left of C, AB is equal to AC, and AD is equal to AE. where point D is in the interior of triangle ABC, point E is to the right of AC, and the length of AD is less than that of AC. connect BD and CE to make an auxiliary line and mark it in red.", "ocr_zh": "有锐角等腰三角形ABC和ADE，点A在BC上方，点B在C左侧，AB等于AC，AD等于AE。其中点D在三角形ABC内部，点E在AC右侧，且AD长度小于AC。连接BD、CE作辅助线并用红色标记。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2027.jpg" }, { "index": 2028, "ocr_en": "There are acute isosceles triangles ABC and ADE, point A is above BC, point B is to the left of C, point A is on BC, point B is to the left of C, AB is equal to AC, and AD is equal to AE, where points D and E are on the right side of AC, and the length of AD is less than that of AC. connect CE and BD to point P, and connect AP. point F is on BP, and connect AF as an auxiliary line, with AF perpendicular to BP, and point H is on PE, and point H is on AH as an auxiliary line. Point F is on BP, connect AF as an auxiliary line, AF is perpendicular to BP at point F. Point H is on PE, connect AH as an auxiliary line, AH is perpendicular to PE at point H. Label the line AP in red, label the lines AF and AH in yellow, and label the angles BAC and DAE with arcs.", "ocr_zh": "有锐角等腰三角形ABC和ADE，点A在BC上方，点B在C左侧，点A在BC上，点B在C左侧，AB等于AC，AD等于AE。其中点D和E在AC的右侧，且AD长度小于AC。连接CE和BD交于点P，连接AP。点F在BP上，连接AF作辅助线，AF垂直BP于点F，点H在PE上，连接AH作辅助线，AH垂直PE于点H。用红色标记线段AP，用黄色标记线段AF和AH，用弧线标记角BAC和角DAE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2028.jpg" }, { "index": 2029, "ocr_en": "There are acute isosceles triangles ABC and ADE, point A is above BC, point B is to the left of C, AB is equal to AC, and AD is equal to AE. where points D and E are to the right of AC, and the length of AD is less than that of AC. connect BD to intersection of AC with point O, connect CE, and the prolongation of BD along the direction of D to intersection of CE with point P, connect DP as an auxiliary line. Mark angles BAC, DAE and BPC with red arcs, and mark segments AB, AC, BD, CP and DP in yellow.", "ocr_zh": "有锐角等腰三角形ABC和ADE，点A在BC上方，点B在C左侧，AB等于AC，AD等于AE。其中点D和E在AC的右侧，且AD长度小于AC。连接BD交AC于点O，连接CE，BD沿D方向延长线交CE于点P，连接DP作辅助线。用红色弧线标记角BAC、角DAE和角BPC，用黄色标记线段AB、AC、BD、CP、DP。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2029.jpg" }, { "index": 2030, "ocr_en": "There are equilateral triangles ABC and BED, point C is above AB, point A is to the left of B, point E is above BD, and point B is to the left of D, where points A, B, and D are collinear, connect CD to intersect BE at point G, connect AE to intersect BC at point F, and CD to intersect AE at point H, connecting FG and HB.", "ocr_zh": "有等边三角形ABC和BED，点C在AB上方，点A在B左侧，点E在BD上方，点B在D左侧，其中点A、B、D共线，连接CD交BE于点G，连接AE交BC于点F，CD与AE交于点H，连接FG、HB。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2030.jpg" }, { "index": 2031, "ocr_en": "In the horizontal direction of the line BC above a point A, point B in the left side of point C, connect AB and AC, BA perpendicular to AC in point A, point D is BC along the direction of the extension of the line on a point, connect AD. AD as the side of the square ADEF, EF in the AD of the lower right, connect DF, AE intersected at point O, connect OC, CF.", "ocr_zh": "在水平方向的直线BC上方有一点A，点B在点C左侧，连接AB和AC，BA垂直AC于点A，点D是BC沿B方向延长线上一点，连接AD。以AD为边作正方形ADEF，EF在AD的右下方，连接DF、AE交于点O，连接OC、CF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2031.jpg" }, { "index": 2032, "ocr_en": "There is a point A above a horizontal line BC, point B is to the left of point C, connect AB and AC, BA is perpendicular to AC at point A, point D is a point on the extension line of BC along the direction of C, connect AD. Make a square ADEF with AD as a side, EF is to the right of AD, connect CF.", "ocr_zh": "在水平方向的直线BC上方有一点A，点B在点C左侧，连接AB和AC，BA垂直AC于点A，点D是BC沿C方向延长线上一点，连接AD。以AD为边作正方形ADEF，EF在AD的右侧，连接CF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2032.jpg" }, { "index": 2033, "ocr_en": "A point A is above the horizontal line BC, point B is to the left of point C. Connect AB and AC, BA is perpendicular to AC at point A. Point D is a point on the line BC, between points B and C. Connect AD. make a square ADEF with AD as a side, EF is to the right of AD, connect CF. point EF is to the right of AC.", "ocr_zh": "在水平方向的直线BC上方有一点A，点B在点C左侧，连接AB和AC，BA垂直AC于点A，点D是直线BC上一点，在点B和C之间，连接AD。以AD为边作正方形ADEF，EF在AD的右侧，连接CF。点EF在AC的右侧。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2033.jpg" }, { "index": 2034, "ocr_en": "In triangle ABC, M is the midpoint of BC,D is a point on MC, and E is a point outside triangle ABC. Connect AD,AE,BE, and DE", "ocr_zh": "三角形ABC中，M为BC中点，D为MC上一点，E为三角形ABC外一点，连接AD,AE,BE,DE", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2034.jpg" }, { "index": 2035, "ocr_en": "Triangle AOB and triangle MON have a common vertex O, and each side extends in different directions. The two triangles have no overlapping parts. Connect AM and BN", "ocr_zh": "三角形AOB和三角形MON有共同的顶点O，各边向不同方向延伸，两个三角形无重叠部分。连接AM,BN", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2035.jpg" }, { "index": 2036, "ocr_en": "There is a point N on one side AB of triangle AOB, and a point M outside triangle ABC. Connect OM,ON, and MN to form a new triangle MON", "ocr_zh": "三角形AOB的一边AB上有一点N，三角形ABC外有一点M，连接OM,ON,MN，形成一个新的三角形MON", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2036.jpg" }, { "index": 2037, "ocr_en": "Triangle ABC and triangle CED share a vertex C, and each side extends in different directions. There is no overlapping part between the two triangles. Connect AD and BE and let them intersect at point O. Point M is on AO and point N is on BO, then connect CM,CN, and MN to form a new triangle MCN", "ocr_zh": "三角形ABC和三角形CED共享一个顶点C，各边向不同方向延伸，两三角形无重叠部分。连接AD和BE,交于点O，M在AO上，N在BO上，连接CM,CN,MN,形成一个新的三角形MCN", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2037.jpg" }, { "index": 2038, "ocr_en": "Triangle ABC and Triangle ADE share the vertex A. Each side extends in different directions. There is no overlapping part between the two triangles. Connect BE and CD. Connect CE and let it intersect AD at point F. Connect BD and let it intersect CE at point G.", "ocr_zh": "三角形ABC和三角形ADE共享一个顶点A，各边向不同方向延伸，两三角形无重叠部分。连接BE,CD，连接CE交AD与点F，连接BD交CE于点G。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2038.jpg" }, { "index": 2039, "ocr_en": "Take points B and C on the straight line BC, and point A outside the line. Connect these points to form a triangle. Take point D on the line segment BC, point E on the right side of line segment AC and outside triangle ABC. Connect AD, AE and DE to form triangle ADE, and connect CE.", "ocr_zh": "在直线BC上取两点B,C，和直线外一点A连成一个三角形。在线段BC上取一点D，线段AC右侧、三角形ABC外侧一点E，连接AD,AE,DE，形成三角形ADE，连接CE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2039.jpg" }, { "index": 2040, "ocr_en": "In the right-angled triangle ABC, angle C is the right angle. A line CE is drawn through point C and intersects line segment AB. Line segment AD is drawn from point A and is perpendicular to CE at point D. A line segment BE is drawn from point B and is also perpendicular to CE at point E. Line segment AD is orange, line segment DE is red, line segment BE is yellow, triangle ACD is blue, and triangle BCE is yellow.", "ocr_zh": "在直角三角形ABC中，角C为直角。过点C作一条直线CE穿过线段AB，过点A作线段AD垂直CE于点D，过点B作一条线段BE垂直CE于点E。线段AD为橙色,线段DE为红色，线段BE为黄色，三角形ACD为蓝色，三角形BCE为黄色。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2040.jpg" }, { "index": 2041, "ocr_en": "In the right-angled triangle ABC, angle C is the right angle. A line CE is drawn through point C and intersects line segment AB. Line segment AD is drawn from point A and is perpendicular to CE at point D. Another line segment BE is drawn from point B and is perpendicular to CE at point E. Line segments AC and BC are colored red.", "ocr_zh": "在直角三角形ABC中，角C为直角。过点C作一条直线CE穿过线段AB，过点A作线段AD垂直CE于点D，过点B作一条线段BE垂直CE于点E。线段AC,BC为红色", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2041.jpg" }, { "index": 2042, "ocr_en": "In triangle ABC, angle C is a right angle. A line l is drawn through point C, and this line only intersects triangle ABC at point C. A line segment AD is drawn through point A and intersects line l at point D. A line segment BE is drawn through point B and intersects line l at point E. Triangle ADC and triangle BCE form two right-angled triangles respectively. Line segment AD and line segment BC are yellow, line segment DE is red, triangle ADC is blue, and triangle BCE is yellow.", "ocr_zh": "三角形ABC的角C为直角，过点C作直线l，l仅与三角形ABC交于C点，过点A作线段AD交l于点D，过点B作线段BE交l于点E，ADC和BCE分别构成两个直角三角形。线段AD，BC为黄色，线段DE为红色，三角形ADC为蓝色，三角形BCE为黄色。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2042.jpg" }, { "index": 2043, "ocr_en": "In triangle ABC, angle C is a right angle. A line l is drawn through point C, and this line only intersects triangle ABC at point C. A line segment AD is drawn through point A and intersects line l at point D. A line segment BE is drawn through point B and intersects line l at point E. Triangle ADC and triangle BCE form two right-angled triangles respectively. Line segment AC and line segment BC are red.", "ocr_zh": "三角形ABC的角C为直角，过点C作直线l，l仅与三角形ABC交于C点，过点A作线段AD交l于点D，过点B作线段BE交l于点E，ADC和BCE分别构成两个直角三角形。线段AC和线段BC为红色", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2043.jpg" }, { "index": 2044, "ocr_en": "In triangle ABC, angle C is a right angle. A line l is drawn through point C, and this line only intersects triangle ABC at point C. A line segment AD is drawn through point A and intersects line l at point D. A line segment BE is drawn through point B and intersects line l at point E. ADC and BCE respectively form two right-angled triangles. All the line segments in the figure are black.", "ocr_zh": "三角形ABC的角C为直角，过点C作直线l，l仅与三角形ABC交于C点，过点A作线段AD交l于点D，过点B作线段BE交l于点E，ADC和BCE分别构成两个直角三角形。图中所有线段均为黑色。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2044.jpg" }, { "index": 2045, "ocr_en": "In triangle ABC, angle C is a right angle. Through point C, draw a line l that passes through the line segment AB. At point A, draw a line segment AD that is perpendicular to l at point D. At point B, draw a line segment BE that is perpendicular to l at point E.", "ocr_zh": "三角形ABC中角C为直角，过点C作直线l穿过线段AB。过点A作线段AD垂直l于点D，过点B作线段BE垂直l于点E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2045.jpg" }, { "index": 2046, "ocr_en": "On the upper and lower sides of the straight line CF, points B and A are taken respectively. Lines AB, BC, and AC are connected to form triangle ABC, where angle C is obtuse. On the left side of line CF and to the right of line segment AB, points D and E are taken, with point D being to the left of point E. Lines AD and BE are connected. Angle ADF is obtuse, and angle BEF is also obtuse.", "ocr_zh": "在直线CF上下两侧分别取点B，A，连接AB,BC,AC形成三角形ABC，其中C为钝角。在CF上、线段AB左侧分别取点D,E，D在E左侧，连接AD,BE。角ADF为钝角，角BEF为钝角。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2046.jpg" }, { "index": 2047, "ocr_en": "Take a point C on line segment DE. Take points A and B on the upper part of line segment DE, where point A is to the left of point B. Connect AD, BE, AC, and BC. Angles E, D, and ACB are all acute angles.", "ocr_zh": "在线段DE上取一点C，在线段DE上侧取点A,B，其中A在B左侧。连接AD,BE,AC,BC,AB。角E，角D，角ACB均为锐角。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2047.jpg" }, { "index": 2048, "ocr_en": "There is a right triangle ABD, with angle ADB being the right angle. From point A, draw line segment AC perpendicular to AB, and ensure that AC = AB. Construct right triangle ACE such that triangle ABD is congruent to triangle ACE, and point D lies on segment AE. Fold triangle ABD along the AB side as the axis of symmetry, resulting in triangle ABM. Fold triangle ACE along the AC side as the axis of symmetry, resulting in triangle ACN.", "ocr_zh": "有一直角三角形ABD，角ADB为直角。以A为顶点作AC垂直AB，且AC=AB。作直角三角形ACE使得三角形ABD与三角形ACE全等，且D点在线段AE上。将三角形ABD以AB边为对称轴翻折，得到三角形ABM，将三角形ACE按AC边为对称轴翻折，得到三角形ACN。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2048.jpg" }, { "index": 2049, "ocr_en": "There is a parallelogram ABDE, where angle D is equal to angle E, and both of these angles are obtuse. On side DE, take point C and connect AC and BC, such that angle ACB is an obtuse angle and angle ACB equals angle D and angle E.", "ocr_zh": "有一平行四边形ABDE，其中角D=角E，且这两个角为钝角。在边DE上取点C，连接AC,BC，使得角ACB为钝角，且角ACB=角D=角E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2049.jpg" }, { "index": 2050, "ocr_en": "On the straight line m, take three points D, A, and E from left to right. On the upper part of m, take points B, F, and C respectively. Connect BD, BA, BF, FD, FA, FE, CF, CA, and CE. Make sure that angle BDA is equal to angle BAC and angle AEC.", "ocr_zh": "在直线m上从左至右取三点D,A,E，在m上方分别取点B,F,C，连接BD,BA,BF,FD,FA,FE,CF,CA,CE，使得角BDA=角BAC=角AEC。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2050.jpg" }, { "index": 2051, "ocr_en": "There is a pentagon ABCDE, where angles A and C are right angles, and angles D and E are acute angles. Draw an auxiliary line passing through points A and C. Draw EF perpendicular to this auxiliary line at point F, draw BG perpendicular to this auxiliary line at point B, and draw DH perpendicular to this auxiliary line at point H. The lengths of line segments EF, BG, and DH are 6, 3, and 4 respectively.", "ocr_zh": "有一五边形ABCDE，其中角A和角C为直角，角D和角E为锐角。做一条辅助线经过点A和点C，过点E作EF垂直于这条辅助线于F，过点B作BG垂直于这条辅助线于点G，过点D作DH垂直于这条辅助线于点H。线段EF,BG,DH的长度分别为6，3，4.", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2051.jpg" }, { "index": 2052, "ocr_en": "There is a right triangle BCE, with angle E being the right angle. On side CE, take a point D and connect BD. At point C, draw line segment AC and connect AD, such that AC is perpendicular to BC and AD is perpendicular to CE.", "ocr_zh": "有一直角三角形BCE，角E为直角。在边CE上取一点D，连接BD。过C点作线段AC并连接AD，使得AC垂直BC，AD垂直CE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2052.jpg" }, { "index": 2053, "ocr_en": "Take points A and D outside line segment BC, and take point E on line segment BC. Connect AB, AE, DE, and DC, such that angles B, angle AED, and angle C are all right angles.", "ocr_zh": "在线段BC外取两点A,D，在线段BC上取一点E，连接AB,AE,DE,DC，使得角B，角AED，角C均为直角。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2053.jpg" }, { "index": 2054, "ocr_en": "Take points A and B on either side of the line CD, and points E and F on the line CD itself. Draw rays CA and CB, and connect BE and AF.", "ocr_zh": "在直线CD两侧分别取点A,B，在直线CD上取点E,F，作射线CA,CB，连接BE,AF", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2054.jpg" }, { "index": 2055, "ocr_en": "On the straight line ED, take points E, C, F from left to right. Take points B and A outside the line ED, with point B above point A. Draw ray CB through point C and point B, and draw ray CA through point C and point A. Connect BE and AF.", "ocr_zh": "在直线ED上从左到右分别取点E,C,F，在直线ED外取点B,A，B在A上方，过点C,B作射线CB，过点C,A作射线CA，连接BE,AF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2055.jpg" }, { "index": 2056, "ocr_en": "On the straight line CD, take points C, E, F from left to right. Take point B on the left side of CD and point A on the right side. Draw ray CB through point C and point B, and draw ray CA through point C and point A. Connect BE and AF.", "ocr_zh": "在直线CD上从左到右分别取点C,E,F，在直线CD左侧取点B，右侧取点A。过点C,B作射线CB，过点C,A作射线CA，连接BE,AF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2056.jpg" }, { "index": 2057, "ocr_en": "There is a square ABCD. Extend DC to point E and connect AE. Extend CB to point F and connect AF, making angle EAF 45°. Connect EF. Take point G on segment DC and draw auxiliary line AG. Segments DG and BF are in red.", "ocr_zh": "有一正方形ABCD，延长DC至点E，连接AE。延长CB至点F，连接AF，使得角EAF为45°，连接EF。在线段DC上取点G，作辅助线AG。线段DG,BF为红色。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2057.jpg" }, { "index": 2058, "ocr_en": "There is a square ABCD. Point E is taken on line segment DC and point F is taken on line segment BC. Connect AE, AF, and EF. Make angle EAF 45°. Extend line segment CB to point G and connect AG and BG with dotted lines. Triangle AEF is blue, triangle AGF is yellow, line segments DE and BF are yellow, and line segment EF is red.", "ocr_zh": "有一正方形ABCD，在线段DC上取点E，在线段BC上取点F，连接AE,AF,EF，使得角EAF为45°。延长线段CB至点G，用虚线连接AG,BG。三角形AEF为蓝色，三角形AGF为黄色，线段DE,BF为黄色，线段EF为红色。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2058.jpg" }, { "index": 2059, "ocr_en": "There is a square ABCD. Extend line segment CB to point F, extend line segment DC to point E, and connect AE, AF, and EF. Make angle EAF 45°. Take point G on line segment DC and connect AG with a dotted line.", "ocr_zh": "有一正方形ABCD，延长线段CB至点F，延长线段DC至点E，连接AE,AF,EF，使得角EAF为45°。在线段DC上取点G，用虚线连接AG。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2059.jpg" }, { "index": 2060, "ocr_en": "There is a right triangle ABC, with angle BAC being the right angle. Point F is taken on line segment BC. Line segment CB is extended to point E, and line segments AE and AF are drawn, such that angle EAF is 45 degrees.", "ocr_zh": "有一直角三角形ABC，角BAC为直角，在线段BC上取点F，延长线段CB至点E，连接AE,AF，使得角EAF为45°。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2060.jpg" }, { "index": 2061, "ocr_en": "There is a right triangle ABC, with angle BAC being the right angle. Points E and F are taken on line segment BC from left to right, and line segments AE and AF are drawn, such that angle EAF is 45°.", "ocr_zh": "有一直角三角形ABC，角BAC为直角。在线段BC上从左至右取点E,F，连接AE,AF，使得角EAF为45°。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2061.jpg" }, { "index": 2062, "ocr_en": "There is a triangle ABC, with angle BAC being a right angle. On the line segment BC, starting from the left to the right, take points E and F. Connect AE and AF, such that angle EAF is 45°. On the left side of triangle ABC, take point D. Use dotted lines to connect DB, DA, and DE, such that DB is perpendicular to BC at point B.", "ocr_zh": "有一三角形ABC，角BAC为直角。在线段BC上从左至右取点E,F，连接AE,AF，使得角EAF为45°。在三角形ABC左侧取点D，用虚线连接DB,DA,DE，使得DB垂直BC于点B。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2062.jpg" }, { "index": 2063, "ocr_en": "There is a quadrilateral ABCD, where angles ABC and D are right angles. Take point E on line segment BC and point F on line segment CD. Connect AE, AF, and EF. Extend line segment CB with a dotted line to point G, and connect AG with a dotted line.", "ocr_zh": "有一四边形ABCD，其中角ABC和角D为直角。在线段BC上取点E，在线段CD上取点F，连接AE,AF,EF。用虚线延长线段CB至点G，用虚线连接AG。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2063.jpg" }, { "index": 2064, "ocr_en": "There is a quadrilateral ABCD. Point E is taken on line segment BC and point F is taken on line segment DC. Lines AE, AF, and EF are drawn. Line segment CD is extended to point M with a dotted line and line AM is drawn. Triangle AFM is yellow, triangle AEF is blue. Line segments BE and dotted line MD are red.", "ocr_zh": "有一四边形ABCD，在线段BC上取点E，在线段DC上取点F，连接AE,AF,EF。用虚线延长CD至点M，连接AM。三角形AFM为黄色，三角形AEF为蓝色。线段BE，虚线MD为红色。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2064.jpg" }, { "index": 2065, "ocr_en": "There is a quadrilateral ABCD. Take point E on line segment BC and point F on line segment DC. Connect AE, AF, and EF. Extend CD with a dotted line to point M and connect AM. Line segments AB and AD are red. Define angle EAF as angle α.", "ocr_zh": "有一四边形ABCD，在线段BC上取点E，在线段DC上取点F，连接AE,AF,EF。用虚线延长CD至点M，连接AM。线段AB,AD为红色，定义角EAF为角α。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2065.jpg" }, { "index": 2066, "ocr_en": "There is a quadrilateral ABCD, where angles B and D are right angles. Take point E on line segment BC and point F on line segment CD, then connect AE, AF, and EF.", "ocr_zh": "有一四边形ABCD，其中角B和角D为直角。在线段BC上取点E，在线段CD上取点F，连接AE,AF,EF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2066.jpg" }, { "index": 2067, "ocr_en": "There is a square ABCD. Point F is taken on side AD and point E is taken on side AB. The line segments CE and CF are connected with a solid line, and the line segment EF is connected with a dotted line. The extension of line segment AD to point G is drawn with a dotted line, and line segment CG is also connected with a dotted line.", "ocr_zh": "有一正方形ABCD，在边AD上取点F，在边AB上取点E，用实线连接CE,CF，用虚线连接EF。用虚线延长线段AD至点G，用虚线连接CG。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2067.jpg" }, { "index": 2068, "ocr_en": "In the square ABCD, take point F on side AD and point E on side AB. Connect CF and CE.", "ocr_zh": "在正方形ABCD中，在边AD上取点F，在边AB上取点E，连接CF,CE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2068.jpg" }, { "index": 2069, "ocr_en": "In the right-angled trapezoid ABCD, angles A and B are right angles. A point E is taken on the hypotenuse AB, and lines DE and CE are drawn.", "ocr_zh": "在直角梯形ABCD中，角A和角B为直角，在直角边AB上取点E，连接DE,CE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2069.jpg" }, { "index": 2070, "ocr_en": "There is a quadrilateral CPMN. On side CN, take a point A. Connect MA, PN. On segment AM, take a point B. Connect CB, PB.", "ocr_zh": "有一四边形CPMN，在边CN上取一点A，连接MA,PN，在线段AM上取一点B，连接CB,PB。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2070.jpg" }, { "index": 2071, "ocr_en": "There is a quadrilateral CPMN. On side CN, take a point A. Connect MA, PN. On segment AM, take a point B. Connect CB, PB.", "ocr_zh": "有一四边形CPMN，在边CN上取一点A，连接MA,PN，在线段AM上取一点B，连接CB,PB。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2071.jpg" }, { "index": 2072, "ocr_en": "There is a right triangle ABC, with angle ACB being the right angle. On the line segment AB, take points M and N from left to right. Connect CM and CN. Extend CM to point D and extend CN to point E. Connect DE, such that angle E is a right angle.", "ocr_zh": "有一直角三角形ACB，角ACB为直角。在线段AB上从左至右取点M,N，连接CM,CN。延长CM至点D，延长CN至点E，连接DE，使得角E为直角。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2072.jpg" }, { "index": 2073, "ocr_en": "In quadrilateral ABCD, angle BCD is a right angle. Connect AC.", "ocr_zh": "在四边形ABCD中，角BCD为直角，连接AC。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2073.jpg" }, { "index": 2074, "ocr_en": "There is a right triangle ABC, with angle BAC being the right angle. A line l is drawn through point C, and this line l does not intersect the triangle ABC. A line segment BE is drawn from point B, perpendicular to line l at point E, and a line segment AD is drawn from point A, also perpendicular to line l at point D. Angle CAD is defined as angle 1, angle ACD as angle 2, and angle BCE as angle 3.", "ocr_zh": "有一直角三角形ABC，角BAC为直角，过点C作直线l，l与三角形ABC无交点。过点B用虚线作线段BE垂直l于点E，过点A用虚线做线段AD垂直l于点D。定义角CAD为角1，角ACD为角2，角BCE为角3。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2074.jpg" }, { "index": 2075, "ocr_en": "There are three parallel lines l1, l2, and l3. The distance between l1 and l2 is greater than that between l2 and l3. Take point B on l3, point A on l2, and point C on l1. Connect AB, AC, and BC, and make angle ABC a right angle.", "ocr_zh": "有三根平行线l1，l2，l3，其中l1与l2的间距大于l2与l3的间距。在l3上取点B，在l2上取点A，在l1上取点C，连接AB,AC,BC，使得角ABC为直角。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2075.jpg" }, { "index": 2076, "ocr_en": "There are three parallel lines l1, l2, and l3, where the distance between l1 and l2 is greater than the distance between l2 and l3. Take point B on l3, point A on l2, and point C on l1. Connect AB, AC, and BC, making angle ABC a right angle. Draw a line segment AD through point A, perpendicular to l3 at point D, and draw a line segment CE through point C, perpendicular to l3 at point E.", "ocr_zh": "有三根平行线l1，l2，l3，其中l1与l2的间距大于l2与l3的间距。在l3上取点B，在l2上取点A，在l1上取点C，连接AB,AC,BC，使得角ABC为直角。过点A用虚线作线段AD垂直l3于点D，过点C用虚线作线段CE垂直l3于点E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2076.jpg" }, { "index": 2077, "ocr_en": "There is a quadrilateral ABCD, where angle ACB is a right angle. Connect AB and CD.", "ocr_zh": "有一四边形ABCD，其中角ACB为直角。连接AB,CD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2077.jpg" }, { "index": 2078, "ocr_en": "There is a quadrilateral ABCD, where angle ACB is a right angle. Connect AB and CD. Draw a line segment AE through point A, which is perpendicular to CD at point E.", "ocr_zh": "有一四边形ABCD，其中角ACB为直角。连接AB,CD。过点A用虚线作线段AE垂直CD于点E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2078.jpg" }, { "index": 2079, "ocr_en": "Triangle AMD and triangle ABC share vertex A. Triangle ABC and triangle BEN share vertex B. Points C and E are on the same side as point D, and points M, A, B, and N are collinear. Angles AMD, DAC, CBE, and BNE are all right angles.", "ocr_zh": "三角形AMD和三角形ABC共享一个顶点A，三角形ABC和三角形BEN共享一个顶点B，C点与D,E两点位于同侧，且点M,A,B,N共线。角AMD,DAC,CBE,BNE均为直角。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2079.jpg" }, { "index": 2080, "ocr_en": "There is a quadrilateral ABCD, with angle ADC being a right angle. Connect BD, and angle ABD is also a right angle.", "ocr_zh": "有一四边形ABCD，角ADC为直角。连接BD，角ABD为直角。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2080.jpg" }, { "index": 2081, "ocr_en": "In the right-angled triangle ABC, angle C is the right angle. Take a point D on side AC and a point E on side AB. Connect BD and DE to form triangle BDE, such that angle BDE is a right angle.", "ocr_zh": "在直角三角形ABC中，角C为直角，在边AC上取一点D，在边AB上取一点E，连接BD,DE，形成三角形BDE，使得角BDE为直角。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2081.jpg" }, { "index": 2082, "ocr_en": "There is a right triangle ABC, with angle BAC being the right angle. Point D is on the extension of side BC. Connect AD. Construct a square ADEF on the right side of triangle ABC with AD as one of its sides. Connect CE and CF.", "ocr_zh": "有一直角三角形ABC，角BAC为直角，D为BC延长线上一点，连接AD，以AD为边在三角形ABC的右侧作正方形ADEF，连接CE,CF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2082.jpg" }, { "index": 2083, "ocr_en": "There is a square ABCD. Point E is on side AB, point F is on side BC, and point G is inside the square ABCD, to the right of point F. Connect line segments EF and FG with dotted lines, and connect line segment DG with a solid line.", "ocr_zh": "有一正方形ABCD，E为AB边上一点，点F在BC边上，点G在正方形ABCD内部，且在F右侧，用虚线连接EF,FG，用实线连接DG。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2083.jpg" }, { "index": 2084, "ocr_en": "There is a quadrilateral ABCD. Select a point O within the quadrilateral ABCD. Connect OA, OB, OC, and OD, ensuring that angles AOB and COD are both right angles. On the side AC, select a point E, and on the side BD, select a point F. Connect EF, ensuring that EF passes through point O. Extend EF with dotted lines to point M, and connect BM with dotted lines.", "ocr_zh": "有一四边形ABCD，在四边形ABCD中取一点O，连接OA,OB,OC,OD，使得角AOB和角COD均为直角，在AC边上取一点E，在BD边上取一点F，连接EF，使得EF穿过点O。用虚线延长EF至点M，并用虚线连接BM。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2084.jpg" }, { "index": 2085, "ocr_en": "There is a quadrilateral ABCD. Take a point O within the quadrilateral ABCD. Connect OA, OB, OC, and OD, such that angles AOB and COD are both right angles. On the side AC, take a point E, and on the side BD, take a point F. Connect EF, and make EF pass through point O. The line segments AB, OA, OB, CD, OC, and OD are all red, and point F is red.", "ocr_zh": "有一四边形ABCD，在四边形ABCD中取一点O，连接OA,OB,OC,OD，使得角AOB和角COD均为直角，在AC边上取一点E，在BD边上取一点F，连接EF，使得EF穿过点O。线段AB,OA,OB,CD,OC,OD均为红色，点F为红色。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2085.jpg" }, { "index": 2086, "ocr_en": "There is a quadrilateral ABCD. Take a point O within the quadrilateral ABCD. Connect OA, OB, OC, and OD, ensuring that angles AOB and COD are both right angles. On the side AC, take a point E, and on the side BD, take a point F. Connect EF, making EF pass through point O. Extend EF with a dotted line to point M, and connect BM with a dotted line. The line segments OF and FM are red, and the line segment BM is yellow.", "ocr_zh": "有一四边形ABCD，在四边形ABCD中取一点O，连接OA,OB,OC,OD，使得角AOB和角COD均为直角，在AC边上取一点E，在BD边上取一点F，连接EF，使得EF穿过点O。用虚线延长EF至点M，并用虚线连接BM。线段OF,FM为红色，线段BM为黄色。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2086.jpg" }, { "index": 2087, "ocr_en": "There is a quadrilateral ABCD. Select a point O within the quadrilateral ABCD. Connect OA, OB, OC, and OD, ensuring that angles AOB and COD are both right angles. On the side AC, select a point E, and on the side BD, select a point F. Connect EF, ensuring that EF passes through point O and that angle CEF is a right angle. Extend EF with a dotted line to point M, and connect BM with a dotted line. Line segments EF and CE are colored red.", "ocr_zh": "有一四边形ABCD，在四边形ABCD中取一点O，连接OA,OB,OC,OD，使得角AOB和角COD均为直角，在AC边上取一点E，在BD边上取一点F，连接EF，使得EF穿过点O，且角CEF为直角。用虚线延长EF至点M，并用虚线连接BM。线段EF,CE为红色。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2087.jpg" }, { "index": 2088, "ocr_en": "There is a quadrilateral ABCD. Select a point O within the quadrilateral ABCD. Connect OA, OB, OC, and OD, ensuring that angles AOB and COD are both right angles. On the AC side, select point E, and on the BD side, select point F. Connect EF, ensuring that EF passes through point O and that angle CEF is a right angle. Draw a dotted line BH perpendicular to EF at point H from point B, and draw a dotted line DK perpendicular to the extended line FK of EF at point K from point D.", "ocr_zh": "有一四边形ABCD，在四边形ABCD中取一点O，连接OA,OB,OC,OD，使得角AOB和角COD均为直角，在AC边上取一点E，在BD边上取一点F，连接EF，使得EF穿过点O，且角CEF为直角。过点B用虚线作BH垂直EF于点H，过点D用虚线作DK垂直于EF的虚线延长线FK于点K。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2088.jpg" }, { "index": 2089, "ocr_en": "There is a quadrilateral ABCD. Take a point O within the quadrilateral ABCD. Connect OA, OB, OC, and OD, such that angles AOB and COD are both right angles. On the side AC, take a point E, and take the midpoint F of side BD. Connect EF, making EF pass through point O and angle CEF being a right angle. Draw a dotted line BH perpendicular to EF at point H from point B, and draw a dotted line DK perpendicular to the extension of EF at point K from point D. The line segment BD is colored red.", "ocr_zh": "有一四边形ABCD，在四边形ABCD中取一点O，连接OA,OB,OC,OD，使得角AOB和角COD均为直角，在AC边上取一点E，取BD边的中点F，连接EF，使得EF穿过点O，且角CEF为直角。过点B用虚线作BH垂直EF于点H，过点D用虚线作DK垂直于EF的虚线延长线FK于点K。线段BD为红色。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2089.jpg" }, { "index": 2090, "ocr_en": "There is a quadrilateral ABCD. Take a point O within the quadrilateral ABCD. Connect OA, OB, OC, and OD, such that angles AOB and COD are both right angles. On the side AC, take a point E, and on the side BD, take a point F. Connect EF, making EF pass through point O and angle CEF being a right angle. Draw a dotted line from point B perpendicular to EF at point H, and draw a dotted line from point D perpendicular to the extended line FK of EF at point K. Line segments BH and DK are colored red.", "ocr_zh": "有一四边形ABCD，在四边形ABCD中取一点O，连接OA,OB,OC,OD，使得角AOB和角COD均为直角，在AC边上取一点E，在BD边上取一点F，连接EF，使得EF穿过点O，且角CEF为直角。过点B用虚线作BH垂直EF于点H，过点D用虚线作DK垂直于EF的虚线延长线FK于点K。线段BH和线段DK为红色。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2090.jpg" }, { "index": 2091, "ocr_en": "Triangle ABC shares vertex A with triangle AEF, and the two triangles do not overlap. Take a point D on side BC and connect AD. Extend AD with a dotted line to point H, where H is below triangle ABC. Connect BH with a dotted line. Extend line segment DA with a dotted line to point P, which is on segment EF. Line segment AH is red.", "ocr_zh": "三角形ABC与三角形AEF共享顶点A且两个三角形无重叠部分。在BC边上取一点D，连接AD。用虚线延长AD至点H，H在三角形ABC下方，用虚线连接BH。用虚线延长线段DA至点P，P在线段EF上。线段AH为红色。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2091.jpg" }, { "index": 2092, "ocr_en": "Triangle ABC and triangle AEF share vertex A and there is no overlapping part between the two triangles. Take a point D on side BC and connect AD.", "ocr_zh": "三角形ABC与三角形AEF共享顶点A且两个三角形无重叠部分。在BC边上取一点D，连接AD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2092.jpg" }, { "index": 2093, "ocr_en": "There is a triangle OAB. Take a point C on the OA side, and take a point D on the OB side. Connect BC and AD, and CD. Take a point H on the line segment BC, and H is to the right of the intersection point of BC and AD. Connect OH.", "ocr_zh": "有一三角形OAB。在OA边上取一点C，，在OB边上取一点D，连接BC,AD,CD。在线段BC上取一点H，H在BC与AD交点的右侧，连接OH。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2093.jpg" }, { "index": 2094, "ocr_en": "There is a triangle OAB. Take a point C inside the triangle OAB. Connect OC and BC. On the right side of triangle OAB, take a point D. Connect OD, AD, and CD. On the line segment BC, take a point H and connect OH.", "ocr_zh": "有一三角形OAB。在三角形OAB内部取一点C，连接OC,BC，在三角形OAB右侧取一点D，连接OD,AD,CD。在线段BC上取一点H，连接OH。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2094.jpg" }, { "index": 2095, "ocr_en": "There is a quadrilateral ABCD. Take a point G on side BC and a point F on side AD. Connect GF. Take a point E on segment GF and connect BE, AE, CE, and DE.", "ocr_zh": "有一四边形ABCD。在BC边上取一点G，在AD边上取一点F，连接GF。在线段GF上取一点E，连接BE,AE,CE,DE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2095.jpg" }, { "index": 2096, "ocr_en": "There is a quadrilateral named BCDE. A point A is taken inside this quadrilateral, and lines AB, AC, AD, and AE are drawn.", "ocr_zh": "有一四边形BCDE，在四边形内部取一点A，连接AB,AC,AD,AE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2096.jpg" }, { "index": 2097, "ocr_en": "There is a triangle ADE. Take a point C inside the triangle, and take a point B on the right side of the triangle. Connect AC, AB, BC, and BD. Take a point F on the line segment BD, and place F on the left side of the intersection point of line segment BD and line segment AC. Connect AF.", "ocr_zh": "有一三角形ADE，在三角形内部取一点C，在三角形右侧取一点B，连接AC,AB,BC,BD.在线段BD上取一点F，F在线段BD与线段AC交点的左侧，连接AF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2097.jpg" }, { "index": 2098, "ocr_en": "There is a quadrilateral ABCD. Take a point G on side BC and a point F on side AD. Connect GF. Take a point E on segment GF and connect BE, AE, CE, and DE. Take a point M on side BC and connect AM.", "ocr_zh": "有一四边形ABCD。在BC边上取一点G，在AD边上取一点F，连接GF。在线段GF上取一点E，连接BE,AE,CE,DE。在BC边上取一点M，连接AM。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2098.jpg" }, { "index": 2099, "ocr_en": "There is a quadrilateral ABCD. Take a point G on side BC and a point F on side AD. Connect GF. Take a point E on segment GF and connect BE, AE, CE, and DE. Draw a line through point A perpendicular to BC at point M. Extend line segment MA and intersect DE at point N.", "ocr_zh": "有一四边形ABCD。在BC边上取一点G，在AD边上取一点F，连接GF。在线段GF上取一点E，连接BE,AE,CE,DE。过点A作AM垂直BC于点M，延长线段MA交DE于点N。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2099.jpg" }, { "index": 2100, "ocr_en": "Take a point P inside the quadrilateral ABCD, and connect PA, PB, PC and PD.", "ocr_zh": "在四边形ABCD内部取一点P，连接PA,PB,PC,PD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2100.jpg" }, { "index": 2101, "ocr_en": "There is a triangle ABC'. Extend line segment BA to point C using a dotted line, extend line segment CA to point B using a dotted line, connect points B and C with a straight line, extend line segment CA to point D, and let D lie on the line segment B'C'. Define angle BAB' as angle α, and define angle ACC' as angle β.", "ocr_zh": "有一三角形AB'C'。用虚线延长B'A至点C，用虚线延长C'A至点B，用直线连接BC，用直线延长CA至点D，D在线段B'C'上。定义角BAB'为角α，定义角CAC'为角β。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2101.jpg" }, { "index": 2102, "ocr_en": "Triangle ABC and triangle AB'C' share the vertex A. Triangle ABC is below and triangle AB'C' is above, and there is no overlapping part between the two triangles. Take a point D on the line B'C', and connect AD.", "ocr_zh": "三角形ABC和三角形AB'C'共享一个顶点A，三角形ABC在下方，三角形AB'C'在上方，两三角形无重叠部分。在B'C'边上取一点D，连接AD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2102.jpg" }, { "index": 2103, "ocr_en": "There is a quadrilateral ABCD. Angle B is acute and angle D is obtuse. Connect AC. Draw a dotted line CF perpendicular to AB at point F from point C. Extend AD with a dotted line to point E, and draw a dotted line CE perpendicular to AE at point E, such that angle AEC is a right angle.", "ocr_zh": "有一四边形ABCD，角B为锐角，角D为钝角。连接AC。过点C用虚线作CF垂直AB于点F。用虚线延长AD至点E，过点C用虚线作CE垂直AE于点E，使得角AEC为直角。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2103.jpg" }, { "index": 2104, "ocr_en": "In quadrilateral ABCD, angle B is an acute angle and angle D is an obtuse angle. Connect AC.", "ocr_zh": "在四边形ABCD中，角B为锐角，角D为钝角。连接AC。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2104.jpg" }, { "index": 2105, "ocr_en": "In quadrilateral ABCD, angle B is acute and angle D is obtuse. Connect AC. Extend AB with a dotted line to point M, and connect CM with a dotted line. Line segments AC and CM are colored red.", "ocr_zh": "在四边形ABCD中，角B为锐角，角D为钝角。连接AC。用虚线延长AB至点M，用虚线连接CM。线段AC,CM为红色", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2105.jpg" }, { "index": 2106, "ocr_en": "In quadrilateral ABCD, angle B is an acute angle and angle D is an obtuse angle. Connect AC. Extend AB with a dotted line to point M, and connect CM with a dotted line. Line segments CD and CB are red, triangle ACD is blue, and triangle BCM is yellow.", "ocr_zh": "在四边形ABCD中，角B为锐角，角D为钝角。连接AC。用虚线延长AB至点M，用虚线连接CM。线段CD,CB为红色，三角形ACD为蓝色，三角形BCM为黄色。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2106.jpg" }, { "index": 2107, "ocr_en": "In quadrilateral ABCD, angle B is an acute angle and angle D is an obtuse angle. Connect AC. Extend AB with a dotted line to point M, and connect CM with a dotted line.", "ocr_zh": "在四边形ABCD中，角B为锐角，角D为钝角。连接AC。用虚线延长AB至点M，用虚线连接CM。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2107.jpg" }, { "index": 2108, "ocr_en": "In quadrilateral ABCD, angle D is an obtuse angle and angle B is an acute angle. Connect AC.", "ocr_zh": "在四边形ABCD中，角D为钝角，角B为锐角。连接AC。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2108.jpg" }, { "index": 2109, "ocr_en": "In quadrilateral ABCD, angle D is obtuse and angle B is acute. Connect AC. Draw a dotted line CF perpendicular to AB at point F from point C. Extend AD with a dotted line to point E, and draw a dotted line CE perpendicular to AE at point E from point C. Line segments CE and CF are colored red.", "ocr_zh": "在四边形ABCD中，角D为钝角，角B为锐角。连接AC。过点C用虚线作CF垂直AB于点F。用虚线延长AD至点E，过点C用虚线作CE垂直AE于点E。线段CE,CF为红色。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2109.jpg" }, { "index": 2110, "ocr_en": "There is a square ABCD. Draw ray AC passing through point A and point C. Extend DC to point Q. Take a point P on ray AC, to the right of point C. Connect BP and PQ, such that angle BPQ is a right angle.", "ocr_zh": "有一正方形ABCD，作射线AC过点A点C。延长DC至点Q，在射线AC上、点C右侧取一点P，连接BP,PQ，使得角BPQ为直角。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2110.jpg" }, { "index": 2111, "ocr_en": "There is an angle AOB. Through point O, draw a ray OP. Point P is on ray OP. Ray PM intersects ray OA at point M, and ray PN intersects ray OB at point N. Connect MN.", "ocr_zh": "有一角AOB，过点O作射线OP，P为OP上一点，射线PM交射线OA于点M，射线PN交射线OB于点N，连接MN。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2111.jpg" }, { "index": 2112, "ocr_en": "There is a triangle ABCD. Connect AC. Take a point P on the line segment AC and a point Q on the side CD. Connect BP and PQ, such that angle BPQ is a right angle.", "ocr_zh": "有一三角形ABCD，连接AC。在线段AC上取一点P，在边CD上取一点Q，连接BP,PQ，使得角BPQ为直角。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2112.jpg" }, { "index": 2113, "ocr_en": "Right triangle ABC and right triangle ADC share the hypotenuse AC, and angles B and D are right angles.", "ocr_zh": "直角三角形ABC和直角三角形ADC共享一条斜边AC，角B和角D为直角。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2113.jpg" }, { "index": 2114, "ocr_en": "A, O, D lie on a straight line, from top to bottom they are A, O, D. Through O, draw rays OB and OC. Angle AOC is an acute angle, and angle AOB is greater than angle AOC. Place a large triangular ruler such that the right-angle vertex touches a point P on ray OC, the shorter leg of the right angle lies on point D, and the longer leg intersects ray OB at point E.", "ocr_zh": "A,O,D在一条直线上，从上至下分别是A,O,D。过O作射线OB,OC，角AOC为锐角，角AOB大于角AOC。将一块足够大的三角尺的直角顶点落在射线OC上一点P上，短的直角边落在点D上，长的直角边交射线OB于点E。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2114.jpg" }, { "index": 2115, "ocr_en": "A, O, D lie on a straight line. From top to bottom, they are A, D, O. Through O, draw rays OB and OC. Angle AOC is an acute angle, and angle AOB is greater than angle AOC. There is a triangular ruler. Let the right-angle vertex of the triangular ruler fall on a point P on ray OC. The shorter leg of the right angle intersects ray OA at point D, and the longer leg intersects ray OB at point E.", "ocr_zh": "A,O,D在一条直线上，从上至下分别是A,D,O。过O作射线OB,OC，角AOC为锐角，角AOB大于角AOC。有一三角尺，令三角尺的直角顶点落在射线OC上一点P上，短的直角边与射线OA交于点D，长的直角边与射线OB交于点E。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2115.jpg" }, { "index": 2116, "ocr_en": "There is a quadrilateral ABCD. Connect AC. Draw a line segment CE passing through point C and perpendicular to AB at point E.", "ocr_zh": "有一四边形ABCD，连接AC，过点C作线段CE垂直AB于点E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2116.jpg" }, { "index": 2117, "ocr_en": "There is a quadrilateral ABCD. The length of side CD is greater than that of side AB, and side DC is above side AB. Connect AC.", "ocr_zh": "有一四边形ABCD，边CD长于边AB，且边DC在上，边AB在下。连接AC。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2117.jpg" }, { "index": 2118, "ocr_en": "There is a quadrilateral ABCD. The length of side CD is greater than that of side AB, and side DC is above side AB. Connect AC.", "ocr_zh": "有一四边形ABCD，边CD长于边AB，且边DC在上，边AB在下。连接AC。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2118.jpg" }, { "index": 2119, "ocr_en": "In quadrilateral ABCD, connect AC.", "ocr_zh": "四边形ABCD中，连接AC。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2119.jpg" }, { "index": 2120, "ocr_en": "There is a quadrilateral ABCD. The lengths of sides AB and AD are similar, and the lengths of sides DC and BC are also similar. After connecting AC, triangle ADC is above and triangle ABC is below.", "ocr_zh": "有一四边形ABCD，边AB与AD长度相近，边DC与BC长度相近，连接AC后三角形ADC在上，三角形ABC在下。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2120.jpg" }, { "index": 2121, "ocr_en": "There is a quadrilateral ABCD, with side AD being the shortest side. By connecting AC, the area of triangle ADC is less than the area of triangle ABC.", "ocr_zh": "有一四边形ABCD，边AD是最短边。连接AC，三角形ADC的面积小于三角形ABC的面积", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2121.jpg" }, { "index": 2122, "ocr_en": "In triangle ABC, take point D and connect DA, DB, and DC. DA, DB, and DC form a radial pattern.", "ocr_zh": "在三角形ABC中取一点D，连接DA,DB,DC。DA,DB,DC成放射状。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2122.jpg" }, { "index": 2123, "ocr_en": "Take a point P in the square ABCD. Connect AP, BP, and CP. Below the square ABCD, take a point P'. Connect PP', BP', and CP' with dotted lines. Triangle ABP is blue, and triangle BCP' is yellow.", "ocr_zh": "在正方形ABCD中取一点P，连接AP,BP,CP。在正方形ABCD下方取一点P'，用虚线连接PP',BP',CP'。三角形ABP为蓝色，三角形BCP'为黄色。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2123.jpg" }, { "index": 2124, "ocr_en": "Take a point P in the square ABCD, and connect AP, BP, and CP.", "ocr_zh": "在正方形ABCD中取一点P，连接AP,BP,CP。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2124.jpg" }, { "index": 2125, "ocr_en": "With side AD of triangle ABD as the base, construct triangle ADE outward such that angles AED and DAE are both acute angles.", "ocr_zh": "以三角形ABD的一边AD为边向外作三角形ADE，使得角AED和角DAE均为锐角。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2125.jpg" }, { "index": 2126, "ocr_en": "Take a point P in the square ABCD, and connect AP, BP, and CP.", "ocr_zh": "在正方形ABCD中取一点P，连接AP,BP,CP。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2126.jpg" }, { "index": 2127, "ocr_en": "Take a point P in the square ABCD, and connect AP, BP, and CP. Below the square ABCD, take a point P'. Connect PP', BP', and CP' with dotted lines. Triangle ABP is yellow, and triangle BCP' is blue.", "ocr_zh": "在正方形ABCD中取一点P，连接AP,BP,CP。在正方形ABCD下方取一点P'，用虚线连接PP',BP',CP'。三角形ABP为黄色，三角形BCP'为蓝色。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2127.jpg" }, { "index": 2128, "ocr_en": "In the right-angled triangle ABC, angle BAC is the right angle. On the right side of triangle ABC, take a point D, and connect AD, BD, and CD such that angles ADC and CAD are both acute angles.", "ocr_zh": "在直角三角形ABC中，角BAC为直角，在三角形ABC右侧取一点D，连接AD,BD,CD，使得角ADC和角CAD均为锐角。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2128.jpg" }, { "index": 2129, "ocr_en": "In triangle ABC, select a point P. Connect PA, PB, and PC, and ensure that PA, PB, and PC are arranged radially.", "ocr_zh": "在三角形ABC中取一点P，连接PA,PB,PC，且PA,PB,PC呈放射状。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2129.jpg" }, { "index": 2130, "ocr_en": "In triangle ABC, take point D and connect DA, DB, and DC. Ensure that DA, DB, and DC are arranged radially. Rotate triangle ABD around point B in a counterclockwise direction by a certain angle to obtain triangle A'D'B. Angle ABA' is an acute angle. The sides of triangle A'D'B are represented by dashed lines, and the dashed lines are used to connect DD' and A'C.", "ocr_zh": "在三角形ABC中取一点D，连接DA,DB,DC，且DA,DB,DC呈放射状。将三角形ABD绕点B逆时针旋转一定角度得到三角形A'D'B，角ABA'为锐角，三角形A'D'B的边为虚线，并用虚线连接DD',A'C。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2130.jpg" }, { "index": 2131, "ocr_en": "In triangle ABC, take point D and connect DA, DB, and DC. Ensure that DA, DB, and DC are arranged radially. Rotate triangle ABD 120° clockwise around point B to obtain triangle A'D'B. The sides of triangle A'D'B are dashed lines, and connect DD' with dashed lines.", "ocr_zh": "在三角形ABC中取一点D，连接DA,DB,DC，且DA,DB,DC呈放射状。将三角形ABD绕点B逆时针旋转120°得到三角形A'D'B。三角形A'D'B的边为虚线，并用虚线连接DD'。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2131.jpg" }, { "index": 2132, "ocr_en": "In triangle ABC, take point D and connect DA, DB, and DC. Ensure that DA, DB, and DC are arranged radially. Rotate triangle ABD 90° counterclockwise around point B to obtain triangle A'D'B. The sides of triangle A'D'B are dashed lines, and connect DD' with dashed lines.", "ocr_zh": "在三角形ABC中取一点D，连接DA,DB,DC，且DA,DB,DC呈放射状。将三角形ABD绕点B逆时针旋转90°得到三角形A'D'B。三角形A'D'B的边为虚线，并用虚线连接DD'。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2132.jpg" }, { "index": 2133, "ocr_en": "In the right-angled triangle ABC, angle BAC is the right angle. Select a point P within triangle ABC and connect PA, PB, and PC, arranging them in a radial pattern.", "ocr_zh": "在直角三角形ABC中，角BAC为直角，在三角形ABC中取一点P，连接PA,PB,PC，使得PA,PB,PC呈放射状。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2133.jpg" }, { "index": 2134, "ocr_en": "In triangle ABC, select a point O. Connect OA, OB, and OC, ensuring that angles AOC, AOB, and BOC are all obtuse, and that OA, OB, and OC are arranged radially.", "ocr_zh": "在三角形ABC中取一点O，连接OA,OB,OC，使得角AOC、角AOB、角BOC均为钝角，且OA,OB,OC成放射状。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2134.jpg" }, { "index": 2135, "ocr_en": "In triangle ABC, select a point D. Connect DA, DB, and DC, ensuring that angle ADC is an acute angle and that DA, DB, and DC are arranged radially.", "ocr_zh": "在三角形ABC中取一点D，连接DA,DB,DC，使得角ADC为锐角，且DA,DB,DC成放射状。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2135.jpg" }, { "index": 2136, "ocr_en": "In the square ABCD, select a point E and connect AE, BE, and CE. Ensure that angles BAE and BCE are both less than 45 degrees.", "ocr_zh": "在正方形ABCD中取一点E，连接AE,BE,CE，使得角BAE和角BCE均小于45°。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2136.jpg" }, { "index": 2137, "ocr_en": "There is a triangle ABC, with vertex B at the top. In triangle ABC, select a point P and connect PA, PB, and PC, arranging them in a radial pattern.", "ocr_zh": "有一三角形ABC，顶点B在最上方。在三角形ABC中取一点P，连接PA,PB,PC，使PA,PB,PC成放射状。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2137.jpg" }, { "index": 2138, "ocr_en": "There is a triangle ABC, with vertex C at the top. In triangle ABC, select a point M and connect MA, MB, and MC, arranging them in a radial pattern.", "ocr_zh": "有一三角形ABC，顶点C在最上方。在三角形ABC中取一点M，连接MA,MB,MC，使MA,MB,MC成放射状。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2138.jpg" }, { "index": 2139, "ocr_en": "There is a triangle ABC, with vertex C at the top. In triangle ABC, select a point M and connect MA, MB, and MC, arranging them in a radial pattern.", "ocr_zh": "有一三角形ABC，顶点C在最上方。在三角形ABC中取一点M，连接MA,MB,MC，使MA,MB,MC成放射状。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2139.jpg" }, { "index": 2140, "ocr_en": "There is a quadrilateral ABCD. Connecting AC divides the quadrilateral into triangles ABC and ACD. In triangle ABC, select a point M, and connect AM, BM, and CM. In triangle ACD, select a point E, and connect ED and AE. Connect EM with a dotted line.", "ocr_zh": "有一四边形ABCD，连接AC将四边形分为三角形ABC和三角形ACD。在三角形ABC中取一点M，连接AM,BM,CM；在三角形ACD中取一点E，连接ED,AE。用虚线连接EM。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2140.jpg" }, { "index": 2141, "ocr_en": "In the Cartesian coordinate system, there is a straight line l: y = (3 * 3^0.5)x. Point A is on the negative x-axis, point B is on the positive x-axis, and point P is a point on the part of line l in the third quadrant. Connect AB, PA, and PB to form triangle PAB.", "ocr_zh": "在平面直角坐标系中有一条直线l：y=(3*3^0.5)x。A为x负半轴上一点，B为x正半轴上一点，P为l在第三象限的部分上的一点，连接AB,PA,PB形成一个三角形PAB。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_2141.jpg" }, { "index": 2142, "ocr_en": "There is a quadrilateral ACDM. Connect CM and extend CM to point E. Connect DE. Take a point B on segment CM and connect AB. On the upper part of segment CE and to the right of segment DE, take a point H. Use dotted lines to connect DH, HM, HE, and DA.", "ocr_zh": "有一四边形ACDM，连接CM，并演唱CM至点E，连接DE。在线段CM上取一点B，连接AB。在线段CE上方、线段DE右侧取一点H，用虚线连接DH,HM,HE,DA。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2142.jpg" }, { "index": 2143, "ocr_en": "There is a quadrilateral ACDM. Connect CM and extend CM to point E. Connect DE. Take a point B on the line segment CM and connect AB.", "ocr_zh": "有一四边形ACDM，连接CM，并演唱CM至点E，连接DE。在线段CM上取一点B，连接AB。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2143.jpg" }, { "index": 2144, "ocr_en": "Triangle ABC and Triangle DEC share the base angle C. Point E is inside triangle ABC, and point D is below line segment BC. Connect BE, and select a point M on line segment BE. Connect AM and DM. Draw auxiliary lines: Extend AM to point H, where H is below line segment BC. Connect AD, DH, and EH. Extend HE to intersect AC at point N.", "ocr_zh": "三角形ABC与三角形DEC共底角C，点E在三角形ABC内，D在BC下方。连接BE，在线段BE上取一点M，连接AM,DM。作辅助线：延长AM至点H，H在BC下方，连接AD,DH,EH，延长HE交AC于点N。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2144.jpg" }, { "index": 2145, "ocr_en": "Triangle ABC and Triangle DEC share the angle C. Point E is inside triangle ABC, and point D is below line segment BC. Connect BE, and select a point M on line segment BE. Connect AM and DM.", "ocr_zh": "三角形ABC与三角形DEC共底角C，点E在三角形ABC内，D在BC下方。连接BE，在线段BE上取一点M，连接AM,DM。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2145.jpg" }, { "index": 2146, "ocr_en": "In the Cartesian coordinate system, A is a point on the positive y-axis, B is a point on the positive x-axis, C is a point on the negative x-axis, E and F are two points in the third quadrant, and B, E, and F are collinear. G is a point in the second quadrant, and G is to the left of E. Connect AC, AF, AB, BE, GF, and GE.", "ocr_zh": "在平面直角坐标系中，A为y轴正半轴上一点，B为x轴正半轴上一点，C为x轴负半轴上一点，E,F为第三象限内两点，且B,E,F共线，G为第二象限内一点，G在E左侧。连接AC,AF,AB,BE,GF,GE。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_2146.jpg" }, { "index": 2147, "ocr_en": "Triangle CED shares side CE with triangle BCE, and triangle BCE shares side BC with triangle ABC. Side BE is at the top, and sides DC and AC are at the bottom. Connect DM and AM.", "ocr_zh": "三角形CED与三角形BCE共用一边CE，三角形BCE与三角形ABC共用一边BC。BE边在上，DC边与AC边在下。连接DM,AM。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2147.jpg" }, { "index": 2148, "ocr_en": "In the Cartesian coordinate system, point D and point A are on the positive y-axis, with D above A. Point B is on the positive x-axis, and point C is on the negative x-axis. Point M is a point in the second quadrant, with its x-coordinate lying between A and C, and its y-coordinate lying between A and D. Connect DM, AM, AC, and AB.", "ocr_zh": "在平面直角坐标系中，点D,A在y轴正半轴，D在A上方，B在x轴正半轴，C在x轴负半轴。M为第二象限内一点，横坐标位于A和C之间，纵坐标位于A和D之间。连接DM,AM,AC,AB。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_2148.jpg" }, { "index": 2149, "ocr_en": "Take a point E on the right side of quadrilateral ABCF. Connect AC, BE, EF, and CE. Let AC intersect BE at point D. On the line segment DE within quadrilateral ABCF, take a point M and connect AM and MF.", "ocr_zh": "在四边形ABCF右侧取一点E，连接AC,BE,EF,CE，其中AC交BE于点D。在线段DE上、四边形ABCF中取一点M，连接AM,MF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2149.jpg" }, { "index": 2150, "ocr_en": "In quadrilateral ABCE, connect AC and BE to intersect at point D. Select a point F on line segment BD and connect AF.", "ocr_zh": "在四边形ABCE中，连接AC,BE交于点D，在线段BD上取一点F，连接AF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2150.jpg" }, { "index": 2151, "ocr_en": "Take a point E on the left side of triangle ABC. Connect BE and CE. Take a point D on segment AC. Take a point M on line segment BE within triangle ABC. Connect DE, AM, and DM.", "ocr_zh": "在三角形ABC左侧取一点E，连接BE,CE，在线段AC上取一点D，在线段BE上、三角形ABC内取一点M，连接DE,AM,DM。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2151.jpg" }, { "index": 2152, "ocr_en": "Take a point E on the left side of triangle ABC. Connect BE and CE. Take a point D on segment AC. Take a point M on line segment BE within triangle ABC. Connect DE, AM, and DM. Draw auxiliary lines to extend DM until it intersects AB at point N.", "ocr_zh": "在三角形ABC左侧取一点E，连接BE,CE，在线段AC上取一点D，在线段BE上、三角形ABC内取一点M，连接DE,AM,DM。作辅助线延长DM交AB于N。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2152.jpg" }, { "index": 2153, "ocr_en": "Triangle ABC shares vertex C with triangle CDE. The two triangles have no intersection points, and angle ACD is an acute angle. Connect BE. Take a point M on segment BE and within triangle ABC, connect AM and DM.", "ocr_zh": "三角形ABC与三角形CDE公用顶点C，两三角形无交点，且角ACD为锐角。连接BE，在线段BE上、三角形ABC内取一点M，连接AM,DM。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2153.jpg" }, { "index": 2154, "ocr_en": "Triangle ABC shares vertex C with triangle CDE. The two triangles have no intersection points, and angle ACD is an acute angle. Line AB is parallel to line EC. Connect BE. Select a point M on segment BE within triangle ABC, and connect AM and DM.", "ocr_zh": "三角形ABC与三角形CDE公用顶点C，两三角形无交点，且角ACD为锐角，AB平行于EC。连接BE，在线段BE上、三角形ABC内取一点M，连接AM,DM。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2154.jpg" }, { "index": 2155, "ocr_en": "In triangle ABC, take the midpoint D of side BC and connect AD.", "ocr_zh": "在三角形ABC中，取BC边中点D，连接AD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2155.jpg" }, { "index": 2156, "ocr_en": "In triangle ABC, take the midpoint D of side BC and connect AD. Draw auxiliary lines: Extend line AD to point E, then connect BE. The line segment DE is red.", "ocr_zh": "在三角形ABC中，取BC边中点D，连接AD。作辅助线：延长AD至点E，连接BE。线段DE为红色。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2156.jpg" }, { "index": 2157, "ocr_en": "In triangle ABC, take the midpoint D of side AB and the midpoint E of side AC. Draw an auxiliary line connecting D and E. The line segment DE and the line segment BC are colored red.", "ocr_zh": "三角形ABC中，取AB边中点D，AC边中点E，作辅助线连接DE。线段DE,BC为红色。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2157.jpg" }, { "index": 2158, "ocr_en": "In triangle ABC, take the midpoint D of side AB and the midpoint E of side AC. Points D and E are colored red.", "ocr_zh": "三角形ABC中，取AB边中点D，AC边中点E。点D,E为红色。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2158.jpg" }, { "index": 2159, "ocr_en": "In triangle ABC, take the midpoint D of side BC and connect AD. Draw auxiliary lines: Extend line AD to point E, then connect BE. The line segment AE is in red.", "ocr_zh": "在三角形ABC中，取BC边中点D，连接AD。作辅助线：延长AD至点E，连接BE。线段AE为红色。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2159.jpg" }, { "index": 2160, "ocr_en": "In the right-angled triangle ABC, angle ACB is the right angle. Take the midpoint D of the hypotenuse AB and draw an auxiliary line connecting CD. The auxiliary line CD is red.", "ocr_zh": "在直角三角形ABC中，角ACB为直角。取斜边AB中点D，作辅助线连接CD。辅助线CD为红色。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2160.jpg" }, { "index": 2161, "ocr_en": "In the isosceles triangle ABC, the two sides AB and AC are red.", "ocr_zh": "等腰三角形ABC的两斜边AB,AC为红色。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2161.jpg" }, { "index": 2162, "ocr_en": "In the isosceles triangle ABC, draw an auxiliary line: Draw a perpendicular line segment AD from vertex A to the base BC. The auxiliary line is the red dotted line.", "ocr_zh": "等腰三角形ABC中，作辅助线：过顶点A作底边BC的垂线段AD，辅助线为红色虚线。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2162.jpg" }, { "index": 2163, "ocr_en": "In the isosceles triangle ABC, take the midpoint D of the base BC, and draw an auxiliary line connecting AD. The base BC is colored red.", "ocr_zh": "等腰三角形ABC中，取底边BC中点D，作辅助线连接AD。底边BC为红色。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2163.jpg" }, { "index": 2164, "ocr_en": "In the right-angled triangle ABC, angle C is the right angle. Take the midpoint D of the hypotenuse AB.", "ocr_zh": "直角三角形ABC中，角C为直角，取斜边AB中点D。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2164.jpg" }, { "index": 2165, "ocr_en": "In the right-angled triangle ABC, angle C is the right angle. Take the midpoint D of the hypotenuse AB and draw an auxiliary line connecting CD. The line segment AB and the auxiliary line CD are colored red.", "ocr_zh": "直角三角形ABC中，角C为直角，取斜边AB中点D，作辅助线连接CD。线段AB和辅助线CD为红色。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2165.jpg" }, { "index": 2166, "ocr_en": "Construct triangle ABD on the left side of side BC of triangle ABC, and construct triangle ACE on the right side of side AC of triangle ABC. Take a point P on segment AD, a point M on segment BC, and a point N on segment CE. Connect line segments PN and MN.", "ocr_zh": "以三角形ABC的BC边为公用边向左侧作三角形ABD，以三角形ABC的AC边为公用边向右侧作三角形ACE。在线段AD上取一点P，在线段BC中取一点M，在线段CE上取一点N，连接PN,MN。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2166.jpg" }, { "index": 2167, "ocr_en": "In trapezoid ABCD, take a point E on side AB, then connect DE, CE, and DB. Draw an auxiliary line: draw line DH perpendicular to BC through point D, and let H be the intersection point.", "ocr_zh": "梯形ABCD中，在边AB上取一点E，连接DE,CE,DB。作辅助线：过点D作DH垂直BC于点H。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2167.jpg" }, { "index": 2168, "ocr_en": "In the trapezoid ABCD, take a point E on side AB, and connect DE, CE, and DB. Draw auxiliary lines: extend BA and CD to intersect at point M.", "ocr_zh": "梯形ABCD中，在边AB上取一点E，连接DE,CE,DB。作辅助线：延长BA和CD交于点M。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2168.jpg" }, { "index": 2169, "ocr_en": "In the trapezoid ABCD, take a point E on side AB, and connect DE, CE and DB.", "ocr_zh": "梯形ABCD中，在边AB上取一点E，连接DE,CE,DB。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2169.jpg" }, { "index": 2170, "ocr_en": "In the right-angled triangle ABC, angle BAC is the right angle. Extend BA to point D, and select a point E on side AC. Connect DE.", "ocr_zh": "直角三角形ABC中，角BAC为直角。延长BA至点D，在边AC上取一点E，连接DE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2170.jpg" }, { "index": 2171, "ocr_en": "Rectangle ABCD. Extend side CB to point P. Connect PA and PD. PD intersects AB at point E.", "ocr_zh": "矩形ABCD，延长边CB至点P，连接PA,PD，PD交AB于点E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2171.jpg" }, { "index": 2172, "ocr_en": "In triangle ABC, draw AD perpendicular to BC at point D. Select the midpoint M of BC.", "ocr_zh": "三角形ABC中，过点A作AD垂直BC于点D。取BC中点M", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2172.jpg" }, { "index": 2173, "ocr_en": "In quadrilateral ABCD, angles BAD and BCD are obtuse angles. Extend BC to point F, and connect AF and BD, intersecting at point H. Make angle AHD a right angle, and let AF intersect DC at point E. Select a point M on segment BC.", "ocr_zh": "四边形ABCD中，角BAD和角BCD为钝角，延长BC至点F，连接AF和BD，交于点H，使得角AHD为直角，AF交DC于点E。在线段BC上取一点M。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2173.jpg" }, { "index": 2174, "ocr_en": "In the quadrilateral ABCE, connect AC and BE. Take a point D on side BC and connect AD and DE. On segment BE, take a point F and connect AF.", "ocr_zh": "四边形ABCE中，连接AC,BE，在边BC上取一点D，连接AD,DE。在线段BE上取一点F，连接AF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2174.jpg" }, { "index": 2175, "ocr_en": "In quadrilateral ABCD, angles BAD and BCD are obtuse angles. Extend BC to point F, and connect AF and BD, intersecting at point H. Make angle AHD a right angle, and let AF intersect DC at point E. Select a point M on segment BC.", "ocr_zh": "四边形ABCD中，角BAD和角BCD为钝角，延长BC至点F，连接AF和BD，交于点H，使得角AHD为直角，AF交DC于点E。在线段BC上取一点M。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2175.jpg" }, { "index": 2176, "ocr_en": "Take a point E on the right side of triangle ABC. Connect AE, AD, DE and BE. Then take a point F on BE and connect AF.", "ocr_zh": "在三角形ABC右侧取一点E，连接AE,AD,DE,BE。在BE上取一点F，连接AF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2176.jpg" }, { "index": 2177, "ocr_en": "Using AB as one side, construct an auxiliary line to form a obtuse triangle ABC, with angle BAC being the obtuse angle. Also, draw line AM perpendicular to BC at point M, passing through point A.", "ocr_zh": "以AB为边，作辅助线构造钝角三角形ABC，角BAC为钝角，并过点A作AM垂直BC于点M。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2177.jpg" }, { "index": 2178, "ocr_en": "In triangle ABC, draw BD perpendicular to AC at point D. Select point E on side AB, point G on the left side of AB, and point H on side BC. Connect GE, GH, and EH. The intersection of GH and BD is point F. Connect EF. Draw auxiliary lines: Draw FN perpendicular to BC at point N, extend BC to point M, and connect FM.", "ocr_zh": "在三角形ABC中，过点B作BD垂直AC于点D。在AB边上取一点E，在AB左侧取一点G，在BC边上取一点H，连接GE,GH,EH，GH交BD于点F，连接EF。作辅助线：过点F作FN垂直BC于点N，延长BC于点M，连接FM。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2178.jpg" }, { "index": 2179, "ocr_en": "At vertex B of triangle ABC, draw BD perpendicular to AC at point D. Take a point G on the left side of triangle ABC, connect GB, GC, and GD. The intersection point of GC and BD is F. Draw an auxiliary line connecting AG.", "ocr_zh": "过三角形ABC一顶点B作BD垂直AC于点D。在三角形ABC左侧取一点G，连接GB,GC,GD，GC交BD于点F。作辅助线连接AG。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2179.jpg" }, { "index": 2180, "ocr_en": "In triangle ABC, draw BD perpendicular to AC at point D. Select point E on side AB, point G on the left side of AB, and point H on side BC. Connect GE, GH, and EH. The intersection point of GH and BD is F. Connect EF.", "ocr_zh": "在三角形ABC中，过点B作BD垂直AC于点D。在AB边上取一点E，在AB左侧取一点G，在BC边上取一点H，连接GE,GH,EH，GH交BD于点F，连接EF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2180.jpg" }, { "index": 2181, "ocr_en": "In triangle ABC, draw BD perpendicular to AC at point D. Select point E on side AB, point G on the left side of AB, and point H on side BC. Connect GE, GH, and EH. The intersection point of GH and BD is F. Connect EF.", "ocr_zh": "在三角形ABC中，过点B作BD垂直AC于点D。在AB边上取一点E，在AB左侧取一点G，在BC边上取一点H，连接GE,GH,EH，GH交BD于点F，连接EF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2181.jpg" }, { "index": 2182, "ocr_en": "In triangle ABC, draw BD perpendicular to AC at point D. Select point E on side AB, point G on the left side of AB, and point H on side BC. Connect GE, GH, and EH. The intersection point of GH and BD is F. Connect EF.", "ocr_zh": "在三角形ABC中，过点B作BD垂直AC于点D。在AB边上取一点E，在AB左侧取一点G，在BC边上取一点H，连接GE,GH,EH，GH交BD于点F，连接EF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2182.jpg" }, { "index": 2183, "ocr_en": "Take a point F on the left side of the square. Connect AF and BF. Angle AFB is an acute angle. Connect DF, AF, and BF. DF intersects AB at point P. Take point G outside of DF and square ABCD. Connect AG, CG, and draw AE perpendicular to DF at point E. Draw auxiliary lines: Draw CH perpendicular to DF through point C and let it intersect at point H. Define angle AGD as angle 1, angle CGD as angle 2, angle ADF as angle 3, angle CDF as angle 4, and angle DCH as angle 5.", "ocr_zh": "在正方形左侧取一点F，连接AF,BF，角AFB为锐角，连接DF,AF,BF，DF交AB于点P。在DF、正方形ABCD外取一点G，连接AG,CG，过点A作AE垂直DF于点E。作辅助线：过点C作CH垂直DF于点H。定义角AGD为角1，角CGD为角2，角ADF为角3，角CDF为角4，角DCH为角5.", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_2183.jpg" }, { "index": 2184, "ocr_en": "Take a point F on the left side of the square. Connect AF and BF. Angle AFB is an acute angle. Connect DF, AF, and BF. DF intersects AB at point P. Take another point G outside DF and square ABCD. Connect AG and CG. Draw a line through point A perpendicular to DF at point E.", "ocr_zh": "在正方形左侧取一点F，连接AF,BF，角AFB为锐角，连接DF,AF,BF，DF交AB于点P。在DF、正方形ABCD外取一点G，连接AG,CG，过点A作AE垂直DF于点E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2184.jpg" }, { "index": 2185, "ocr_en": "Take a point F on the left side of the square. Connect AF and BF. Angle AFB is an acute angle. Connect DF, AF, and BF. DF intersects AB at point P. Take another point G outside DF and square ABCD. Connect AG, CG, and draw a line through point A perpendicular to DF at point E. Draw an auxiliary line extending GA to point M and connect DM. Make angle CDM a right angle.", "ocr_zh": "在正方形左侧取一点F，连接AF,BF，角AFB为锐角，连接DF,AF,BF，DF交AB于点P。在DF、正方形ABCD外取一点G，连接AG,CG，过点A作AE垂直DF于点E。作辅助线延长GA至点M，连接DM，使得角CDM为直角。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2185.jpg" }, { "index": 2186, "ocr_en": "On the side AG of the quadrilateral AGCD, select two points B and F from left to right on the side AG, and take point E on the side AD. Connect AC, CE, BC, and CF.", "ocr_zh": "在四边形AGCD的边AG上从左至右取两点B,F，在边AD上取一点E，连接AC,CE,BC,CF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2186.jpg" }, { "index": 2187, "ocr_en": "In the square ABCD, point P is on side CD. Connect BP. Draw DE perpendicular to PB, with E being the intersection point of DE and BP. Connect CE. Point E is on the right side of the square ABCD.", "ocr_zh": "正方形ABCD中，P是CD上一点，连接BP，过D作DE垂直PB，交BP于点E，连接CE，E在正方形ABCD右侧。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2187.jpg" }, { "index": 2188, "ocr_en": "In the square ABCD, the diagonals AC and BD intersect at point O. On side AB, take a point E, connect DE, draw CG perpendicular to DE at point G through point C, and draw AF perpendicular to DE at point F through point A. Connect OG.", "ocr_zh": "正方形ABCD中，对角线AC与BD交于点O。在边AB上取一点E，连接DE，过点C作CG垂直DE于点G，过点A作AF垂直DE于点F，连接OG。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2188.jpg" }, { "index": 2189, "ocr_en": "In the square ABCD, extend CD to point P, connect BP, and draw DE perpendicular to BP at point E (where E is above the square ABCD), then connect CE.", "ocr_zh": "正方形ABCD中，延长CD至点P，连接BP，过点D作DE垂直BP于点E，E在正方形ABCD上方，连接CE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2189.jpg" }, { "index": 2190, "ocr_en": "Square ABCD, extend DC to point P, connect BP. Draw DE perpendicular to BP at point E, then connect CE.", "ocr_zh": "正方形ABCD，延长DC至点P，连接BP。过点D作DE垂直BP于点E，连接CE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2190.jpg" }, { "index": 2191, "ocr_en": "In the right-angled triangle ABC, angle C is the right angle. Take a point D on the left side of triangle ABC. Connect AD and BD. Draw DE perpendicular to AC at point E. On line BD within triangle ABC, take a point F. Connect EF.", "ocr_zh": "直角三角形ABC中角C为直角，在三角形ABC左侧取一点D，连接AD,BD，过点D作DE垂直AC于点E，在BD上、三角形ABC内部取一点F，连接EF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2191.jpg" }, { "index": 2192, "ocr_en": "In the square ABCD, take a point E on side BC and a point F on side CD. Connect AE, AF and EF to form triangle AEF.", "ocr_zh": "正方形ABCD中，在边BC上取一点E，在边CD上取一点F，连接AE,AF,EF形成一个三角形AEF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2192.jpg" }, { "index": 2193, "ocr_en": "In the square ABCD, take a point E on side BC and a point F on side CD. Connect AE, AF and EF to form triangle AEF. Extend AE to point G and connect GF.", "ocr_zh": "正方形ABCD中，在边BC上取一点E，在边CD上取一点F，连接AE,AF,EF形成一个三角形AEF。延长AE至点G，连接GF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2193.jpg" }, { "index": 2194, "ocr_en": "In triangle ABC, let angle B be α and angle C be 2α. Draw an auxiliary line: Draw AE perpendicular to BC at point E. Select a point D on segment BE, and connect AD.", "ocr_zh": "三角形ABC中，令角B为α，角C为2α。作辅助线：过点A作AE垂直BC于点E，在线段BE上取一点D，连接AD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2194.jpg" }, { "index": 2195, "ocr_en": "In triangle ABC, let angle B be α and angle C be 2α.", "ocr_zh": "三角形ABC中，令角B为α，角C为2α。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2195.jpg" }, { "index": 2196, "ocr_en": "Take the midpoint O of line segment BC. Draw a perpendicular line l from O to BC. On line l, above BC, take a point A. Draw auxiliary lines connecting AB and AC.", "ocr_zh": "取线段BC中点O，过O作BC的垂线l，在l上、BC的上方取一点A。作辅助线连接AB,AC。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2196.jpg" }, { "index": 2197, "ocr_en": "Take the midpoint O of line segment BC. Draw a perpendicular line l from O to BC. Then, on line l and above BC, select a point A.", "ocr_zh": "取线段BC中点O，过O作BC的垂线l，在l上、BC的上方取一点A。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2197.jpg" }, { "index": 2198, "ocr_en": "In triangle ABC, the perpendicular line l1 of AB and the perpendicular line l2 of AC intersect at point O within triangle ABC. Connect OB and OC.", "ocr_zh": "三角形ABC中，AB的垂线l1和AC的垂线l2交于三角形ABC中一点O，连接OB,OC。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2198.jpg" }, { "index": 2199, "ocr_en": "In quadrilateral ABCD, select point B. Point B is located to the lower left of the intersection point of AD and CE. Connect AB, BC, BD, and BE.", "ocr_zh": "在四边形ABCD中取一点B，B在AD与CE交点的左下方，连接AB,BC,BD,BE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2199.jpg" }, { "index": 2200, "ocr_en": "In triangle ABC, with B and C as the centers and radii greater than (1/2)BC, arcs are drawn. These two arcs intersect at points M and N. A straight line MN is drawn and it intersects side AB at point E.", "ocr_zh": "三角形ABC中，分别以B和C为圆心，以大于（1/2）BC的长为半径作弧，两弧交于点M,N；作直线MN交边AB于点E。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2200.jpg" }, { "index": 2201, "ocr_en": "In quadrilateral BFDG, angles G and BFD are right angles. Draw ray BD through point B. Take a point A on BG. Extend BF to point C and connect AC. Draw line DE perpendicular to AC through point D and let it intersect AC at point E. Draw auxiliary lines connecting AD and CD.", "ocr_zh": "四边形BFDG中，角G和角BFD为直角。过点B作射线BD，在BG上取一点A，延长BF至点C，连接AC，过点D作直线DE垂直AC于点E。作辅助线连接AD,CD。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2201.jpg" }, { "index": 2202, "ocr_en": "Segment AC intersects segment BD at point O. Connect AB and CE. Angle ABE is a right angle. Through point E, draw EO perpendicular to AC at point O. Connect BO. Extend BO to point D, which is inside triangle OCE. Connect CD.", "ocr_zh": "线段AC与BD交于点O，连接AB,CE，角ABE为直角。过点E作EO垂直AC于点O，连接BO，延长BO至点D，D位于三角形OCE内部，连接CD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2202.jpg" }, { "index": 2203, "ocr_en": "In triangle ABC, take point F on side AC and point E on side BC. Fold angle C along EF so that point C coincides with point O. Connect OA. Draw OD perpendicular to AB at point D through point O.", "ocr_zh": "在三角形ABC中，在AC上取一点F，在BC上取一点E，将角C沿着EF折叠，使得C点与O点重合。连接OA，过点O作OD垂直AB于点D。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2203.jpg" }, { "index": 2204, "ocr_en": "BD bisects angle ABC. On side AB, take a point E. Draw a line through point E that is parallel to BC and intersects BD at point F.", "ocr_zh": "BD平分角ABC，在AB边上取一点E，过点E平行BC交BD于点F。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2204.jpg" }, { "index": 2205, "ocr_en": "BD bisects angle ABC.", "ocr_zh": "BD平分角ABC。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2205.jpg" }, { "index": 2206, "ocr_en": "BD bisects angle ABC. On side AB, take a point E. Draw a line through point E that is parallel to BC and intersects BD at point F. Mark triangle BEF in blue.", "ocr_zh": "BD平分角ABC，在AB边上取一点E，过点E平行BC交BD于点F。将三角形BEF标为蓝色。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2206.jpg" }, { "index": 2207, "ocr_en": "BD bisects angle ABC. On side AB, take a point E. Draw a line through point E that is parallel to BC and intersects BD at point F.", "ocr_zh": "BD平分角ABC，在AB边上取一点E，过点E平行BC交BD于点F。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2207.jpg" }, { "index": 2208, "ocr_en": "BD bisects angle ABC. Take a point E on the straight line BC, to the left of point B. Then take a point F on line BA. Connect points E and F.", "ocr_zh": "BD平分角ABC。在直线BC上、B点左边取一点E，在BA上取一点F，连接EF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2208.jpg" }, { "index": 2209, "ocr_en": "BD bisects angle ABC. On the line BC, to the left of point B, select a point E. Then, draw a line EF parallel to BD and let it intersect AB at point F.", "ocr_zh": "BD平分角ABC。在直线BC上、B点左边取一点E，过点E作EF平行BD交AB于点F。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2209.jpg" }, { "index": 2210, "ocr_en": "BD bisects angle ABC. On the line BC, to the left of point B, select a point E. Then, draw a line EF parallel to BD and let it intersect AB at point F. Define angle ABD as angle 1, angle CBD as angle 2, and angle EFB as angle 3.", "ocr_zh": "BD平分角ABC。在直线BC上、B点左边取一点E，过点E作EF平行BD交AB于点F。定义角ABD为角1，角CBD为角2，角EFB为角3.", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2210.jpg" }, { "index": 2211, "ocr_en": "In triangle ABC, draw BF passing through point B and intersecting AC at point F. Draw CG passing through point C and intersecting AB at point G. The intersection of BF and CG is point H. Draw DE passing through point H and intersecting AB at point D and AC at point E. Point D is below point G and point E is below point F.", "ocr_zh": "在三角形ABC中，过点B作BF交AC于点F，过点C作CG交AB于点G，BF与CG交于点H，过点H作DE交AB于点D，交AC于点E，D在G下方，E在F下方。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2211.jpg" }, { "index": 2212, "ocr_en": "Point D lies on the extension of side BC of triangle ABC. The angle bisectors of angles ABC and ACD intersect at point O. Through point O, a parallel line to BC is drawn, intersecting sides AB and AC at points E and F respectively.", "ocr_zh": "D为三角形ABC的边BC的延长线上一点，角ABC与角ACD的平分线交于点O，过点O作BC的平行线，分别角AB,AC于点E,F。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2212.jpg" }, { "index": 2213, "ocr_en": "CD is the angle bisector of angle C in triangle ABC. Through point D, draw parallel lines to AC and BC respectively, which intersect BC at point E and AC at point F.", "ocr_zh": "CD是三角形ABC中角C的平分线，过点D分别作AC,BC的平行线交BC于点E，交AC于点F。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2213.jpg" }, { "index": 2214, "ocr_en": "In triangle ABC, ED is parallel to BC. ED intersects AB and AC at points E and D respectively. Through point B, draw BG intersecting ED at point G. Through point C, draw CF intersecting ED at point F.", "ocr_zh": "在三角形ABC中，ED平行于BC，ED分别与AB,AC交于点E,D，过点B作BG交ED于点G，过点C作CF交ED于点F。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2214.jpg" }, { "index": 2215, "ocr_en": "In triangle ABC, extend side BC to point G. Draw line BO through point B and line CO through point C. The two lines BO and CO intersect at point O, which is on the right side of triangle ABC. Select a point D on side AB and connect OD. This line OD intersects side AC at point E.", "ocr_zh": "三角形ABC中，延长BC至点G，过点B作BO，过点C作CO，BO与CO交于点O，O在三角形ABC右侧，在AB上取一点D，连接OD交AC于点E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2215.jpg" }, { "index": 2216, "ocr_en": "In the right trapezoid ABCD, take a point E on the slant side CD, and connect AE and BE.", "ocr_zh": "在直角梯形ABCD中，在斜腰CD上取一点E，连接AE,BE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2216.jpg" }, { "index": 2217, "ocr_en": "In the rectangle ABCD, fold the angle B along the diagonal AC, so that point B coincides with point B' on the other side.", "ocr_zh": "矩形ABCD，将角B沿对角线AC对折，使得B点落在B'点处。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2217.jpg" }, { "index": 2218, "ocr_en": "OC is the diagonal of angle AOB. Take a point D on OA. Draw DP passing through point D and intersect OC at point P.", "ocr_zh": "OC为角AOB对角线，在OA上取一点D，过点D作DP交OC于点P。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2218.jpg" }, { "index": 2219, "ocr_en": "In triangle ABC, angle B is 45 degrees and angle C is 30 degrees.", "ocr_zh": "三角形ABC中，角B等于45°，角C等于30°。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2219.jpg" }, { "index": 2220, "ocr_en": "In triangle ABC, angle B is 45 degrees and angle C is 30 degrees. Draw an auxiliary line AD through point A, which is perpendicular to BC at point D.", "ocr_zh": "三角形ABC中，角B等于45°，角C等于30°。过点A作辅助线AD垂直BC于点D。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2220.jpg" }, { "index": 2221, "ocr_en": "In triangle ABC, angle B is 45 degrees and angle C is 30 degrees. Draw an auxiliary line AD through point A, which is perpendicular to BC at point D. Angle BAD is 45°, Angle DAC is 60°, AB is (2^0.5)a, AC is 2a, BD is a, CD is (3^0.5)a, and AD is a.", "ocr_zh": "三角形ABC中，角B等于45°，角C等于30°。过点A作辅助线AD垂直BC于点D。角BAD为45°，角DAC为60°，AB为（2^0.5）a，AC为2a，BD为a，CD为（3^0.5）a，AD为a。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2221.jpg" }, { "index": 2222, "ocr_en": "In triangle ABC, draw an auxiliary line AD through point A, which is perpendicular to BC at point D.", "ocr_zh": "三角形ABC中，过点A作辅助线AD垂直BC于点D。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2222.jpg" }, { "index": 2223, "ocr_en": "In triangle ABC, angle A is 120°, angle B is 45°, and angle C is 15°. Draw an auxiliary line: extend BA to point D, and draw CD perpendicular to BD through point C at point D.", "ocr_zh": "三角形ABC中，角A为120°，角B为45°，角C为15°。作辅助线：延长BA至点D，过点C作CD垂直BD于点D。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2223.jpg" }, { "index": 2224, "ocr_en": "In triangle ABC, angle A measures 150°. Draw an auxiliary line: Draw BD perpendicular to the extension of CA through point B, and let point D be the intersection. Then, angle BAD is 30°.", "ocr_zh": "三角形ABC中，角A为150°。作辅助线：过点B作BD垂直CA的延长线于点D，角BAD为30°.", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2224.jpg" }, { "index": 2225, "ocr_en": "In triangle ABC, draw AD perpendicular to BC at point D. Angle B is 45°, angle C is 60°, angle BAD is 45°, and angle CAD is 30°.", "ocr_zh": "三角形ABC中，过点A作AD垂直BC于点D。角B为45°，角C为60°，角BAD为45°，角CAD为30°。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2225.jpg" }, { "index": 2226, "ocr_en": "In triangle ABC, angle A measures 135°. Draw an auxiliary line: Draw BD perpendicular to the extension of CA through point B, and let it intersect at point D.", "ocr_zh": "三角形ABC中，角A为135°。作辅助线：过点B作BD垂直CA的延长线于点D。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2226.jpg" }, { "index": 2227, "ocr_en": "There is a triangle ABC, with vertex A located at the top.", "ocr_zh": "有一三角形ABC，顶点A在上方。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2227.jpg" }, { "index": 2228, "ocr_en": "In triangle ABC, vertex B is at the top, with AB measuring 20 meters and BC measuring 30 meters.", "ocr_zh": "三角形ABC中，顶点B在上方，AB为20m，BC为30m。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2228.jpg" }, { "index": 2229, "ocr_en": "In quadrilateral ABCD, take the midpoint E of AB, and connect CA and CE.", "ocr_zh": "四边形ABCD中，取AB中点E，连接CA,CE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2229.jpg" }, { "index": 2230, "ocr_en": "Point P is located outside the parallelogram ABCD. Connect AC and BD to intersect at point O. Then connect PA, PB, PC, and PD.", "ocr_zh": "P为平行四边形ABCD外一点，连接AC、BD交于点O，连接PA,PB,PC,PD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2230.jpg" }, { "index": 2231, "ocr_en": "There is a right triangle ABC, with angle C being the right angle and vertex A at the top.", "ocr_zh": "有一直角三角形ABC，角C为直角，顶点A在上端。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2231.jpg" }, { "index": 2232, "ocr_en": "There is a segment MON. Points A and B are located on sides OM and ON respectively. Connect AB. Take point C on the right side of AB. Connect AC and BC to form triangle ABC.", "ocr_zh": "有一角MON，点A,B分别在边OM,ON上，连接AB，在AB右侧取一点C，连接AC,BC，形成三角形ABC。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2232.jpg" }, { "index": 2233, "ocr_en": "In triangle ABC, take the midpoint E of AB, and draw CD perpendicular to AB through point C, intersecting at point D.", "ocr_zh": "三角形ABC中，取AB中点E，过点C作CD垂直AB于点D。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2233.jpg" }, { "index": 2234, "ocr_en": "In triangle ABC, draw AD perpendicular to BC at point D. Draw E as the median of side AB from point C. Draw DG perpendicular to CE at point G.", "ocr_zh": "三角形ABC中，过点A作AD垂直BC于点D，过点C作AB边上的中线E，过点D作DG垂直CE于点G。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2234.jpg" }, { "index": 2235, "ocr_en": "In the parallelogram ABCD, connect BD. From point A, draw AF perpendicular to BC at point F, and AF intersects BD at point E.", "ocr_zh": "平行四边形ABCD中，连接BD，过点A作AF垂直BC于点F，AF交BD于点E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2235.jpg" }, { "index": 2236, "ocr_en": "In triangle ABC, draw BM perpendicular to AC through point B. Select a point P inside triangle ABC, with P located above BM. Draw PE perpendicular to AB at point E, PG perpendicular to AC at point G, and PF perpendicular to BC at point F. Add auxiliary lines: connect PA, PB, and PC.", "ocr_zh": "三角形ABC中，过点B作BM垂直AC于点M。在三角形ABC内部取一点P，P在BM上方，过点P作PE垂直AB于点E，PG垂直AC于点G，PF垂直BC于点F。作辅助线：连接PA,PB,PC。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2236.jpg" }, { "index": 2237, "ocr_en": "In triangle ABC, draw BM perpendicular to AC at point M. Select a point P inside triangle ABC, with P located above BM. Draw PE perpendicular to AB at point E, PG perpendicular to AC at point G, and PF perpendicular to BC at point F. BM is colored red.", "ocr_zh": "三角形ABC中，过点B作BM垂直AC于点M。在三角形ABC内部取一点P，P在BM上方，过点P作PE垂直AB于点E，PG垂直AC于点G，PF垂直BC于点F。BM为红色。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2237.jpg" }, { "index": 2238, "ocr_en": "Take a point P inside triangle ABC. Point P is above line segment BM. Draw line segments PE, PG, and PF perpendicular to AB, AC, and BC respectively at points E, G, and F.", "ocr_zh": "在三角形ABC内部取一点P，P在BM上方，过点P作PE垂直AB于点E，PG垂直AC于点G，PF垂直BC于点F。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2238.jpg" }, { "index": 2239, "ocr_en": "In triangle ABC, draw CE perpendicular to AB at point E. Draw an auxiliary line: extend BC to point P, connect PA, extend AC to point H such that PH is perpendicular to AH. Draw DP perpendicular to AB at point D.", "ocr_zh": "三角形ABC中，过点C作CE垂直AB于点E。作辅助线：延长BC至点P，连接PA，延长AC至点H，使得PH垂直AH。过点P作DP垂直AB于点D。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2239.jpg" }, { "index": 2240, "ocr_en": "In triangle ABC, draw CM perpendicular to AB at point M. On the right side of triangle ABC, take point P. Draw PG perpendicular to AC at point G, draw PE perpendicular to the extension of BA at point E, and draw PF perpendicular to the extension of BC at point F. Add auxiliary lines: connect PA, PB, and PC.", "ocr_zh": "三角形ABC中，过点C作CM垂直AB于点M。在三角形ABC右侧取一点P，过点P作PG垂直AC于点G，作PE垂直BA的延长线于点E，作PF垂直BC的延长线于点F。作辅助线：连接PA,PB,PC。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2240.jpg" }, { "index": 2241, "ocr_en": "In triangle ABC, draw CM perpendicular to AB at point M. Extend MC to point P. Draw PE perpendicular to AB at point E. Draw PF perpendicular to the extension of BC at point F. Draw PG perpendicular to the extension of AC at point G. Add auxiliary lines: Connect PA, PB, and PC.", "ocr_zh": "三角形ABC中，过点C作CM垂直AB于点M。延长MC至点P，过点P作PE垂直AB于点E，过点P作PF垂直BC的延长线于点F，过点P作PG垂直AC的延长线于点G。作辅助线：连接PA,PB,PC。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2241.jpg" }, { "index": 2242, "ocr_en": "In triangle ABC, point P is on the base BC. Line segment CE is drawn from point C perpendicular to AB at point E. Line segment PD is drawn from point P perpendicular to AB at point E. Line segment PH is drawn from point P perpendicular to AC at point H. Auxiliary line PA is drawn.", "ocr_zh": "三角形ABC中，P为底边BC上一点，过点C作CE垂直AB于点E，过点P作PD垂直AB于点E，过点P作PH垂直AC于点H。作辅助线连接PA。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2242.jpg" }, { "index": 2243, "ocr_en": "In the square ABCD, E is a point on diagonal BD, and F is a point on CE. A line is drawn through point F, which is perpendicular to BC at point M, and another line is drawn perpendicular to BD at point N.", "ocr_zh": "正方形ABCD中，E为对角线BD上一点，F为CE上一点，过点F作FM垂直BC于点M，FN垂直BD于点N。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2243.jpg" }, { "index": 2244, "ocr_en": "In the square ABCD, E is a point on diagonal BD, and F is a point on CE. A line is drawn through point F, which is perpendicular to BC at point M, and another line is drawn perpendicular to BD at point N. Draw auxiliary lines: Connect BF. Draw line EH passing through point E and perpendicular to BC at point H.", "ocr_zh": "正方形ABCD中，E为对角线BD上一点，F为CE上一点，过点F作FM垂直BC于点M，FN垂直BD于点N。作辅助线：连接BF，过点E作EH垂直BC于点H。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2244.jpg" }, { "index": 2245, "ocr_en": "In triangle ABC, point P is on the base BC. Line segment BG is drawn from point B perpendicular to AC at point G. Line segment PD is drawn from point P perpendicular to AB at point D. Line segment PE is drawn from point P perpendicular to AC at point E.", "ocr_zh": "三角形ABC中，P为底边BC上一点，过点B作BG垂直AC于点G，过点P作PD垂直AB于点D，过点P作PE垂直AC于点E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2245.jpg" }, { "index": 2246, "ocr_en": "In triangle ABC, draw line segment AI perpendicular to BC at point I. Draw line segment BG perpendicular to AC at point G. Draw line segment CH perpendicular to AB at point H. Point P is a point inside triangle ABC. Draw line segment PD perpendicular to AB at point D, PE perpendicular to AC at point E, and PF perpendicular to BC at point F.", "ocr_zh": "三角形ABC中，过点A作AI垂直BC于点I，过点B作BG垂直AC于点G，过点C作CH垂直AB于点H。P为三角形ABC内一点，过点P作PD垂直AB于点D，PE垂直AC于点E，PF垂直BC于点F。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2246.jpg" }, { "index": 2247, "ocr_en": "In triangle ABC, draw line segment BG perpendicular to AC at point G. Point P is a point within triangle ABC. Draw line segment PD perpendicular to AB at point D, line segment PE perpendicular to AC at point E, and line segment PF perpendicular to BC at point F.", "ocr_zh": "三角形ABC中，过点B作BG垂直AC于点G。P为三角形ABC内一点，过点P作PD垂直AB于点D，PE垂直AC于点E，PF垂直BC于点F。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2247.jpg" }, { "index": 2248, "ocr_en": "In the Cartesian coordinate system, line l1: y = (-3/4)x + 3 intersects the positive x-axis at point A and the positive y-axis at point C. Line l2: y = 3x + 3 intersects the positive y-axis at point C and the negative x-axis at point B.", "ocr_zh": "平面直角坐标系中，l1：y = （-3/4）x + 3与x轴正半轴交于点A，与y轴正半轴交于点C，l2：y = 3x + 3与y轴正半轴交于点C，与x轴负半轴交于点B。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_2248.jpg" }, { "index": 2249, "ocr_en": "In triangle ABC, extend BC to point P. Draw PD perpendicular to AB at point D. Draw PE perpendicular to the extension of AC at point E. Draw CF perpendicular to AB at point F.", "ocr_zh": "三角形ABC中，延长BC至点P，过点P作PD垂直AB于点D，过点P作PE垂直AC的延长线于点E。过点C作CF垂直AB于点F。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2249.jpg" }, { "index": 2250, "ocr_en": "In triangle ABC, draw CF perpendicular to AB at point F. Let P be a point on the base BC. Draw PD perpendicular to AB at point D. Also, draw PE perpendicular to AC at point E.", "ocr_zh": "三角形ABC中，过点C作CF垂直AB于点F，P为底边BC上一点，过点P作PD垂直AB于点D，过点P作PE垂直AC于点E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2250.jpg" }, { "index": 2251, "ocr_en": "In triangle ABC, draw CF perpendicular to AB at point F. Let P be a point on the base BC. Draw PD perpendicular to AB at point D. Also, draw PE perpendicular to AC at point E. Draw auxiliary lines to connect AP.", "ocr_zh": "三角形ABC中，过点C作CF垂直AB于点F，P为底边BC上一点，过点P作PD垂直AB于点D，过点P作PE垂直AC于点E。作辅助线连接AP。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2251.jpg" }, { "index": 2252, "ocr_en": "In quadrilateral ABCD, angles C and D are right angles. Connect AB. Take a point O on AB and draw auxiliary lines connecting OC and OD. Define angle BOC as angle 1, angle BOD as angle 2, angle BAC as angle 3, angle OCA as angle 4, angle BAD as angle 5, and angle ODA as angle 6.", "ocr_zh": "四边形ABCD中，角C和角D为直角，连接AB。在AB上取一点O，作辅助线连接OC,OD。定义角BOC为角1，角BOD为角2，角BAC为角3，角OCA为角4，角BAD为角5，角ODA为角6.", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2252.jpg" }, { "index": 2253, "ocr_en": "In quadrilateral ABCD, angles C and D are right angles, and line AB is drawn.", "ocr_zh": "四边形ABCD中，角C和角D为直角，连接AB。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2253.jpg" }, { "index": 2254, "ocr_en": "Right triangle ABC and right triangle ABD share the hypotenuse AB, with angles C and D being right angles. The two triangles have an overlapping part. Take the midpoint O of AB, and draw auxiliary lines connecting OC and OD. Mark angles CAD and COD.", "ocr_zh": "直角三角形ABC和直角三角形ABD共享一条斜边AB，角C和角D为直角，两个三角形有重叠部分。取AB中点O，作辅助线连接OC,OD。标出角CAD和角COD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2254.jpg" }, { "index": 2255, "ocr_en": "Right-angled triangle ABC and right-angled triangle ABD share the common hypotenuse AB, with angles C and D being right angles. The two triangles have an overlapping part.", "ocr_zh": "直角三角形ABC和直角三角形ABD共享一条斜边AB，角C和角D为直角，两个三角形有重叠部分。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2255.jpg" }, { "index": 2256, "ocr_en": "In quadrilateral ABCD, angles C and D are right angles. Connect AB. Take the midpoint O of AB, and draw auxiliary lines connecting OC and OD.", "ocr_zh": "四边形ABCD中，角C和角D为直角，连接AB。取AB中点O，作辅助线连接OC,OD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2256.jpg" }, { "index": 2257, "ocr_en": "Right-angled triangle ABC and right-angled triangle ABD share the common hypotenuse AB, with angles C and D being right angles. The two triangles have an overlapping part. Take the midpoint O of AB, and draw auxiliary lines connecting OC and OD.", "ocr_zh": "直角三角形ABC和直角三角形ABD共享一条斜边AB，角C和角D为直角，两个三角形有重叠部分。取AB中点O，作辅助线连接OC,OD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2257.jpg" }, { "index": 2258, "ocr_en": "In quadrilateral ABCD, AC and BD intersect at point E. G and H are the midpoints of AC and BD respectively. Connect GH.", "ocr_zh": "四边形ABCD中，AC,BD交于点E，G,H分别是AC,BD中点，连接GH。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2258.jpg" }, { "index": 2259, "ocr_en": "In quadrilateral ABCD, connect AC and BD. Angles ADB and ACB are right angles. Take the midpoint E of side AB, and connect DE and CE.", "ocr_zh": "四边形ABCD中，连接AC,BD，角ADB和角ACB为直角。取边AB中点E，连接DE,CE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2259.jpg" }, { "index": 2260, "ocr_en": "In triangle ABC, draw BF perpendicular to AC at point F, and draw CE perpendicular to AB at point E. Connect EF. Take the midpoint D of EF, take the midpoint G of BC, and connect DG. Draw auxiliary lines EG and FG.", "ocr_zh": "三角形ABC中，过点B作BF垂直AC于点F，过点C作CE垂直AB于点E，连接EF，取EF中点D，取BC中点G，连接DG。作辅助线连接EG,FG。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2260.jpg" }, { "index": 2261, "ocr_en": "In triangle ABC, draw BF perpendicular to AC at point F, and draw CE perpendicular to AB at point E. Connect EF. Take the midpoint D of EF, and take the midpoint G of BC. Connect DG.", "ocr_zh": "三角形ABC中，过点B作BF垂直AC于点F，过点C作CE垂直AB于点E，连接EF，取EF中点D，取BC中点G，连接DG。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2261.jpg" }, { "index": 2262, "ocr_en": "In the quadrilateral ABCE, AC and BE intersect at point F. Take the midpoint O of AB and connect OF.", "ocr_zh": "四边形ABCE中，AC与BE交于点F，取AB中点O，连接OF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2262.jpg" }, { "index": 2263, "ocr_en": "Point P is located inside the angle AOB.", "ocr_zh": "点P在角AOB内部。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2263.jpg" }, { "index": 2264, "ocr_en": "Point P is located inside the angle AOB. Draw auxiliary lines: Draw line PP' passing through point P and perpendicular to OB. Point P' is below point O. Draw a line perpendicular to point C from point P', and intersect OB at point D.", "ocr_zh": "点P在角AOB内部。作辅助线：过点P作PP'垂直OB，P'在OB下方，过点P'垂直于点C，交OB于点D。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2264.jpg" }, { "index": 2265, "ocr_en": "In triangle AOB, select points P and Q, where P is to the upper left of Q. Connect PQ. Draw auxiliary lines: Through point P, draw PP' perpendicular to OA, with P' located above OA. Through point Q, draw QQ' perpendicular to OB, with Q' located below OB. Connect P'Q' and it intersects OA at point M and OB at point N. Connect PM and QN.", "ocr_zh": "在角AOB中取点P,Q，其中P在Q左上方，连接PQ。作辅助线：过点P作PP'垂直OA，P'在OA左上方，过点Q作QQ'垂直OB，Q'在OB下方。连接P'Q'交OA于点M，交OB雨点N，连接PM,QN。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_2265.jpg" }, { "index": 2266, "ocr_en": "On the straight line l, select two points P and Q from left to right. Take point A above point P on the left side, and connect PA. Take point B above point Q on the right side, and connect BQ. Draw auxiliary lines: B' and B are symmetrical to l, connect BB' and B'Q; through point A, draw AA' parallel and equal to PQ, connect QA', A', Q, and B' so that the three points are collinear.", "ocr_zh": "在直线l上从左至右取两点P,Q，在P左上方取一点A，连接PA，在Q右上方取一点B，连接BQ。作辅助线：B'和B关于l对称，连接BB',B'Q；过点A作AA'平行且等于PQ，连接QA'，A',Q,B'三点共线。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2266.jpg" }, { "index": 2267, "ocr_en": "Take points P and Q in the angle AOB.", "ocr_zh": "在角AOB中取点P,Q.", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2267.jpg" }, { "index": 2268, "ocr_en": "Take a point P inside the angle AOB. Points P1 and P are symmetrical to P with respect to OA, and points P2 and P are symmetrical to P with respect to OB. Connect P1P2, intersecting OA at point M and OB at point N. Connect PM and PN. Draw auxiliary lines connecting OP1, OP2, PP1, PP2, and OP.", "ocr_zh": "在角AOB内部取一点P，P1和P关于OA对称，P2和P关于OB对称，连接P1P2，交OA于点M，交OB于点N。连接PM,PN。作辅助线连接OP1,OP2,PP1,PP2,OP。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2268.jpg" }, { "index": 2269, "ocr_en": "In triangle ABC, draw AD perpendicular to BC at point D. Select a point P on AD. Connect CP. Select point E on AC. Connect PE. Ensure that the extension of line EP passes through point B.", "ocr_zh": "三角形ABC中，过点A作AD垂直BC于点D。在AD上取一点P，连接CP，在AC上取一点E，连接PE，使得EP的延长线经过点B。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2269.jpg" }, { "index": 2270, "ocr_en": "In triangle AOB, select a point P. Let M and N be located on sides OA and OB respectively. Connect PM, PN and MN.", "ocr_zh": "在角AOB中取一点P，M,N分别在边OA,OB上，连接PM,PN,MN。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2270.jpg" }, { "index": 2271, "ocr_en": "In triangle ABC, take the midpoint E of AB. Draw a line through point E and intersect AC at point F. Select a point M on line segment EF. Connect BM and BD.", "ocr_zh": "三角形ABC中，取AB中点E，过点E作直线EF交AC于点F，在线段EF上取一点M，连接BM,BD。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2271.jpg" }, { "index": 2272, "ocr_en": "At vertex C of triangle ABC, draw a line l perpendicular to BC. Construct the triangle A'B'C that is symmetrical to triangle ABC with respect to line l. Let P be a point on segment A'C. Connect AP and BP.", "ocr_zh": "过三角形ABC顶点C作直线l垂直BC，作三角形ABC关于直线l对称的三角形A'B'C，P为线段A'C上一点，连接AP,BP。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2272.jpg" }, { "index": 2273, "ocr_en": "In triangle ABC, take the midpoint E of AB. Draw a line EF passing through point E and intersecting AC at point F. Select a point D on line segment EF. Take the midpoint D of BC and connect DB and DM.", "ocr_zh": "三角形ABC中，取AB中点E，过点E作直线EF交AC于点F，在线段EF上取一点D，取BC中点D，连接DB,DM。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2273.jpg" }, { "index": 2274, "ocr_en": "Point P is a point on ray BM. Point A is a point directly above P on the right side. Connect PA. Rotate ray BM clockwise by a certain angle (an acute angle) to obtain ray BN. Extend AP and intersect BN at point C, such that AC is perpendicular to BN.", "ocr_zh": "P为射线BM上一点，A为P右上方一点，连接PA。将射线BM顺时针旋转一定角度（锐角）得到射线BN，延长AP交BN于点C，使得AC垂直BN。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2274.jpg" }, { "index": 2275, "ocr_en": "Point P is a point on the ray BM, and point A is a point outside the ray BM, connected to PA; Take B as the endpoint to make the red imaginary ray BN (BN and point A are on both sides of the ray BM, and the angle NBM is an acute angle), C is a point on the BN, and make the yellow auxiliary line PC, which satisfies that the PC is perpendicular to the BN and the vertical foot is C.", "ocr_zh": "点P是射线BM上一点，点A是射线BM外一点，连接PA；以B为端点做红色虚射线BN（BN与点A在射线BM两侧，且角NBM为锐角），C是BN上一点，做黄色辅助线PC，满足PC垂直于BN，垂足为C。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2275.jpg" }, { "index": 2276, "ocr_en": "Point P is a point on the ray BM, and point A is a point outside the ray BM, connected to PA; Take B as the endpoint to make the virtual ray BN (BN and point A are on both sides of the ray BM, and the angle NBM is an acute angle), C is a point on the BN, and the auxiliary line PC is used to meet the PC perpendicular to the BN, and the vertical foot is C; C' is also a point on the BN, do the red auxiliary line AC', meet AC' perpendicular to BN, and perpendicular to C'.", "ocr_zh": "点P是射线BM上一点，点A是射线BM外一点，连接PA；以B为端点做虚射线BN（BN与点A在射线BM两侧，且角NBM为锐角），C是BN上一点，做辅助线PC，满足PC垂直于BN，垂足为C；C'也是BN上一点，做红色辅助线AC'，满足AC'垂直于BN，垂足为C'。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2276.jpg" }, { "index": 2277, "ocr_en": "Point P is the point on the ray BM, and point A is a point outside the ray BM, connecting the line segment PA.", "ocr_zh": "点P是射线BM上一点，点A是射线BM外一点，连接线段PA。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2277.jpg" }, { "index": 2278, "ocr_en": "In the Cartesian coordinate system xOy, the straight lines l1 and l2 are perpendicular to each other, the vertical foot is the point B, the point B is on the positive half axis of the y axis, the l1, l2 and x axes intersect at the points A and C respectively, and the points A and C are on the negative and positive semi-axes of the x-axis respectively. Point E is on the x-axis (to the right of point C), point F is in the first quadrant part of l1, do the auxiliary line EF and extend the intersection with l2 at the point C', do the auxiliary line CF.", "ocr_zh": "直角坐标系xOy中，直线l1与l2相互垂直，垂足为点B，点B在y轴正半轴，l1、l2与x轴分别交于点A和点C，点A与点C分别在x轴负半轴与正半轴上。点E在x轴上（点C右侧），点F在l1的第一象限部分，做辅助线EF并延长与l2交于点C'，做辅助线CF。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_2278.jpg" }, { "index": 2279, "ocr_en": "In the Cartesian coordinate system xOy, the straight lines l1 and l2 are perpendicular to each other, the vertical foot is the point B, the point B is on the positive half axis of the y axis, the l1, l2 and the x-axis intersect at the points A and C respectively, the points A and C are on the negative and positive semi-axes of the x-axis, and the points E are on the x-axis (to the right of the point C).", "ocr_zh": "直角坐标系xOy中，直线l1与l2相互垂直，垂足为点B，点B在y轴正半轴，l1、l2与x轴分别交于点A和点C，点A与点C分别在x轴负半轴与正半轴上，点E在x轴上（点C右侧）。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_2279.jpg" }, { "index": 2280, "ocr_en": "In the Cartesian coordinate system xOy, the straight lines l1 and l2 are perpendicular to each other, the vertical foot is the point B, the point B is on the positive half axis of the y axis, the l1, l2 and x axes intersect at the points A and C respectively, and the points A and C are on the negative and positive semi-axes of the x-axis respectively. Point E is on the x-axis (to the right of point C) and point F is in the first quadrant of l1, making the auxiliary line EF and extending the intersection with l2 at point C'. Make a red dotted line l3 bisect the second quadrant, make a red dotted line segment (auxiliary line) FG perpendicular to l3, and perpendicular to the point G. In addition, the black dashed segment FQ is perpendicular to the x-axis and intersects with l3 at the point Q.", "ocr_zh": "直角坐标系xOy中，直线l1与l2相互垂直，垂足为点B，点B在y轴正半轴，l1、l2与x轴分别交于点A和点C，点A与点C分别在x轴负半轴与正半轴上。点E在x轴上（点C右侧），点F在l1的第一象限部分，做辅助线EF并延长与l2交于点C'。做红色虚线直线l3平分第二四象限，做红色虚线段（辅助线）FG垂直于l3，垂足为点G。另外做黑色虚线段FQ垂直于x轴且与l3交于点Q。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_2280.jpg" }, { "index": 2281, "ocr_en": "In the Cartesian coordinate system xoy, the intersection points of the line l with the x-axis and the y-axis are point B and point C, respectively, B is on the positive half axis of x, C is on the negative half axis of y, point A is on the line segment OB, and the point P is on BC, connecting the line segment AP", "ocr_zh": "直角坐标系xoy中，直线l与x轴和y轴的交点分别为点B和点C，B在x正半轴，C在y负半轴，点A在线段OB上，点P在BC上，连接线段AP", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_2281.jpg" }, { "index": 2282, "ocr_en": "In the isosceles triangle ABC, AB is equal to AC, AD is the midline on the side BC, AD is perpendicular to BC, and the perpendicular foot is D", "ocr_zh": "等腰三角形ABC中，AB等于AC，AD为边BC上的中线，AD垂直于BC，且垂足为D", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2282.jpg" }, { "index": 2283, "ocr_en": "In the Cartesian coordinate system xoy, A is on the x positive semi-axis, C is on the y positive semi-axis, and B is in the first quadrant, OABC forms a rectangle, and the OC length is greater than OA. The straight line passes through the rectangle, and the intersection points with the OC, AB, and x-axis are D, E, and F respectively, the point F is on the right side of the x-axis A, and OM is perpendicular to this straight line, and the perpendicular foot is M inside the rectangle.", "ocr_zh": "直角坐标系xoy中，A在x正半轴，C在y正半轴，B在第一象限，OABC构成长方形，且OC长度要大于OA。一直线穿过长方形，与OC、AB、x轴的交点分别为D、E、F，点F在x轴A的右侧，OM垂直于此直线，垂足为M在长方形内部。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_2283.jpg" }, { "index": 2284, "ocr_en": "In an acute triangle ACE, point B is on AC, point D is on AE, and the red dotted line connects the BD, which is parallel to the CE.", "ocr_zh": "锐角三角形ACE中，点B在AC上，点D在AE上，红色虚线连接BD，BD平行于CE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2284.jpg" }, { "index": 2285, "ocr_en": "In the acute triangle ABC, point D is on AB, point E is at AC, and BC is parallel to DE. where DE is the red line segment and BC is the yellow line segment.", "ocr_zh": "锐角三角形ABC中，点D在AB上，点E在AC，BC平行于DE。其中DE为红色线段，BC为黄色线段。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2285.jpg" }, { "index": 2286, "ocr_en": "In the acute triangle ABC, point D is on AB, point E is at AC, and BC is parallel to DE.", "ocr_zh": "锐角三角形ABC中，点D在AB上，点E在AC，BC平行于DE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2286.jpg" }, { "index": 2287, "ocr_en": "In the acute triangle ABD, extend AB to point C, extend AD to E, and connect the CE with the red dotted line so that the CE is parallel to the BD.", "ocr_zh": "在锐角三角形ABD，延长AB到点C，延长AD到E，红色虚线连接CE，使得CE平行于BD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2287.jpg" }, { "index": 2288, "ocr_en": "Left: In an acute triangle ACE, point B is on AC, point D is on AE, and the red dotted line connects BD, which is parallel to CE. Right: In an acute triangle ABD, AB is extended to point C, AD is extended to E, and the red dotted line connects CE, which is parallel to BD.", "ocr_zh": "左图：锐角三角形ACE中，点B在AC上，点D在AE上，红色虚线连接BD，BD平行于CE。 右图：锐角三角形ABD中，延长AB到点C，延长AD到E，红色虚线连接CE，CE平行于BD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2288.jpg" }, { "index": 2289, "ocr_en": "In triangle ABC, D is on AB, E is on AC, and M is on BC, connecting AM and DE, and the intersection of the two is F, and the DE is parallel to BC. Among them, DF and BM are red line segments, and CM and EF are yellow line segments.", "ocr_zh": "三角形ABC中，D在AB上，E在AC上，M在BC上，连接AM，DE，两者交点为F，且满足DE平行于BC。其中DF，BM为红色线段，CM，EF为黄色线段。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2289.jpg" }, { "index": 2290, "ocr_en": "In the triangle ABC, AD is the midline on the side of BC, E is a point close to A on AB, and the intersection of CE and AD is F.", "ocr_zh": "三角形ABC中，AD为BC边上的中线，E为AB上靠近A一点，连接CE与AD的交于F。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2290.jpg" }, { "index": 2291, "ocr_en": "In the parallelogram ABCD, the angle A is an acute angle, the line segment BD is a diagonal, the point E is on AD (near D), and F is on the BD, connecting the line segment EF, satisfying EF is parallel to AB. Among them, the line segments AB, BD, AD, EF are all solid red lines.", "ocr_zh": "平行四边形ABCD中，角A为锐角，线段BD为一条对角线，点E在AD上（靠近D），F在BD上，连接线段EF，满足EF平行于AB。其中，线段AB，BD，AD，EF均为红色实线。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2291.jpg" }, { "index": 2292, "ocr_en": "In the quadrilateral ABCD, angle ABC and C are acute angles, and angle A and ADC are obtuse angles. P is the point on the diagonal BD, E is the point on AD, F is the point on BC, connecting PE and PF. It is satisfied that PE is parallel to AB, and PF is parallel to CD.", "ocr_zh": "四边形ABCD中，角ABC和角C为锐角，角A和角ADC为钝角。P为对角线BD上一点，E为AD上一点，F为BC上一点，连接PE，PF。满足PE于AB平行，PF与CD平行。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2292.jpg" }, { "index": 2293, "ocr_en": "In the triangle ABC, AD is the midline on the side of BC, E is a point close to A on AB, and the intersection of CE and AD is F. Pass D to make the auxiliary line, DG is parallel to CF, and AB is at point G. Among them, the three sides of the triangle BEC and the dotted line DG are all red.", "ocr_zh": "三角形ABC中，AD为BC边上的中线，E为AB上靠近A一点，连接CE与AD的交于F。过D做辅助线DG平行于CF交AB于点G。其中三角形BEC三条边与虚线DG均为红色。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2293.jpg" }, { "index": 2294, "ocr_en": "In the parallelogram ABCD, angle ABC is an acute angle. AC and BD intersect at point O, with point E on AO, G on AD, and F on BC, forming triangle EFG, satisfying that EG is parallel to BD and EG is perpendicular to EF, with the foot of the altitude being E.", "ocr_zh": "平行四边形ABCD中，角ABC为锐角。AC与BD交于点O，点E在AO上，G在AD上，F在BC上，做三角形EFG，满足EG平行于BD且EG垂直于EF，垂足为E。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2294.jpg" }, { "index": 2295, "ocr_en": "In the right triangle ABC, the angle C is the right angle. Point D on AB, point E on BC, point F on AC, connect the line segments DE, EF. EF is parallel and equal to AD.", "ocr_zh": "直角三角形ABC中，角C为直角。点D在AB上，点E在BC上，点F在AC上，连接线段DE，EF。满足EF平行且等于AD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2295.jpg" }, { "index": 2296, "ocr_en": "In the obtuse triangle ABC, angle C is an obtuse angle. AF is the middle line on the BC side, point D is on AB, point E is on AC, and the intersection point of line segment DE and AF is G. Meet DE parallel to BC.", "ocr_zh": "钝角三角形ABC中，角C为钝角。AF为BC边上的中线，点D在AB上，点E在AC上，线段DE与AF交点为G。满足DE平行于BC。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2296.jpg" }, { "index": 2297, "ocr_en": "In the triangular ABD, extend the AB to point C. The dashed CE is parallel to AD and the extension line of DB is intersected with E.", "ocr_zh": "三角形ABD中，延长AB到点C。过C做虚线CE平行于AD，且与DB的延长线交于E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2297.jpg" }, { "index": 2298, "ocr_en": "The line segment AB is parallel to the line segment CD, connecting BC and AD, and the two intersect at the point O. The triangular ABO area is blue and the triangular CDO area is yellow.", "ocr_zh": "线段AB平行于线段CD，连接BC与AD，两者交于点O。三角形ABO面积为蓝色，三角形CDO面积为黄色。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2298.jpg" }, { "index": 2299, "ocr_en": "The triangular ACD, point B on AC and point E on the CD extension line, connects BE, and BE and AD intersect at point O.", "ocr_zh": "三角形ACD，点B在AC上，点E在CD延长线上，连接BE，BE与AD交于点O。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2299.jpg" }, { "index": 2300, "ocr_en": "The triangular ACD, point B on AC and point E on the CD extension line, connects BE, and BE and AD intersect at point O. The red dotted line DF is parallel to BE, and AC is crossed at the point F.", "ocr_zh": "三角形ACD，点B在AC上，点E在CD延长线上，连接BE，BE与AD交于点O。过D做红色虚线DF平行于BE，交AC于点F。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2300.jpg" }, { "index": 2301, "ocr_en": "In the parallelogram ABCD, extend BA to point E (AE length is less than AB), and connect EC to AD at point F. where AE and CD are red line segments.", "ocr_zh": "平行四边形ABCD中，延长BA到点E（AE长度小于AB），连接EC于AD交于点F。其中AE，CD为红色线段。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2301.jpg" }, { "index": 2302, "ocr_en": "The line segment AB is parallel to the line segment CD, E is on AB, the point F is on the CD, and the three line segments AD, BC and EF intersect at the point O. Among them, AE and FE are red lines, and EB and CF are yellow lines.", "ocr_zh": "线段AB平行于线段CD，E在AB上，点F在CD上，AD、BC与EF三条线段交于点O。其中AE、FE为红色线段，EB、CF为黄色线段。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2302.jpg" }, { "index": 2303, "ocr_en": "There is an inverted semicircle with the diameter of the line segment MN, the midpoint of the arc MN is P, which connects the PM and extends to the point A, and AB is perpendicular to the straight line where the MN is located, and the vertical foot is B; Connect the PN and extend it to point C, making the line segment CD perpendicular to the straight line where the MN is located, and the vertical foot is D.", "ocr_zh": "有以线段MN为直径的倒置半圆，圆弧MN中点为P，连接PM并延长至点A，做AB垂直于MN所在直线，垂足为B；连接PN并延长至点C，做线段CD垂直于MN所在直线，垂足为D。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2303.jpg" }, { "index": 2304, "ocr_en": "In the parallelogram ABCD, the angle ABC is an acute angle, AC and BD intersect at the point O, the point E is on AD, and the connection AE and AO intersect at the point G. Point F is on BC, connecting DF and CO to point H. and BE is parallel to DF.", "ocr_zh": "平行四边形ABCD中，角ABC为锐角，AC与BD交于点O，点E在AD上，连接AE与AO交于点G；点F在BC上，连接DF与CO交于点H。且BE平行于DF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2304.jpg" }, { "index": 2305, "ocr_en": "In the rectangular ABCD, the line segment AC is a diagonal, E is a point close to A on AD, and the connecting line segment BE and AC intersect at the point F.", "ocr_zh": "长方形ABCD中，线段AC为一条对角线，E为AD上靠近A的一点，连接线段BE与AC交于点F。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2305.jpg" }, { "index": 2306, "ocr_en": "There is an inverted semicircle with the diameter of the line segment MN, the midpoint of the arc MN is P, which connects the PM and extends to the point A, and AB is perpendicular to the straight line where the MN is located, and the vertical foot is B; Connect the PN and extend it to point C, making the line segment CD perpendicular to the straight line where the MN is located, and the vertical foot is D. Finally, the red dotted line PH is perpendicular to MN, and the vertical foot is H. (AB, CD, AP, CP are solid red lines)", "ocr_zh": "有以线段MN为直径的倒置半圆，圆弧MN中点为P，连接PM并延长至点A，做AB垂直于MN所在直线，垂足为B；连接PN并延长至点C，做线段CD垂直于MN所在直线，垂足为D。最后做红色虚线PH垂直于MN，垂足为H。（AB、CD、AP、CP均为红色实线）", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2306.jpg" }, { "index": 2307, "ocr_en": "In triangle ABC, D, E, and F are the midpoints of AC, AB, BC, respectively, and the intersection of AF, CE, and BD is O.", "ocr_zh": "三角形ABC中，D、E、F分别为AC、AB、BC的中点，线段AF、CE、BD的交点为O。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2307.jpg" }, { "index": 2308, "ocr_en": "In the acute triangle ABC, point D is a point on AB close to B, connecting CD.", "ocr_zh": "锐角三角形ABC中，点D为AB上靠近B的一点，连接CD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2308.jpg" }, { "index": 2309, "ocr_en": "In the acute triangle ABC, the point D is a point AB, and E is a point close to C on AC, connecting DE.", "ocr_zh": "锐角三角形ABC中，点D为AB一点，E为AC上靠近C一点，连接DE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2309.jpg" }, { "index": 2310, "ocr_en": "In triangle ABC, point D is a point near A and E is a point on AB, connecting DE.", "ocr_zh": "三角形ABC中，点D为AC靠近A一点，E为AB上一点，连接DE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2310.jpg" }, { "index": 2311, "ocr_en": "In the diamond-shaped ABCD, the angle D is an acute angle, extending the diagonal AC to the point E, connecting BE, satisfying that AE and BE are equal in length", "ocr_zh": "菱形ABCD中，角D为锐角，延长对角线AC至点E，连接BE，满足AE与BE长度相等", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2311.jpg" }, { "index": 2312, "ocr_en": "In triangle ABC, the angle BAC is an obtuse angle, M is a point close to B on BC, and N is a point near C on BC, connecting AM and AN.", "ocr_zh": "三角形ABC中，角BAC为钝角，M为BC上靠近B一点，N为BC上靠近C一点，连接AM、AN。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2312.jpg" }, { "index": 2313, "ocr_en": "In the square ABCD, AC and BD are diagonal, E is a point close to O on OC, connecting BE, AF is perpendicular to BE, and the vertical foot is F.", "ocr_zh": "正方形ABCD中，AC、BD为对角线，E为OC上靠近O一点，连接BE，过A做AF垂直于BE，垂足为F。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2313.jpg" }, { "index": 2314, "ocr_en": "In the isosceles triangle ABC, AB is equal to AC, D is a point close to B on BC, and E is a point close to B on AB, connecting AD and DE.", "ocr_zh": "等腰三角形ABC中，AB等于AC，D为BC上靠近B一点，E为AB上靠近B一点，连接AD、DE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2314.jpg" }, { "index": 2315, "ocr_en": "The line segment AC and the line segment DB intersect at the point O, the angular AOD is an obtuse angle, the solid line connects AB and DC, the red dotted line connects AD and BC, and the areas of the triangle ADO and triangle BCO are blue and yellow, respectively.", "ocr_zh": "线段AC与线段DB交于点O，角AOD为钝角，实线连接AB、DC，红色虚线连接AD、BC，三角形ADO和三角形BCO的面积分别为蓝色和黄色。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2315.jpg" }, { "index": 2316, "ocr_en": "The line segment AC and the line segment DB intersect at the point O, the angular AOD is an obtuse angle, the solid line connects AB and DC, and the areas of the triangle ABO and the triangle DCO are blue and yellow, respectively.", "ocr_zh": "线段AC与线段DB交于点O，角AOD为钝角，实线连接AB、DC，三角形ABO和三角形DCO的面积分别为蓝色和黄色。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2316.jpg" }, { "index": 2317, "ocr_en": "In triangle ABC, E is a point on AB near B, and D is a point on AC near A, connecting DE, BD, CE.BD and EC intersect at point O.", "ocr_zh": "三角形ABC中，E为AB上靠近B一点，D为AC上靠近A一点，连接DE、BD、CE。BD与EC交于点O", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2317.jpg" }, { "index": 2318, "ocr_en": "There is a semicircular OAB with O as the center of the circle, line segment AB as the diameter, point C as the midpoint of arc AB, D as a point on the arc AC close to A, red solid line connecting AD, BD, AC, and red dotted line connecting BC. BD and AC intersect at point E.", "ocr_zh": "有以O为圆心，线段AB为直径的半圆OAB，点C为弧AB的中点，D为弧AC上靠近A的一点，红色实线连接AD、BD、AC，红色虚线连接BC。BD与AC交于点E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2318.jpg" }, { "index": 2319, "ocr_en": "There is a semicircular OAB with O as the center of the circle, the line segment AB as the diameter, the point C is the midpoint of the arc AB, and D is the point on the arc AC close to A, connecting AD, BD, AC, and BD and AC at the point E.", "ocr_zh": "有以O为圆心，线段AB为直径的半圆OAB，点C为弧AB的中点，D为弧AC上靠近A的一点，连接AD、BD、AC，BD与AC交于点E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2319.jpg" }, { "index": 2320, "ocr_en": "In the acute triangle ABC, the length of AB is less than AC, and the triangle ADE is obtained by rotating the triangle ABC counterclockwise around point A, and D is the rotation B, which is exactly on BC.", "ocr_zh": "锐角三角形ABC中，AB长度小于AC，绕A点逆时针旋转三角形ABC得到三角形ADE，D是旋转后B，恰好在BC边上。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2320.jpg" }, { "index": 2321, "ocr_en": "In the circle with the line segment AB as the diameter and the point O as the center of the circle, C and D are the two points on the arc, and the point A is the midpoint of the inferior arc CD, and the connecting chord CD and AB intersect at the point E.", "ocr_zh": "以线段AB为直径，以点O为圆心的圆中，C、D为圆弧上两点，点A是劣弧CD的中点，连接弦CD与AB交于点E。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2321.jpg" }, { "index": 2322, "ocr_en": "The circle O is the circumscribed circle of the rectangular ABCD, the center of the circle is O, the line segment AB should be shorter than the line segment AD, and E is the point on the line segment BC close to B, which connects AE and extends it, and intersects the circle O at the point F.", "ocr_zh": "圆O是矩形ABCD的外接圆，圆心为O，线段AB要短于线段AD，E为线段BC上靠近B的一点，连接AE并延长，与圆O交于点F。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2322.jpg" }, { "index": 2323, "ocr_en": "In the irregular quadrilateral ABCD, the angle BAD and the angular ADC are obtuse angles, and the diagonal is the line segment AC and BD, where AC is perpendicular to AB at point A, and BD is perpendicular to CD at point D.", "ocr_zh": "不规则四边形ABCD中，角BAD与角ADC为钝角，对角线为线段AC与BD，其中，AC垂直于AB于点A，BD垂直于CD于点D。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2323.jpg" }, { "index": 2324, "ocr_en": "In the triangle ABC, the angle BAC is a right angle, D is a point on BC, the line segment AD is perpendicular to BC, the vertical foot is the point D, and F is a point close to A on AC, connecting the line segment BF.BF and AD intersect at the point E.", "ocr_zh": "三角形ABC中，角BAC为直角，D为BC上一点，线段AD垂直于BC，垂足为点D，F为AC上靠近A的一点，连接线段BF，BF与AD交于点E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2324.jpg" }, { "index": 2325, "ocr_en": "In triangle ABC, the angle C is a right angle, and the line segment AC is slightly shorter than the line segment BC.", "ocr_zh": "三角形ABC中，角C是直角，且线段AC略短于线段BC。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2325.jpg" }, { "index": 2326, "ocr_en": "In triangle ABC, the angle ACB is a right angle, and the line segment AC is slightly shorter than the line segment BC. D is a point on AB, connect the CD, meet the CD perpendicular to AB, and the vertical foot is the point D. The angle ACD and the corner CBD are marked with a red arc, and the angle A and the angle DCB are marked with a yellow double arc.", "ocr_zh": "三角形ABC中，角ACB为直角，且线段AC略短于线段BC。D为AB上一点，连接CD，满足CD垂直于AB，垂足为点D。角ACD与角CBD用红色弧线标出，角A与角DCB用黄色双弧线标出。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2326.jpg" }, { "index": 2327, "ocr_en": "In triangle ABC, the angle ACB is a right angle, and the line segment AC is slightly shorter than the line segment BC. D is a point on AB, connecting CD, satisfying CD perpendicular to AB, perpendicular to AB, vertical foot is point D, and line segment CD is a red solid line segment.", "ocr_zh": "三角形ABC中，角ACB为直角，且线段AC略短于线段BC。D为AB上一点，连接CD，满足CD垂直于AB，垂足为点D，线段CD为红色实线段。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2327.jpg" }, { "index": 2328, "ocr_en": "In the isosceles triangle ABC, AB is the same length as AC and is longer than BC, AD is the midline on the side BC, E is a point on AB, the connecting DE meets DE perpendicular to AB, and the vertical foot is E.", "ocr_zh": "等腰三角形ABC中，AB与AC长度相同且比BC长，AD为边BC上的中线，E为AB上一点，连接DE满足DE垂直于AB，垂足为E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2328.jpg" }, { "index": 2329, "ocr_en": "In the rectangular ABCD, AD is longer than AB, the line segment AC is a diagonal, E is a point on AD, the connection BE and AC intersect at the point G, and the BE is perpendicular to AC", "ocr_zh": "矩形ABCD中，AD比AB长，线段AC为一条对角线，E为AD上一点，连接BE与AC相交于点G，满足BE垂直于AC", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2329.jpg" }, { "index": 2330, "ocr_en": "In the right triangle ABC, the angle ACB is a right angle, AC is longer than BC, D is a point above AB, CD is perpendicular to AB, the vertical foot is D, E is a point on AD, and CE is connected.", "ocr_zh": "直角三角形ABC中，角ACB为直角，AC比BC要长，D为AB上一点，CD垂直于AB，垂足为D，E为AD上一点，连接CE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2330.jpg" }, { "index": 2331, "ocr_en": "In triangle ABC, the angle BAC is a right angle, D is a point on BC, the line segment AD is perpendicular to BC, and the vertical foot is the point D.", "ocr_zh": "三角形ABC中，角BAC为直角，D为BC上一点，线段AD垂直于BC，垂足为点D。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2331.jpg" }, { "index": 2332, "ocr_en": "In the right triangle ABC, the angle C is the right angle, and the circle O is made with AC as the diameter, O is the midpoint of AC and the center of the circle O, the circle O and AB intersect at the point D, and the point E is the midpoint of BC, connecting DE.", "ocr_zh": "直角三角形ABC中，角C为直角，以AC为直径做圆O，O为AC中点也是圆O的圆心，圆O与AB交于点D，点E为BC的中点，连接DE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2332.jpg" }, { "index": 2333, "ocr_en": "In the square ABCD, AC and BD are diagonals, and AC and BD intersect at the point O, the point E is a point close to C on the CD, the connecting line segment BE, F is the point on the BE, the CF is perpendicular to BE, the vertical foot is F, and finally the OF is connected.", "ocr_zh": "正方形ABCD中，AC、BD为对角线，且AC与BD交于点O，点E为CD上靠近C的一点，连接线段BE，F为BE上一点，满足CF垂直于BE，垂足为F，最后连接OF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2333.jpg" }, { "index": 2334, "ocr_en": "In the right triangle ABC, the angle ACB is a right angle, AC is longer than BC, D is a point above AB, CD is perpendicular to AB, and the vertical foot is D.", "ocr_zh": "直角三角形ABC中，角ACB为直角，AC比BC要长，D为AB上一点，CD垂直于AB，垂足为D。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2334.jpg" }, { "index": 2335, "ocr_en": "In the acute triangle ABC, the point E is a point on the edge of AC, which is connected to BE and extended to point F, AF and CF are connected to meet that AF is parallel to BC, and there is a point D on AB, which is connected to DE and extended to intersect with FC at point G, and DG is parallel to BC. Among them, CF is a black dotted line, DE is a red solid line, and EG is a red dashed line.", "ocr_zh": "锐角三角形ABC中，点E为AC边上一点，连接BE并延长至点F，连接AF、CF，满足AF平行于BC，另外AB上有一点D，连接DE并延长与FC交于点G，满足DG平行于BC。其中，CF为黑色虚线，DE为红色实线，EG为红色虚线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2335.jpg" }, { "index": 2336, "ocr_en": "In the triangular BED, the angle BED is an obtuse angle, extending DE to point A, extending BE to point C, and the red solid line connects AB and CD, satisfying that AB is parallel to CD, and AB is longer than CD. In addition, connect the BD and make a solid yellow line segment EF and BD intersect at the point F, and EF is parallel to AB.", "ocr_zh": "三角形BED中，角BED为钝角，延长DE至点A，延长BE至点C，红色实线连接AB、CD，满足AB平行于CD，且AB长于CD。另外连接BD，做黄色实线段EF与BD交于点F，EF平行于AB。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2336.jpg" }, { "index": 2337, "ocr_en": "In the triangular BED, the angle BED is an obtuse angle, which extends DE to point A and BE to point C, connecting AB and CD, satisfying that AB is parallel to CD, and AB is longer than CD. In addition, the BD is connected and the line segment EF and BD intersect at the point F, and the EF is parallel to AB.", "ocr_zh": "三角形BED中，角BED为钝角，延长DE至点A，延长BE至点C，连接AB、CD，满足AB平行于CD，且AB长于CD。另外连接BD，做线段EF与BD交于点F，EF平行于AB。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2337.jpg" }, { "index": 2338, "ocr_en": "In the triangular BCF, the angle BFC is an obtuse angle, extending CF to point A, extending BF to point D, connecting AB and CD, satisfying that AB is parallel to CD, and AB is shorter than CD. In addition, connect BC and make the line segment EF and BC intersect at point E, and EF is parallel to AB.", "ocr_zh": "三角形BCF中，角BFC为钝角，延长CF至点A，延长BF至点D，连接AB、CD，满足AB平行于CD，且AB短于CD。另外连接BC，做线段EF与BC交于点E，EF平行于AB。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2338.jpg" }, { "index": 2339, "ocr_en": "In the triangular ACE, the angular AEC is an obtuse angle, extending CE to point B and AE to point D, connecting AB and CD, satisfying that AB is parallel to CD, and AB is longer than CD. In addition, AC is connected and the line segment EF and AC intersect at point F, and EF is parallel to AB. Among them, AB, EF, CD are yellow line segments, and BC and AD are red line segments.", "ocr_zh": "三角形ACE中，角AEC为钝角，延长CE至点B，延长AE至点D，连接AB、CD，满足AB平行于CD，且AB长于CD。另外连接AC，做线段EF与AC交于点F，EF平行于AB。其中，AB、EF、CD为黄色线段，BC、AD为红色线段。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2339.jpg" }, { "index": 2340, "ocr_en": "In the triangular BED, the angle BED is an obtuse angle, which extends DE to point A and BE to point C, connecting AB and CD, satisfying that AB is parallel to CD, and AB is longer than CD. In addition, the BD is connected and the line segment EF and BD intersect at the point F, and the EF is parallel to AB.", "ocr_zh": "三角形BED中，角BED为钝角，延长DE至点A，延长BE至点C，连接AB、CD，满足AB平行于CD，且AB长于CD。另外连接BD，做线段EF与BD交于点F，EF平行于AB。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2340.jpg" }, { "index": 2341, "ocr_en": "In the triangular BED, the angle BED is an obtuse angle, which extends DE to point A and BE to point C, connecting AB and CD, satisfying that AB is parallel to CD, and AB is longer than CD. In addition, connect the BD to meet the requirements that AB is perpendicular to BD, and make the line segment EF and BD intersect at the point F, and EF is parallel to AB.", "ocr_zh": "三角形BED中，角BED为钝角，延长DE至点A，延长BE至点C，连接AB、CD，满足AB平行于CD，且AB长于CD。另外连接BD，满足AB垂直于BD，做线段EF与BD交于点F，EF平行于AB。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2341.jpg" }, { "index": 2342, "ocr_en": "The line segment AB is parallel to the CD, connecting AD, BC, AC, BD, AC and BD at the point O, crossing the point O to make the line segment EF parallel to AB, EF and AD at the point E, and BC at the point F.", "ocr_zh": "线段AB平行于CD，连接AD、BC、AC、BD，AC与BD交于点O，过点O做线段EF平行于AB，EF与AD交于点E，与BC交于点F。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2342.jpg" }, { "index": 2343, "ocr_en": "The ray OB and the ray OA form an acute angle AOB, the ray OC divides the angle AOB into two angles, the point M is on the OC, the MN is parallel to the OB, and the MN and OA intersect the point N. There is a point E on the OB, connect the EM and extend the intersection with the OA at point D.", "ocr_zh": "射线OB与射线OA组成锐角AOB，射线OC将角AOB分为两个角，点M在OC上，做MN平行于OB，MN与OA交于点N。OB上有一点E，连接EM并延长与OA交于点D。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2343.jpg" }, { "index": 2344, "ocr_en": "In trapezoidal ABCD, AB is parallel to CD, AC and BD are diagonals and intersect at the point O, and the line segment EF and AD intersect F and BC at E, satisfying EF to be parallel to AB", "ocr_zh": "梯形ABCD中，AB平行于CD，AC、BD为对角线且相交于点O，过O做线段EF与AD交于F，与BC交于E，满足EF平行于AB", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2344.jpg" }, { "index": 2345, "ocr_en": "In the rectangular ABCD, AD is longer than AB, the line segments AC and BD are diagonal, the intersection point is the point O, the crossing point O is perpendicular to the OE of BC, and the vertical foot is the point E.", "ocr_zh": "矩形ABCD中，AD比AB长，线段AC、BD为对角线，交点为点O，过点O做OE垂直于BC，垂足为点E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2345.jpg" }, { "index": 2346, "ocr_en": "In triangle ABC, the point G is on AB and the point F is on AC, connecting GF, GF is parallel to BC, doing GD is perpendicular to BC, perpendicular to point D, doing FE perpendicular to BC, and perpendicular to point E. AH is perpendicular to BC through A, the vertical foot is H, and AH and GF intersect at the point K. where AH is a black dotted line.", "ocr_zh": "三角形ABC中，点G在AB上，点F在AC上，连接GF，GF平行于BC，做GD垂直于BC，垂足为点D，做FE垂直于BC，垂足为点E。过A做AH垂直于BC，垂足为H，且AH与GF交于点K。其中AH为黑色虚线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2346.jpg" }, { "index": 2347, "ocr_en": "In triangle ABC, the point G is on AB and the point F is on AC, connecting GF, GF is parallel to BC, doing GD is perpendicular to BC, perpendicular to point D, doing FE perpendicular to BC, and perpendicular to point E. AH is perpendicular to BC through A, the vertical foot is H, and AH and GF intersect at the point K. where AH is a red dotted line.", "ocr_zh": "三角形ABC中，点G在AB上，点F在AC上，连接GF，GF平行于BC，做GD垂直于BC，垂足为点D，做FE垂直于BC，垂足为点E。过A做AH垂直于BC，垂足为H，且AH与GF交于点K。其中AH为红色虚线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2347.jpg" }, { "index": 2348, "ocr_en": "In the triangle ABC, the point E is on AB, the point H is on AC, the connection EH, EH is parallel to BC, EF is perpendicular to BC, the perpendicular foot is the point F, the perpendicular foot is the point F, the perpendicular foot is perpendicular to BC, and the perpendicular foot is the point G. AD is perpendicular to BC through A, the vertical foot is D, and AD and EH intersect at the point M.", "ocr_zh": "三角形ABC中，点E在AB上，点H在AC上，连接EH，EH平行于BC，做EF垂直于BC，垂足为点F，做HG垂直于BC，垂足为点G。过A做AD垂直于BC，垂足为D，且AD与EH交于点M。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2348.jpg" }, { "index": 2349, "ocr_en": "In the right triangle ABC, the angle C is the right angle, the point D is on AC, the point E is on AB, the connection DE, DE is parallel to AB, DM is perpendicular to AB, the perpendicular foot is the point M, the perpendicular foot is perpendicular to AB, and the perpendicular foot is the point N. The midpoint of DM is G, the midpoint of EN is F, and the FG is connected.", "ocr_zh": "直角三角形ABC中，角C为直角，点D在AC上，点E在AB上，连接DE，DE平行于AB，做DM垂直于AB，垂足为点M，做EN垂直于AB，垂足为点N。DM中点为G，EN中点为F，连接FG。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2349.jpg" }, { "index": 2350, "ocr_en": "In the acute triangle ABC, the point P is on AB, the point Q is on AC, the PQ is connected, the PQ is parallel to BC, the PN is perpendicular to BC, the perpendicular foot is the point N, the QM is perpendicular to BC, and the perpendicular foot is the point M. AD is perpendicular to BC through A, the vertical foot is D, and AD and PQ intersect at the point H. Among them, except for the line segment AD, which is a solid black line, the rest of the line segments are solid red lines.", "ocr_zh": "锐角三角形ABC中，点P在AB上，点Q在AC上，连接PQ，PQ平行于BC，做PN垂直于BC，垂足为点N，做QM垂直于BC，垂足为点M。过A做AD垂直于BC，垂足为D，且AD与PQ交于点H。其中，除了线段AD为黑色实线，其余线段均为红色实线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2350.jpg" }, { "index": 2351, "ocr_en": "In the acute triangle ABC, there are several small rectangles stacked inside the triangle, the bottom edge of the first rectangle is on the edge of BC, one of the vertices is connected to AB, the second rectangle is above the first rectangle, the left vertex is connected to AB, and so on.", "ocr_zh": "锐角三角形ABC中，有若干个小矩形堆积在三角形内部，第一个矩形底边在BC边上，其中一个顶点与AB相接，第二个矩形在第一个矩形上方，左侧顶点与AB相接，以此类推。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2351.jpg" }, { "index": 2352, "ocr_en": "In the acute triangle ABC, the point P is on AB, the point N is on AC, the PN is connected to the PN, the PN is parallel to BC, the PQ is perpendicular to BC, the perpendicular foot is the point Q, the NM is perpendicular to BC, and the perpendicular foot is the point M. AD is perpendicular to BC through A, D is perpendicular to the vertical foot, and AD and PN intersect at point E. In addition, connect the PN midpoint with the QM midpoint to divide the rectangular PQMN into two small squares.", "ocr_zh": "锐角三角形ABC中，点P在AB上，点N在AC上，连接PN，PN平行于BC，做PQ垂直于BC，垂足为点Q，做NM垂直于BC，垂足为点M。过A做AD垂直于BC，垂足为D，AD与PN交于点E。另外，连接PN中点与QM中点，将矩形PQMN分为两个小的正方形。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2352.jpg" }, { "index": 2353, "ocr_en": "In the acute triangle ABC, the point P is on AB, the point N is on AC, the PN is connected to the PN, the PN is parallel to BC, the PQ is perpendicular to BC, the perpendicular foot is the point Q, the NM is perpendicular to BC, and the perpendicular foot is the point M. AD is perpendicular to BC through A, D is perpendicular to the vertical foot, and AD and PN intersect at point E. PN is longer than PQ", "ocr_zh": "锐角三角形ABC中，点P在AB上，点N在AC上，连接PN，PN平行于BC，做PQ垂直于BC，垂足为点Q，做NM垂直于BC，垂足为点M。过A做AD垂直于BC，垂足为D，AD与PN交于点E。PN比PQ长", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2353.jpg" }, { "index": 2354, "ocr_en": "There are right-angled triangle APC and right-angled triangle BPD, A, P, B in a straight line, CA perpendicular to AP, perpendicular foot A, and CA longer than AP. DB is perpendicular to PB, vertical foot is B, and DB is shorter than PB. The triangular APC area is blue and the triangular BPD area is yellow.", "ocr_zh": "存在直角三角形APC与直角三角形BPD，A、P、B依次在一条直线上，CA垂直于AP，垂足为A，且CA长于AP。DB垂直于PB，垂足为B，且DB短于PB。三角形APC面积为蓝色，三角形BPD面积为黄色。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2354.jpg" }, { "index": 2355, "ocr_en": "There are right-angled triangle APC and right-angled triangle BPD, A, B, P in a straight line, CA perpendicular to AP, perpendicular foot A, and CA shorter than AP. DB is perpendicular to PB, vertical foot is B, and DB is longer than PB. The triangle APC area is yellow, the triangle BPD area is blue, and the two triangles are located on either side of the AP.", "ocr_zh": "存在直角三角形APC与直角三角形BPD，A、B、P依次在一条直线上，CA垂直于AP，垂足为A，且CA短于AP。DB垂直于PB，垂足为B，且DB长于PB。三角形APC面积为黄色，三角形BPD面积为蓝色，两个三角形位于AP两侧。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2355.jpg" }, { "index": 2356, "ocr_en": "There are right-angled triangle APC and right-angled triangle BPD, A, B, P in a straight line, CA perpendicular to AP, perpendicular foot A, and CA shorter than AP. DB is perpendicular to PB, vertical foot is B, and DB is longer than PB. Two triangles are located on either side of the AP, and the angular CPD is at right angles.", "ocr_zh": "存在直角三角形APC与直角三角形BPD，A、B、P依次在一条直线上，CA垂直于AP，垂足为A，且CA短于AP。DB垂直于PB，垂足为B，且DB长于PB。两个三角形位于AP两侧，且角CPD为直角。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2356.jpg" }, { "index": 2357, "ocr_en": "In the rectangular ABCD, AD is the long side, and the point E is on the AD, connecting BE and CE, satisfying that BE is perpendicular to CE, and the vertical foot is the point E. The triangle ABE area is blue, the triangle BCE area is pink, and the triangle CDE area is yellow.", "ocr_zh": "矩形ABCD中，AD为长边，点E在AD上，连接BE、CE，满足BE垂直于CE，垂足为点E。三角形ABE面积为蓝色，三角形BCE面积为粉色，三角形CDE面积为黄色。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2357.jpg" }, { "index": 2358, "ocr_en": "In the isosceles acute triangle ABC, the length of AB and AC is equal, D is the point on BC, E is the point on AC, and AD and DE are connected, and the angle ABD, angle ADE, and angle DCE are equal. The triangular ABD area is blue, and the triangular CDE area is yellow.", "ocr_zh": "等腰锐角三角形ABC中，AB与AC长度相等，D为BC上一点，E为AC上一点，连接AD、DE，满足角ABD、角ADE、角DCE三者相等。其中三角形ABD面积为蓝色，三角形CDE面积为黄色。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2358.jpg" }, { "index": 2359, "ocr_en": "In the rectangular ABCD, AB is the long side, the point E is on AB, and F is on BC, connecting DE and EF, satisfying DE perpendicular to EF, and perpendicular to E. Among them, the line segments CD and CF are black, and the other line segments are solid red lines.", "ocr_zh": "矩形ABCD中，AB为长边，点E在AB上，F在BC上，连接DE、EF，满足DE垂直于EF，垂足为E。其中线段CD、CF为黑色，其他线段均为红色实线。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2359.jpg" }, { "index": 2360, "ocr_en": "In the square ABCD, the point E is on AB and F is on BC, connecting DE and EF, satisfying DE perpendicular to EF and perpendicular to E. Among them, the line segments CD and CF are black, and the other line segments are solid red lines.", "ocr_zh": "正方形ABCD中，点E在AB上，F在BC上，连接DE、EF，满足DE垂直于EF，垂足为E。其中线段CD、CF为黑色，其他线段均为红色实线。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2360.jpg" }, { "index": 2361, "ocr_en": "In the isosceles right-angled triangle ABC, AC and BC are equal in length, the angle ACB is a right angle, E is a point on AB close to A, and F is a point on BC. Except for the line segment CF, which is black, the rest of the line segments are red.", "ocr_zh": "等腰直角三角形ABC中，AC与BC长度相等，角ACB为直角，E为AB上靠近A的一点，F为BC上一点，连接CE、EF，满足角A、角B与角CEF相等。其中除了线段CF为黑色，其余线段均为红色。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2361.jpg" }, { "index": 2362, "ocr_en": "In the acute isosceles triangle ABC, AB and AC are equal in length, point D is a point on BC close to C, and point E is a point on AB near B, connecting AD and DE.", "ocr_zh": "锐角等腰三角形ABC中，AB和AC长度相等，点D为BC上靠近C的一点，点E为AB上靠近B的一点，连接AD、DE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2362.jpg" }, { "index": 2363, "ocr_en": "In the rectangular ABCD, AD is the long side, the point E is on AB, the connection DE, the F is on BC, the connection EF and DF, and the triangle DEF and the triangle DEA congruence. Among them, AE and AD are dashed lines.", "ocr_zh": "矩形ABCD中，AD为长边，点E在AB上，连接DE，F在BC上，连接EF、DF，三角形DEF与三角形DEA全等。其中AE、AD为虚线。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2363.jpg" }, { "index": 2364, "ocr_en": "In the rectangular ABCD, AB is the short side, there is a little M on AD, and there is a little N on BC, (closer to CD than M), connect MN, and use MN as the axis of symmetry to make a symmetrical figure of ABNM, the symmetry point B' of B falls on CD, and the symmetry point A' of A falls outside the rectangle, where AM, AB, BN are dashed lines, and the rest are solid lines.", "ocr_zh": "矩形ABCD中，AB为短边，AD上有一点M，BC上有一点N，（比M更靠近CD），连接MN，以MN为对称轴做体型ABNM的对称图形，B的对称点B'落在CD上，A的对称点A'落在矩形外，其中AM、AB、BN为虚线，其余均为实线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2364.jpg" }, { "index": 2365, "ocr_en": "A, P, and B are the points on the straight line l, and the right-angled triangle CDP falls on the same side of l with P as the right-angled vertice, and the dotted line connects AC and BD, satisfying that AC is perpendicular to l, the perpendicular foot is the point A, the BD is perpendicular to l, and the perpendicular foot is the point B.", "ocr_zh": "A、P、B依次为直线l上的点，以P为直角顶点做直角三角形CDP落在l同侧，虚线连接AC、BD，满足AC垂直于l，垂足为点A，BD垂直于l，垂足为点B。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2365.jpg" }, { "index": 2366, "ocr_en": "There are right-angled triangle APC and right-angled triangle BPD, A, B, P in a straight line l, CA perpendicular to AP, perpendicular to A, and CA shorter than AP. DB is perpendicular to PB, vertical foot is B, and DB is longer than PB. Connected DC, the two triangles are located on both sides of l, and the angle CPD is at right angles. where AC and BD are dashed lines.", "ocr_zh": "存在直角三角形APC与直角三角形BPD，A、B、P依次在一条直线l上，CA垂直于AP，垂足为A，且CA短于AP。DB垂直于PB，垂足为B，且DB长于PB。连接DC，两个三角形位于l两侧，且角CPD为直角。其中AC和BD为虚线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2366.jpg" }, { "index": 2367, "ocr_en": "The point P is a point on the red straight line l, with P as the right-angled vertex to make a right-angled triangle CPD, points C and D on both sides of the line, and the angle CPD is right-angled.", "ocr_zh": "点P是红色直线l上一点，以P为直角顶点做直角三角形CPD，点C和点D在直线两侧，角CPD为直角。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2367.jpg" }, { "index": 2368, "ocr_en": "The point P is a point on the red straight line l, with P as the right-angled vertex to make a right-angled triangle CPD, points C and D on both sides of the line, and the angle CPD is right-angled.", "ocr_zh": "点P是红色直线l上一点，以P为直角顶点在l一侧做直角三角形CPD，角CPD为直角。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2368.jpg" }, { "index": 2369, "ocr_en": "A, P, and B are the points on the straight line l, and the right-angled triangle CDP falls on the same side of l, and the red dotted line connects AC and BD, satisfying that AC is perpendicular to l, the perpendicular foot is the point A, the BD is perpendicular to l, and the perpendicular foot is the point B.", "ocr_zh": "A、P、B依次为直线l上的点，以P为直角顶点做直角三角形CDP落在l同侧，红色虚线连接AC、BD，满足AC垂直于l，垂足为点A，BD垂直于l，垂足为点B。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2369.jpg" }, { "index": 2370, "ocr_en": "A, P, and B are the points on the straight line l, and the right-angled triangle CDP falls on the same side of l with P as the right-angled vertice, and the dotted line connects AC and BD, satisfying that AC is perpendicular to l, the perpendicular foot is the point A, the BD is perpendicular to l, and the perpendicular foot is the point B. In addition, the area of the triangle ACP is blue, and the area of the triangle BPD is yellow.", "ocr_zh": "A、P、B依次为直线l上的点，以P为直角顶点做直角三角形CDP落在l同侧，虚线连接AC、BD，满足AC垂直于l，垂足为点A，BD垂直于l，垂足为点B。另外三角形ACP的面积为蓝色，三角形BPD的面积为黄色。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2370.jpg" }, { "index": 2371, "ocr_en": "Left: In the Cartesian coordinate system XOY, the right-angled triangle OAB is made with O as the right-angled vertice, OB is in the first quadrant, OA is in the fourth quadrant, and OA is longer than OB. Right: In the Cartesian coordinate system XOY, the right-angled triangle OAB is made with O as the right-angled vertex, OB is in the first quadrant, OA is in the fourth quadrant, and OA is longer than OB. Do the dashed line BC perpendicular to the y-axis at the point C, and do the dashed line AD perpendicular to the y-axis at the point D.", "ocr_zh": "左图：直角坐标系XOY中，以O为直角顶点做直角三角形OAB，OB在第一象限，OA在第四象限并且OA长于OB。 右图：直角坐标系XOY中，以O为直角顶点做直角三角形OAB，OB在第一象限，OA在第四象限并且OA长于OB。做虚线BC垂直于y轴于点C，做虚线AD垂直y轴于点D。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_2371.jpg" }, { "index": 2372, "ocr_en": "Left: In the square ABCD, E is the midpoint of AD, connected to BE. Right: In the square ABCD, E is the midpoint of AD, connected to BE. The point F is on the DC, the dotted line connects EF, and the EF is perpendicular to BE, and the vertical foot is E.", "ocr_zh": "左图：正方形ABCD中，E为AD中点，连接BE。 右图：正方形ABCD中，E为AD中点，连接BE。点F在DC上，虚线连接EF，满足EF垂直于BE，垂足为E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2372.jpg" }, { "index": 2373, "ocr_en": "In the right triangle ABC, the angle C is the right angle, BC is slightly longer than AC, the point D is the point on BC, and the point E is the point on AB, connecting AD and DE, satisfying AD perpendicular to DE, and perpendicular to the foot D.", "ocr_zh": "直角三角形ABC中，角C为直角，BC略长于AC，点D为BC上一点，点E为AB上一点，连接AD、DE，满足AD垂直于DE，垂足为D。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2373.jpg" }, { "index": 2374, "ocr_en": "In the rectangular ABCD, BD is the diagonal, F is a point outside BC, connecting BF and DF, and DF and BC intersect at the point E.", "ocr_zh": "矩形ABCD中，BD为对角线，F为 BC外一点，连接BF、DF，DF与BC交于点E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2374.jpg" }, { "index": 2375, "ocr_en": "In the right triangle ABC, the angle C is the right angle, BC is slightly longer than AC, the point D is the point on BC, and the point E is the point on AB, connecting AD and DE, satisfying AD perpendicular to DE, and perpendicular to the foot D. Do the dashed line EM perpendicular to BC, and the vertical foot is M. Among them, except for BE and BM, the rest of the solid lines (dashed lines) are red.", "ocr_zh": "直角三角形ABC中，角C为直角，BC略长于AC，点D为BC上一点，点E为AB上一点，连接AD、DE，满足AD垂直于DE，垂足为D。做虚线段EM垂直于BC，垂足为M。其中，除BE和BM，其余实线（虚线）均为红色。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2375.jpg" }, { "index": 2376, "ocr_en": "In the Cartesian coordinate system xOy, the straight line l passes through two points, (0,3) and (3,0), (0,3) is the point B, (3,0) is the point A, AB is an edge in the first quadrant to make a rectangular ABCD, and the points C and D are concave arcs.", "ocr_zh": "直角坐标系xOy中，直线l过（0，3）和（3，0）两个点，（0，3）为点B，（3，0）为点A，以AB为一条边在第一象限做矩形ABCD，做过点C和点D向下凹的圆弧。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_2376.jpg" }, { "index": 2377, "ocr_en": "In a semicircle with O as the center of the circle and BC as the diameter, there is a point E on the arc close to C, and the tangent line of the arc is made through the point E, BA and CD are perpendicular to the tangent line, and the vertical feet are point A and point D respectively. Connect to the CE.", "ocr_zh": "以O为圆心，BC为直径的半圆中，靠近C的圆弧上有一点E，过点E做圆弧的切线，分别做BA、CD垂直于切线，垂足分别为点A和点D。连接CE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2377.jpg" }, { "index": 2378, "ocr_en": "In the right triangle ABC, the angle C is a right angle, BC is longer than AC, E is a point on AB, D is a point on BC, connecting AD and DE, satisfying AD perpendicular to DE, and perpendicular to D is D.", "ocr_zh": "直角三角形ABC中，角C为直角，BC长于AC，E为AB上一点，D为BC上一点，连接AD、DE，满足AD垂直于DE，垂足为D。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2378.jpg" }, { "index": 2379, "ocr_en": "In the parallelogram ABCD, angle A is an obtuse angle, and E is a point on BC, connecting AE and DE, with AE perpendicular to DE at point E.", "ocr_zh": "平行四边形ABCD中，角A为钝角，E为BC上一点，连接AE、DE，满足AE垂直于DE于点E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2379.jpg" }, { "index": 2380, "ocr_en": "D, C, and E are three points in a straight line l, making BD perpendicular to l to point D, doing AE perpendicular to l to point E, AE longer than BD, connecting AB, AC, BC, satisfying AC perpendicular to BC, and perpendicular to C.", "ocr_zh": "D、C、E三点依次在直线l上，做BD垂直于l于点D，做AE垂直于l于点E，AE长于BD，连接AB、AC、BC，满足AC垂直于BC，垂足为C。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2380.jpg" }, { "index": 2381, "ocr_en": "In the isosceles acute triangle ABE,AE is equal to AB， DE is perpendicular to BE to point E, and BC is perpendicular to AB to point B (outside the triangle), so that D, A, and C are in a straight line, connecting DC and CE.", "ocr_zh": "等腰锐角三角形ABE中，AE等于AB，做DE垂直于BE于点E，做BC垂直于AB于点B（三角形外部），使得D、A、C三点在一条直线上，连接DC、CE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2381.jpg" }, { "index": 2382, "ocr_en": "M, B, and N are in the same straight line in turn, doing AM perpendicular to the straight line at the point M, doing CN perpendicular to the straight line at the point N, AM longer than CN, connecting AB, AC, BC, satisfying AB perpendicular to BC, and perpendicular to B.", "ocr_zh": "M、B、N三点依次在同一直线上，做AM垂直于直线于点M，做CN垂直于直线于点N，AM长于CN，连接AB、AC、BC，满足AB垂直于BC，垂足为B。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2382.jpg" }, { "index": 2383, "ocr_en": "In the right triangle ABC, angle B is a right angle, AB is shorter than BC, and P is a point on BC close to B, which connects AP.", "ocr_zh": "直角三角形ABC中，角B为直角，AB短于BC，P为BC上靠近B的一点，连接AP。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2383.jpg" }, { "index": 2384, "ocr_en": "In the acute triangle ABC, AB is longer than AC, D is the inner point of the triangle, connecting AD, E is the outer point of AC, connecting AE and DE, satisfying the angle DAE and the angle BAC, and the triangle DAE is similar to the triangle BAC. Among them, AB and AD are red line segments, and AC and AE are yellow line segments.", "ocr_zh": "锐角三角形ABC中，AB长于AC，D为三角形内部一点，连接AD，E为AC外部一点，连接AE、DE，满足角DAE和角BAC相等，且三角形DAE相似于三角形BAC。其中，AB、AD为红色线段，AC、AE为黄色线段。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_2384.jpg" }, { "index": 2385, "ocr_en": "In the acute triangle ABC, AB is longer than AC, D is the inner point of the triangle, connecting AD, E is the outer point of AC, connecting AE and DE, satisfying the angle DAE and the angle BAC, and the triangle DAE is similar to the triangle BAC. The red dotted line connects BD and CE.", "ocr_zh": "锐角三角形ABC中，AB长于AC，D为三角形内部一点，连接AD，E为AC外部一点，连接AE、DE，满足角DAE和角BAC相等，且三角形DAE相似于三角形BAC。红色虚线连接BD、CE。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_2385.jpg" }, { "index": 2386, "ocr_en": "In the acute triangle ABC, AB is longer than AC, D is a point outside AC, AD is connected, E is an outside point of AC, AE and DE are connected, and the angle DAE and angle BAC are equal, and the triangle DAE is similar to the triangle BAC, and the point D is closer to the triangle. After connecting the CE, connect the BD and extend the intersection with the CE at point P. Angular BAC, angular DAE, angular BPC are equal.", "ocr_zh": "锐角三角形ABC中，AB长于AC，D为AC外部一点，连接AD，E为AC外部一点，连接AE、DE，满足角DAE和角BAC相等，且三角形DAE相似于三角形BAC，点D距离三角形更近。连接CE之后，连接BD并延长与CE交于点P。角BAC、角DAE、角BPC相等。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_2386.jpg" }, { "index": 2387, "ocr_en": "In the obtuse triangle ABC, the angle A is an obtuse angle, AC is longer than AB, D is the inner point of the triangle, connecting AD, E is the outer point of AC, connecting AE and DE, satisfying the angle DAE and the angle BAC are equal, and the triangle DAE is similar to the triangle BAC. After connecting the CE, connect the BD and extend the intersection with the CE at point P. Angular BAC, angular DAE, angular BPC are equal.", "ocr_zh": "钝角三角形ABC中，角A为钝角，AC长于AB，D为三角形内部一点，连接AD，E为AC外部一点，连接AE、DE，满足角DAE和角BAC相等，且三角形DAE相似于三角形BAC。连接CE之后，连接BD并延长与CE交于点P。角BAC、角DAE、角BPC相等。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_2387.jpg" }, { "index": 2388, "ocr_en": "In the right-angled triangle ADE, the angular AED is a right angle, AE is equal to DE, DA is extended to point B, BC is perpendicular to BD, vertical foot is B, connecting CD and BE at point P, connecting AP and AC, AE and CD at point M, AC and BE at point F.", "ocr_zh": "直角三角形ADE中，角AED为直角，AE等于DE，延长DA至点B，做BC垂直于BD，垂足为B，连接CD与BE交于点P，连接AP、AC，AE与CD交于点M，AC与BE交于点F。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2388.jpg" }, { "index": 2389, "ocr_en": "Make an acute triangle ABC around the point A counterclockwise rotation of the triangle AB'C', (the rotation angle is greater than the angle CAB), connect BB' and CC', the three sides of the triangle ABC are dotted lines.", "ocr_zh": "做锐角三角形ABC绕点A逆时针旋转后的三角形AB'C',(旋转角度大于角CAB)，连接BB'和CC'，三角形ABC三条边均为虚线。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_2389.jpg" }, { "index": 2390, "ocr_en": "The right triangle ABC, the angle C is the right angle, AC is longer than BC, rotate this triangle counterclockwise around A and enlarge it in equal proportions (the rotation angle is greater than the angle BAC) to obtain the triangle AB'C', connect CC', translate CC' to B'E, and connect BE.", "ocr_zh": "直角三角形ABC，角C为直角，AC长于BC，将此三角形绕A逆时针旋转并等比例放大（旋转角度大于角BAC），得到三角形AB'C'，连接CC'，将CC'平移至B'E，连接BE。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_2390.jpg" }, { "index": 2391, "ocr_en": "The right triangle ABC, the angle C is the right angle, AC is longer than BC, the triangle is rotated clockwise around B (the rotation angle is greater than the angle ABC), the triangle EBD is obtained, the AE is connected, the CD is connected and the extension intersects with AE at the point F, with AB at the point G, and finally connects BF with DE at the point H.", "ocr_zh": "直角三角形ABC，角C为直角，AC长于BC，将此三角形绕B顺时针旋转（旋转角度大于角ABC），得到三角形EBD，连接AE，连接CD并延长与AE交于点F，与AB交于点G，最后连接BF与DE交于点H。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_2391.jpg" }, { "index": 2392, "ocr_en": "In the rectangular ABCD, BD is the diagonal, extending the CB to the point E, doing FE perpendicular to BE, and perpendicular foot to E, satisfying the triangle BEF is similar to the triangle BAD. Connect AE and DF.", "ocr_zh": "矩形ABCD中，BD为对角线，延长CB至点E，做FE垂直于BE，垂足为E，满足三角形BEF相似于三角形BAD。连接AE、DF。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_2392.jpg" }, { "index": 2393, "ocr_en": "In the rectangular ABCD, AB is the long side, BD is the diagonal, the triangle ABD is rotated clockwise around B and reduced in equal proportions, (the rotation angle is greater than 90 degrees), the triangle EBF is obtained, and AE and DF are connected, and the two intersect at the point H, and AE and BD intersect at the point O.", "ocr_zh": "矩形ABCD中，AB为长边，BD为对角线，将三角形ABD绕B顺时针旋转并等比例缩小，（旋转角度大于90度），得到三角形EBF，连接AE、DF，两者交于点H，AE与BD交于点O。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_2393.jpg" }, { "index": 2394, "ocr_en": "Rectangular BACD, AB is the long side, BD is the diagonal.", "ocr_zh": "矩形BACD，AB为长边 ，BD为对角线。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2394.jpg" }, { "index": 2395, "ocr_en": "Rectangular BACD, AB is the long side, BD is the diagonal. E is the midpoint of AB, EF is perpendicular to AB, and the vertical foot is E, which intersects with BD at the point F.", "ocr_zh": "矩形BACD，AB为长边 ，BD为对角线。E为AB中点，做EF垂直于AB，垂足为E，与BD交于点F。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2395.jpg" }, { "index": 2396, "ocr_en": "In the equilateral triangle ADE, extend DB to point B and DE to point C, connect AB and AC, and satisfy that the angle ABD is equal to the angle CAE and the angle BAD is equal to the angle ACE. Among them, the triangle ABD area is blue, the triangle ADE area is light khaki, and the triangle ACE area is pink.", "ocr_zh": "等边三角形ADE中，延长DB到点B，延长DE到点C，连接AB、AC，满足角ABD等于角CAE，角BAD等于角ACE。其中三角形ABD面积为蓝色，三角形ADE面积为浅卡其色，三角形ACE面积为粉色。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2396.jpg" }, { "index": 2397, "ocr_en": "In triangle ABC, angle A is a right angle, and AB and AC are equal in length. AB and AC are red lines.", "ocr_zh": "三角形ABC中，角A为直角，AB与AC长度相等。AB、AC均为红色线段。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2397.jpg" }, { "index": 2398, "ocr_en": "In the isosceles right-angled triangle ABC, the angle BAC is a right angle, and D and E are the upper points of BC in turn, connecting AD and AE, satisfying the angle DAE and angle B and angle C are all 45 degrees.", "ocr_zh": "等腰直角三角形ABC中，角BAC为直角，D、E依次为BC上一点，连接AD、AE，满足角DAE与角B和角C都是45度。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2398.jpg" }, { "index": 2399, "ocr_en": "In the isosceles right-angled triangle ABC, the angle BAC is a right angle, and D and E are the upper points of BC in turn, connecting AD and AE, and satisfying the angle DAE is 45 degrees.", "ocr_zh": "等腰直角三角形ABC中，角BAC为直角，D、E依次为BC上一点，连接AD、AE，满足角DAE是45度。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2399.jpg" }, { "index": 2400, "ocr_en": "In the isosceles right triangle ABC, the angle BAC is a right angle, and D and E are the upper points of BC, connecting AD and AE, satisfying the angle DAE, angle B and angle C are 45 degrees. , extend AE to point F, extend AD to point G, and connect FG so that the triangle AFG is an isosceles right triangle and the angle F is a right angle.", "ocr_zh": "等腰直角三角形ABC中，角BAC为直角，D、E依次为BC上一点，连接AD、AE，满足角DAE、角B和角C是45度。，延长AE至点F，延长AD至点G，连接FG，使得三角形AFG为等腰直角三角形，角F为直角。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2400.jpg" }, { "index": 2401, "ocr_en": "In the isosceles right triangle ABC, the angle BAC is a right angle, and D and E are the upper points of BC, connecting AD and AE, satisfying the angle DAE, angle B and angle C are 45 degrees. , extend AE to point F, extend AD to point G, and connect FG so that the triangle AFG is an isosceles right triangle and the angle F is a right angle", "ocr_zh": "等腰直角三角形ABC中，角BAC为直角，D、E依次为BC上一点，连接AD、AE，满足角DAE、角B和角C是45度。，延长AE至点F，延长AD至点G，连接FG，使得三角形AFG为等腰直角三角形，角F为直角", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2401.jpg" }, { "index": 2402, "ocr_en": "In the square ABCD, BD is the diagonal, E is a point near B on BC, and F is a point near C on CD, connecting AE, AF, EF, AE and BD at the point M, AF and BD at the point N.", "ocr_zh": "正方形ABCD中，BD为对角线，E为BC上靠近B一点，F为CD上靠近C一点，连接AE、AF、EF，AE与BD相交于点M，AF与BD相交于点N。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2402.jpg" }, { "index": 2403, "ocr_en": "In the obtuse triangle PAB, the angle APB is an obtuse angle, and C and D are the two points on the line segment AB in turn, connecting PC and PD, and satisfying the triangle PCD is an equilateral triangle.", "ocr_zh": "钝角三角形PAB中，角APB为钝角，C、D依次为线段AB上两点，，连接PC、PD，满足三角形PCD为等边三角形。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2403.jpg" }, { "index": 2404, "ocr_en": "In the right triangle ABC, E is the point above BC, connects AE and extends to point G so that AG is equal to AB, and in addition FG is perpendicular to AG at point G, and FG length is equal to AB, connecting AF and extending the extension line with CB to intersect point D.", "ocr_zh": "直角三角形ABC中，E为BC上一点，连接AE并延长至点G，使得AG等于AB，另外做FG垂直于AG于点G，并且FG长度等于AB，连接AF并延长与CB的延长线交于点D。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2404.jpg" }, { "index": 2405, "ocr_en": "In the isosceles right triangle ABC, the angle BAC is a right angle, and D and E are the upper points of BC, connecting AD and AE, satisfying the angle DAE, angle B and angle C are 45 degrees. , extend AE to point G, extend AD to point F, and connect FG so that the triangle AFG is an isosceles right triangle and the angle G is a right angle", "ocr_zh": "等腰直角三角形ABC中，角BAC为直角，D、E依次为BC上一点，连接AD、AE，满足角DAE、角B和角C是45度。，延长AE至点G，延长AD至点F，连接FG，使得三角形AFG为等腰直角三角形，角G为直角", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2405.jpg" }, { "index": 2406, "ocr_en": "In the right triangle ABC, the angle C is the right angle, BC is longer than AC, O is the midpoint of the hypotenuse AB, the dashed line OD is perpendicular to AC, the perpendicular foot is D, the dashed line OH is perpendicular to BC, and the perpendicular foot is H. E is the last point on AD, F is one point on CH, and it is connected to OE and OF.", "ocr_zh": "直角三角形ABC中，角C为直角，BC长于AC，O为斜边AB的中点，做虚线OD垂直于AC，垂足为D，做虚线OH垂直于BC，垂足为H。E为AD上一点，F为CH上一点，连接OE、OF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2406.jpg" }, { "index": 2407, "ocr_en": "In the right triangle ABC, the angle C is the right angle, BC is longer than AC, O is the midpoint of the hypotenuse AB, the dashed line OM is perpendicular to AB, the vertical foot is O, and the intersection with BC is the point M. E is the upper point of AC and F is the upper point of BC, connecting OE and OF so so that OE is perpendicular to OF and vertical foot is O. The area of the triangular AEO is blue.", "ocr_zh": "直角三角形ABC中，角C为直角，BC长于AC，O为斜边AB的中点，做虚线OM垂直于AB，垂足为O，与BC交于点M。E为AC上一点，F为BC上一点，连接OE、OF，使得OE垂直于OF，垂足为O。三角形AEO的面积为蓝色。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2407.jpg" }, { "index": 2408, "ocr_en": "In the right triangle ABC, the angle C is the right angle, BC is longer than AC, O is the midpoint of the hypotenuse AB, the red dotted line OM is perpendicular to AB, the vertical foot is O, and the intersection with BC is the point M. E is the upper point of AC and F is the upper point of BC, connecting OE and OF so so that OE is perpendicular to OF and vertical foot is O.", "ocr_zh": "直角三角形ABC中，角C为直角，BC长于AC，O为斜边AB的中点，做红色虚线OM垂直于AB，垂足为O，与BC交于点M。E为AC上一点，F为BC上一点，连接OE、OF，使得OE垂直于OF，垂足为O。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2408.jpg" }, { "index": 2409, "ocr_en": "In the right triangle ABC, the angle C is the right angle, BC is longer than AC, O is the midpoint of the hypotenuse AB, the dashed line OD is perpendicular to AC, the perpendicular foot is D, the dashed line OH is perpendicular to BC, and the perpendicular foot is H. E is the last point of AD, F is the last point of CH, connecting OE and OF, and satisfying OE perpendicular to OF and point O. The triangular ODE area is blue.", "ocr_zh": "直角三角形ABC中，角C为直角，BC长于AC，O为斜边AB的中点，做虚线OD垂直于AC，垂足为D，做虚线OH垂直于BC，垂足为H。E为AD上一点，F为CH上一点，连接OE、OF，满足OE垂直于OF与点O。三角形ODE面积为蓝色。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2409.jpg" }, { "index": 2410, "ocr_en": "In the right triangle ABC, the angle C is the right angle, BC is longer than AC, O is the midpoint of the hypotenuse AB, E is the point on AC, F is the point on BC, and OE and OF are connected to meet the OE perpendicular to OF and the point O.", "ocr_zh": "直角三角形ABC中，角C为直角，BC长于AC，O为斜边AB的中点，E为AC上一点，F为BC上一点，连接OE、OF，满足OE垂直于OF与点O。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2410.jpg" }, { "index": 2411, "ocr_en": "In the right triangle ABC, the angle C is the right angle, BC is longer than AC, O is the midpoint of the hypotenuse AB, the dashed line OM is perpendicular to AB, the vertical foot is O, and the intersection with BC is the point M. E is the upper point of AC and F is the upper point of BC, connecting OE and OF so so that OE is perpendicular to OF and vertical foot is O.", "ocr_zh": "直角三角形ABC中，角C为直角，BC长于AC，O为斜边AB的中点，做虚线OM垂直于AB，垂足为O，与BC交于点M。E为AC上一点，F为BC上一点，连接OE、OF，使得OE垂直于OF，垂足为O。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2411.jpg" }, { "index": 2412, "ocr_en": "In the right triangle ABC, the angle C is the right angle, BC is longer than AC, O is a point on the hypotenuse AB, E is a point on AC, F is a point on BC, OE and OF are connected, and OE is perpendicular to OF and point O.", "ocr_zh": "直角三角形ABC中，角C为直角，BC长于AC，O为斜边AB上一点，E为AC上一点，F为BC上一点，连接OE、OF，满足OE垂直于OF与点O。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2412.jpg" }, { "index": 2413, "ocr_en": "In the right triangle ABC, the angle C is the right angle, BC is longer than AC, O is a point above the hypotenuse AB, the dashed line OD is perpendicular to AC, the perpendicular foot is D, the dashed line OH is perpendicular to BC, and the perpendicular foot is H. E is the last point of AD, F is the last point of CH, connecting OE and OF, and satisfying OE perpendicular to OF and point O. The triangular ODE area is blue.", "ocr_zh": "直角三角形ABC中，角C为直角，BC长于AC，O为斜边AB上一点，做虚线OD垂直于AC，垂足为D，做虚线OH垂直于BC，垂足为H。E为AD上一点，F为CH上一点，连接OE、OF，满足OE垂直于OF与点O。三角形ODE面积为蓝色。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2413.jpg" }, { "index": 2414, "ocr_en": "In the right triangle ABC, the angle C is the right angle, BC is longer than AC, O is a point on the hypotenuse AB, the dotted line OM is perpendicular to AB, the vertical foot is O, and the intersection with BC is the point M. E is the upper point of AC and F is the upper point of BC, connecting OE and OF so so that OE is perpendicular to OF and vertical foot is O. The area of the triangular AEO is blue.", "ocr_zh": "直角三角形ABC中，角C为直角，BC长于AC，O为斜边AB上一点，做虚线OM垂直于AB，垂足为O，与BC交于点M。E为AC上一点，F为BC上一点，连接OE、OF，使得OE垂直于OF，垂足为O。三角形AEO的面积为蓝色。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2414.jpg" }, { "index": 2415, "ocr_en": "In the right triangle ABC, the angle C is the right angle, BC is longer than AC, O is a point above the hypotenuse AB, the red dashed line OD is perpendicular to AC, the vertical foot is D, the red dashed line OH is perpendicular to BC, and the vertical foot is H. E is the last point on AD, F is one point on CH, and it is connected to OE and OF.", "ocr_zh": "直角三角形ABC中，角C为直角，BC长于AC，O为斜边AB上一点，做红色虚线OD垂直于AC，垂足为D，做红色虚线OH垂直于BC，垂足为H。E为AD上一点，F为CH上一点，连接OE、OF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2415.jpg" }, { "index": 2416, "ocr_en": "In the right triangle ABC, the angle C is the right angle, BC is longer than AC, O is a point on the hypotenuse AB, the red dotted line OM is perpendicular to AB, the vertical foot is O, and the intersection with BC is the point M. E is the upper point of AC and F is the upper point of BC, connecting OE and OF so so that OE is perpendicular to OF and vertical foot is O.", "ocr_zh": "直角三角形ABC中，角C为直角，BC长于AC，O为斜边AB上一点，做红色虚线OM垂直于AB，垂足为O，与BC交于点M。E为AC上一点，F为BC上一点，连接OE、OF，使得OE垂直于OF，垂足为O。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2416.jpg" }, { "index": 2417, "ocr_en": "In the rectangular ABCD, AD is the long side, AC and BD are diagonals and intersect at the point P, E is a point close to B on AB, and F is a point close to C on BC. Among them, the line segments AB and BC and ray PE and PF are all solid red lines.", "ocr_zh": "矩形ABCD中，AD为长边，AC、BD为对角线且交于点P，E为AB上靠近B一点，F为BC上靠近C的一点，做射线PE、PF，满足PE垂直于PF，垂足为P。其中，线段AB、BC和射线PE、PF均为红色实线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2417.jpg" }, { "index": 2418, "ocr_en": "In the right triangle ABC, the angle C is the right angle, BC is the long side, O is the hypotenuse AB a point, the dotted line OD is perpendicular to AC, the perpendicular foot is D, the dashed line OH is perpendicular to BC, and the perpendicular foot is H. E is the last point on AD, F is one point on CH, and it is connected to OE and OF. Angle B is α.", "ocr_zh": "直角三角形ABC中，角C为直角，BC为长边，O为斜边AB上一点，做虚线OD垂直于AC，垂足为D，做虚线OH垂直于BC，垂足为H。E为AD上一点，F为CH上一点，连接OE、OF。角B为α。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2418.jpg" }, { "index": 2419, "ocr_en": "In the rectangular ABCD, AD is the long side, AC and BD are diagonals and intersect at the point P, E is the midpoint of AB, and F is the midpoint of BC. Among them, the line segments AB and BC and ray PE and PF are all solid red lines.", "ocr_zh": "矩形ABCD中，AD为长边，AC、BD为对角线且交于点P，E为AB上中点，F为BC中点，做射线PE、PF，满足PE垂直于PF，垂足为P。其中，线段AB、BC和射线PE、PF均为红色实线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2419.jpg" }, { "index": 2420, "ocr_en": "In the right triangle ABC, the angle C is the right angle, BC is longer than AC, O is a point on the hypotenuse AB, the dotted line OM is perpendicular to AB, the vertical foot is O, and the intersection with BC is the point M. E is the upper point of AC and F is the upper point of BC, connecting OE and OF so so that OE is perpendicular to OF and vertical foot is O. Angle A is α.", "ocr_zh": "直角三角形ABC中，角C为直角，BC长于AC，O为斜边AB上一点，做虚线OM垂直于AB，垂足为O，与BC交于点M。E为AC上一点，F为BC上一点，连接OE、OF，使得OE垂直于OF，垂足为O。角A 为α。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2420.jpg" }, { "index": 2421, "ocr_en": "In the rectangular ABCD, AD is the long side, AC and BD are diagonals and intersect at the point P, E is a point close to B on AB, and F is a point close to C on BC. The red dotted line PM is perpendicular to AB and the vertical foot is M, and the red dotted line PN is perpendicular to BC and the vertical foot is N. The area of the triangular PNF is blue.", "ocr_zh": "矩形ABCD中，AD为长边，AC、BD为对角线且交于点P，E为AB上靠近B一点，F为BC上靠近C的一点，做射线PE、PF，满足PE垂直于PF，垂足为P。做红色虚线PM垂直于AB，垂足为M，做红色虚线PN垂直于BC，垂足为N。三角形PNF面积为蓝色。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_2421.jpg" }, { "index": 2422, "ocr_en": "In the rectangular ABCD, AD is the long side, AC and BD are diagonals, E is a point on the Ab extension line, F is a point on the BC extension line, and P is a point close to A on the AC. Among them, the ray PE, PF and the line segment BC and CD are all red.", "ocr_zh": "矩形ABCD中，AD为长边，AC、BD为对角线，E为Ab延长线上一点，F为BC延长线上一点，P为AC上靠近A的一点，连接PE、PF并延长，满足PE垂直于PF，垂足为P。其中射线PE、PF和线段BC、CD均为红色。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2422.jpg" }, { "index": 2423, "ocr_en": "In the right triangle ABC, the angle C is the right angle, BC is slightly longer than AC, O is a point above AB, P is a point on AC, Q is a point on BC, OP is connected and extended to point M, OQ is connected and extended to point N.", "ocr_zh": "直角三角形ABC中，角C为直角，BC略长于AC，O为AB上一点，P为AC上一点，Q为BC上一点，连接OP并延长至点M，连接OQ并延长至点N。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2423.jpg" }, { "index": 2424, "ocr_en": "In the right triangle ABC, the angle BAC is a right angle, AB is longer than AC, the line segment AD is perpendicular to the point BC, the vertical foot is D, E is a point on AB, F is a point on AC, connect DE and extend to one point, connect DF and extend to another point, meet DE perpendicular to DF, vertical foot is D, and connect the end of the extension line to form a triangle.", "ocr_zh": "直角三角形ABC中，角BAC为直角，AB长于AC，做线段AD垂直于点BC，垂足为D，E为AB上一点，F为AC上一点，连接DE并延长至一点，连接DF并延长至另一点，满足DE垂直于DF，垂足为D，连接延长线端点构成三角形。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2424.jpg" }, { "index": 2425, "ocr_en": "In the isosceles right triangle ABC, the angle BAC is a right angle, AB is equal to AC, D is the midpoint of BC, E is a point on AB, F is a point on AC, connect DE and extend to one point, connect DF and extend to another point, meet DE perpendicular to DF, perpendicular to D, connect the extension line ends to form a triangle, and finally connect EF.", "ocr_zh": "等腰直角三角形ABC中，角BAC为直角，AB等于AC，D为BC的中点，E为AB上一点，F为AC上一点，连接DE并延长至一点，连接DF并延长至另一点，满足DE垂直于DF，垂足为D，连接延长线端点构成三角形，最后连接EF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2425.jpg" }, { "index": 2426, "ocr_en": "In the isosceles right triangle ABC, the angle BAC is a right angle, AB is equal to AC, D is the midpoint of BC, E is the point on AB, F is a point on AC, connect DE and extend to one point, connect DF and extend to another point, meet DE perpendicular to DF, perpendicular to D, connect the extension line ends to form a triangle.", "ocr_zh": "等腰直角三角形ABC中，角BAC为直角，AB等于AC，D为BC的中点，E为AB上一点，F为AC上一点，连接DE并延长至一点，连接DF并延长至另一点，满足DE垂直于DF，垂足为D，连接延长线端点构成三角形。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2426.jpg" }, { "index": 2427, "ocr_en": "In the right triangle ABC, the angle ACB is a right angle, CB is slightly longer than AC, and CD is perpendicular to AB at the point D. Extend AC to point E and CB to point F, connect DE and DF, so that DE is perpendicular to DF and vertical foot is D.", "ocr_zh": "直角三角形ABC中，角ACB为直角，CB略长于AC，做CD垂直于AB于点D。延长AC至点E，延长CB至点F，连接DE、DF，使得DE垂直于DF，垂足为D。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2427.jpg" }, { "index": 2428, "ocr_en": "In the isosceles right triangle ABC, the angle ACB is a right angle, CB is equal to AC, and CD is perpendicular to AB to the point D. Point E is on AC and point F is on BC, connecting DE and DF so that DE is perpendicular to DF and vertical to D.", "ocr_zh": "等腰直角三角形ABC中，角ACB为直角，CB等于AC，做CD垂直于AB于点D。点E在AC上，点F在BC上，连接DE、DF，使得DE垂直于DF，垂足为D。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2428.jpg" }, { "index": 2429, "ocr_en": "In the right triangle ABC, the angle ACB is a right angle, the CB is slightly shorter than the AC, and the CD is perpendicular to AB to the point D. Point E is on AC and point F is on BC, connecting DE and DF so that DE is perpendicular to DF and vertical to D.", "ocr_zh": "直角三角形ABC中，角ACB为直角，CB略短于AC，做CD垂直于AB于点D。点E在AC上，点F在BC上，连接DE、DF，使得DE垂直于DF，垂足为D。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2429.jpg" }, { "index": 2430, "ocr_en": "In the square ABCD, AC and BD are diagonal lines and intersect at the point O, the point P is on the OB, the point M is on the AB, and the point N is on the AD, connecting PM and PN so that the PM is perpendicular to the PN and the vertical foot is P.", "ocr_zh": "正方形ABCD中，AC、BD为对角线且相交于点O，点P在OB上，点M在AB上，点N在AD上，连接PM、PN，使得PM垂直于PN，垂足为P。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2430.jpg" }, { "index": 2431, "ocr_en": "In the square ABCD, AC and BD are diagonal lines and intersect at the point O, the point P is on OC, the point M is on AB, and the point N is on AD, connecting PM and PN so that PM is perpendicular to PN and perpendicular to P.", "ocr_zh": "正方形ABCD中，AC、BD为对角线且相交于点O，点P在OC上，点M在AB上，点N在AD上，连接PM、PN，使得PM垂直于PN，垂足为P。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2431.jpg" }, { "index": 2432, "ocr_en": "In the acute triangle ABC, AD is the angle bisector of the angle BAC, which intersects with BC at the point D, the line segments AB and BD are red, and the line segments AC and CD are yellow.", "ocr_zh": "锐角三角形ABC中，AD为角BAC的角平分线，与BC相交于点D，线段AB、BD为红色，线段AC、CD为黄色。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2432.jpg" }, { "index": 2433, "ocr_en": "In the acute triangle ABC, AD is the angle bisector of the angle BAC, which intersects with BC at the point D, and the line segment AD is red.", "ocr_zh": "锐角三角形ABC中，AD为角BAC的角平分线，与BC相交于点D，线段AD为红色。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2433.jpg" }, { "index": 2434, "ocr_en": "In the acute triangle ABC, AD is the angle bisector of the angle BAC, which intersects with BC at the point D, and the parallel line of the AD through the point C intersects the extension line of the BA at the point E. Among them, AE is a black dotted line, and CE is a red dashed line.", "ocr_zh": "锐角三角形ABC中，AD为角BAC的角平分线，与BC相交于点D，过点C做AD的平行线与BA的延长线交于点E。其中，AE为黑色虚线，CE为红色虚线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2434.jpg" }, { "index": 2435, "ocr_en": "In the acute triangle ABC, AD is the angle bisector of the angle BAC, which intersects with BC at the point D, and the parallel line of the AD through the point C intersects the extension line of the BA at the point E. Among them, AE and CE are black dotted lines.", "ocr_zh": "锐角三角形ABC中，AD为角BAC的角平分线，与BC相交于点D，过点C做AD的平行线与BA的延长线交于点E。其中，AE和CE为黑色虚线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2435.jpg" }, { "index": 2436, "ocr_en": "In the acute triangle ABC, AD is the angle bisector of the angle BAC, which intersects with BC at the point D, the point D is perpendicular to AB at the point E, and the DF is perpendicular to AC at the point F, and the high AH on the edge of BC is the intersection of the point D. Among them, AH, DE, DF are all black dotted lines.", "ocr_zh": "锐角三角形ABC中，AD为角BAC的角平分线，与BC相交于点D，过点D做DE垂直于AB于点E，过点D做DF垂直于AC于点F，做BC边上的高AH。其中，AH、DE、DF均为黑色虚线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2436.jpg" }, { "index": 2437, "ocr_en": "Angle B is an acute angle, E and A are two points on one side, C and D are two points on another edge, the yellow dotted line connects EC, the black line connects AC, and the red line connects AD. Satisfying EC is parallel to AD.", "ocr_zh": "角B为锐角，E、A依次为一条边上的两点，C、D依次为另一条边上两点，黄色虚线连接EC，黑色线段连接AC，红色线段连接AD。满足EC平行于AD.", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2437.jpg" }, { "index": 2438, "ocr_en": "In the obtuse triangle ABC, the angle ACB is an obtuse angle, extending BA and BC, the bisector of the outer angle of the angle BAC and the BC extension line intersect at the point D, the perpendicular line DE is made from the point D to the BA extension line, the perpendicular foot is E, the perpendicular line is DF from D to AC, the perpendicular foot is F, and finally the perpendicular line AH is made from the point A to CD, and the perpendicular foot is H. Among them, AD is a red solid line, DE and DF are yellow dotted lines, and AH is a blue dashed line.", "ocr_zh": "钝角三角形ABC中，角ACB为钝角，延长BA、BC，做角BAC外角的平分线与BC延长线交于点D，过点D向BA延长线做垂线DE，垂足为E，过D向AC作垂线DF，垂足为F，最后过A点向CD做垂线AH，垂足为H。其中，AD为红色实线，DE、DF为黄色虚线，AH为蓝色虚线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2438.jpg" }, { "index": 2439, "ocr_en": "In triangle ABC, angle A is an obtuse angle, and the bisector of angle ACB intersects AB at point D.", "ocr_zh": "三角形ABC中，角A为钝角，角ACB的平分线与AB交于点D。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2439.jpg" }, { "index": 2440, "ocr_en": "In the right triangle ABC, the angle B is a right angle, AB is slightly shorter than BC, and the angle bisector of the angle BAC intersects BC at the point D.", "ocr_zh": "直角三角形ABC中，角B为直角，AB略短于BC，角BAC的角平分线与BC交于点D。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2440.jpg" }, { "index": 2441, "ocr_en": "In triangle ABC, angle A is an obtuse angle, and the bisector of angle ACB intersects AB at point D. The angle DCB is angle 1, and the angle DCA is angle 2. The DQ is perpendicular to BC and the vertical foot is Q, and the DN is perpendicular to CA, and the extension line of CA intersects at point N. In addition, the CH is perpendicular to BA through point C, and the extension line of BA intersects point H. Among them, CD is a red solid line, DQ and DN are red dotted lines, and AH and CH are black dashed lines.", "ocr_zh": "三角形ABC中，角A为钝角，角ACB的平分线与AB交于点D。角DCB为角1，角DCA为角2。过点D做DQ垂直于BC，垂足为Q，过点D做DN垂直于CA，与CA的延长线交于点N。另外过点C做CH垂直于BA，与BA的延长线交于点H。其中，CD为红色实线，DQ、DN为红色虚线，AH、CH为黑色虚线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2441.jpg" }, { "index": 2442, "ocr_en": "In the obtuse triangle ABC, the angle ACB is an obtuse angle, which extends BA and BC, and the bisector AD and BC extension line of the angle BAC complement angle intersect at the point D.", "ocr_zh": "钝角三角形ABC中，角ACB为钝角，延长BA、BC，做角BAC补角的平分线AD与BC延长线交于点D。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2442.jpg" }, { "index": 2443, "ocr_en": "In the right triangle ABC, the angle BAC is a right angle, and the bisector AD and BC of the angle BAC intersect at the point D.", "ocr_zh": "直角三角形ABC中，角BAC为直角，做角BAC的平分线AD与BC交于点D。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2443.jpg" }, { "index": 2444, "ocr_en": "In the acute triangle ABC, the bisector AD and BC of the angle BAC intersect at the point D.", "ocr_zh": "锐角三角形ABC中，做角BAC的平分线AD与BC交于点D。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2444.jpg" }, { "index": 2445, "ocr_en": "In the obtuse triangle ABC, the angle ACB is an obtuse angle, which extends BA to point F and extends BC, the bisector AD and BC of the angle BAC intersect at the point D, and the angle bisector of the angle CAF intersects the BC extension line at the point E.", "ocr_zh": "钝角三角形ABC中，角ACB为钝角，延长BA至点F，延长BC，做角BAC的平分线AD与BC交于点D，做角CAF的角平分线与BC延长线交于点E。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2445.jpg" }, { "index": 2446, "ocr_en": "In the obtuse triangle ABC, the angle BAC is the obtuse angle, the angle bisector of the angle BAC intersects with BC at the point D, E is the midpoint of BC, and F is the point on AC, connecting EF to satisfy EF parallel to AD.", "ocr_zh": "钝角三角形ABC中，角BAC为钝角，做角BAC的角平分线与BC相交于点D，E为BC的中点，F为AC上一点，连接EF，满足EF平行于AD。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2446.jpg" }, { "index": 2447, "ocr_en": "In the right triangle ABC, the angle B is a right angle, and the bisector of the angle BAC intersects with BC at the point D, satisfying that BD is shorter than CD.", "ocr_zh": "直角三角形ABC中，角B为直角，做角BAC的平分线与BC交于点D，满足BD短于CD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2447.jpg" }, { "index": 2448, "ocr_en": "In the 5 times 4 grid, each grid is a square, with the vertex in the lower left corner as the reference origin (0, 0), point A coordinates (2, 1), point B coordinates (4, 2), point C coordinates (2, 2), point D coordinates (1, 3), point E coordinates (1, 2), and point F coordinates (2, 3), respectively connecting AB and BD, and the dotted line connecting AD, the angle ABC is α, and the angle DBE is β", "ocr_zh": "5乘4的网格内，每个格子为正方形，以左下角的顶点为参考原点（0，0），点A坐标（2，1），点B坐标（4，2），点C坐标（2，2），点D坐标（1，3），点E坐标（1，2），点F坐标（2，3），分别连接AB、BD，虚线连接AD，角ABC为α，角DBE为β", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2448.jpg" }, { "index": 2449, "ocr_en": "In the 5 times 4 grid, each grid is a square, with the vertex in the lower left corner as the reference origin (0,0), the point A coordinates (2,1), the point B coordinates (4,2), the point C coordinates (2,2), the point D coordinates (1,3), and the point E coordinates (1,2), which are connected to AB and BD respectively, and the dotted line is connected to AD, the angle ABC is α, and the angle DBE is β, and these two corners are marked with red arcs.", "ocr_zh": "5乘4的网格内，每个格子为正方形，以左下角的顶点为参考原点（0，0），点A坐标（2，1），点B坐标（4，2），点C坐标（2，2），点D坐标（1，3），点E坐标（1，2），分别连接AB、BD，虚线连接AD，角ABC为α，角DBE为β，这两个角用红色弧线标出。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2449.jpg" }, { "index": 2450, "ocr_en": "In the 5 times 4 grid, each grid is a square, with the vertex in the lower left corner as the reference origin (0,0), the point A coordinate (2,1), the point B coordinate (4,2), the point C coordinate (2,2), the red solid line connects AC and BC, and the corner ABC is the α, which is marked with a red arc.", "ocr_zh": "5乘4的网格内，每个格子为正方形，以左下角的顶点为参考原点（0，0），点A坐标（2，1），点B坐标（4，2），点C坐标（2，2），红色实线连接AC、BC，角ABC为α，用红色弧线标出。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2450.jpg" }, { "index": 2451, "ocr_en": "In the 5 times 4 grid, each grid is a square, with the vertex in the lower left corner as the reference origin (0, 0), the point A coordinate (2, 1), the point B coordinate (4, 2), the point C coordinate (2, 2), the point D coordinate (1, 3), and the point E coordinate (1, 2), which are respectively connected to AB and BD, where BE and DE are red solid lines, angle ABC is α, and angle DBE is β marked with a red arc", "ocr_zh": "5乘4的网格内，每个格子为正方形，以左下角的顶点为参考原点（0，0），点A坐标（2，1），点B坐标（4，2），点C坐标（2，2），点D坐标（1，3），点E坐标（1，2），分别连接AB、BD，其中BE、DE为红色实线，角ABC为α，角DBE为β用红色弧线标出", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2451.jpg" }, { "index": 2452, "ocr_en": "In the 5 times 4 grid, each grid is a square, with the vertex in the lower left corner as the reference origin (0, 0), point A coordinates (2, 1), point B coordinates (11), point C coordinates (4, 1), point D coordinates (1, 3), point E coordinates (4, 2), point F coordinates (4, 3), and point M coordinates (2, 3), respectively, connecting AB, AE, DE, in which each side of the rectangular BCFD is a red solid line, the angle ADB is α, the angle EDF is β, and the angle ADE is 45 degrees. β marked red.", "ocr_zh": "5乘4的网格内，每个格子为正方形，以左下角的顶点为参考原点（0，0），点A坐标（2，1），点B坐标（1，1），点C坐标（4，1），点D坐标（1，3），点E坐标（4，2），点F坐标（4，3），点M坐标（2，3），分别连接AB、AE、DE，其中矩形BCFD各边为红色实线，角ADB为α，角EDF为β，角ADE为45度。β标红。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_2452.jpg" }, { "index": 2453, "ocr_en": "In the 5 times 4 grid, each grid is a square, with the vertex in the lower left corner as the reference origin (0, 0), point A coordinates (2, 1), point B coordinates (11), point C coordinates (4, 1), point D coordinates (1, 3), point E coordinates (4, 2), point F coordinates (4, 3), and point H coordinates (1, 2), respectively, connecting AB, AE, DE, in which each side of the rectangular BCFD is a red solid line, the angle ADB is α, the angle EDF is β, and the angle ADE is 45 degrees. α marked red.", "ocr_zh": "5乘4的网格内，每个格子为正方形，以左下角的顶点为参考原点（0，0），点A坐标（2，1），点B坐标（1，1），点C坐标（4，1），点D坐标（1，3），点E坐标（4，2），点F坐标（4，3），点H坐标（1，2），分别连接AB、AE、DE，其中矩形BCFD各边为红色实线，角ADB为α，角EDF为β，角ADE为45度。α标红。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2453.jpg" }, { "index": 2454, "ocr_en": "In the Cartesian coordinate system xOy, the origin is O, point B is on the negative half axis of the y axis, point A is on the positive half axis of the x axis, point C is on the positive half axis of the x axis and OC is longer than OA, a straight line passes through A and B, and another straight line passes through B and C, the angle OBA is angle 1, the angle ABC is 45 degrees, and the angle OCB is angle 2", "ocr_zh": "直角坐标系xOy中，原点为O，点B在y轴的负半轴，点A在x轴的正半轴，点C在x轴正半轴且OC长于OA，一条直线经过A、B，另一条直线经过B、C，角OBA为角1，角ABC为45度，角OCB为角2", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_2454.jpg" }, { "index": 2455, "ocr_en": "In the 4 times 3 grid, each grid is a square, with the vertex in the lower left corner as the reference origin (0,0), point A coordinate (0,3), point B coordinate (2,3), point C coordinate (4,1), point D coordinate (1,0), and point E coordinate (3,0), connecting CA, CB, CD, CE respectively.", "ocr_zh": "4乘3的网格内，每个格子为正方形，以左下角的顶点为参考原点（0，0），点A坐标（0，3），点B坐标（2，3），点C坐标（4，1），点D坐标（1，0），点E坐标（3，0），分别连接CA、CB、CD、CE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2455.jpg" }, { "index": 2456, "ocr_en": "In the square ABCD, the point G is the midpoint of BC, which connects AG, and makes a triangle ABG about the axisymmetric graphic triangle AFG of the line segment AG, F is the symmetry point of B, and the extension of GF and CD intersects at the point E. Among them, AB and BG are dashed lines.", "ocr_zh": "正方形ABCD中，点G为BC中点，连接AG，做三角形ABG关于线段AG的轴对称图形三角形AFG，F为B的对称点，延长GF与CD交于点E。其中，AB、BG为虚线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2456.jpg" }, { "index": 2457, "ocr_en": "In the Cartesian coordinate system xOy, A and B are two points on the negative half axis of the x-axis, OA is longer than OB, C is one point on the positive half axis of the y-axis, and an opening upward parabola passes through three points A, B, and C, and the vertex is P. A straight line passes through the first, third, and fourth quadrants and intersects the parabola at the vertex P and the right point Q, and the point M is the point on the PA segment of the parabola, connecting AC, CM, AP, AQ.", "ocr_zh": "直角坐标系xOy中，A、B依次为x轴负半轴上两点，OA长于OB，C为y轴正半轴上一点，一开口向上抛物线过A、B、C三点，且顶点为P。一条直线经过一、三、四象限并与抛物线交于顶点P和右侧一点Q，点M为抛物线PA段上一点，连接AC、CM、AP、AQ。", "class": "Analytic Geometry", "difficulty": 3, "image_path": "fig_2457.jpg" }, { "index": 2458, "ocr_en": "In the Cartesian coordinate system xOy, A is a point on the negative half axis of the x-axis, B is a point on the negative half axis of the y-axis, the length of OB is greater than OA, and a straight line passes through two points of AB.", "ocr_zh": "直角坐标系xOy中，A为x轴负半轴上一点，B为y轴负半轴上一点，OB长度大于OA，一条直线经过AB两点。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_2458.jpg" }, { "index": 2459, "ocr_en": "In the square ABCD, point G is near D on AD, point E is near A on AB, and point F is near B on BC, connecting DE and FG, both of which intersect at point H.", "ocr_zh": "正方形ABCD中，点G在AD上靠近D，点E在AB上靠近A，点F在BC上靠近B，连接DE和FG，两者交于点H。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2459.jpg" }, { "index": 2460, "ocr_en": "In triangle ABC, P is a point close to B on BC and connects AP, and in addition, the triangle AMN is similar to triangle ABC and rotates counterclockwise, AQ is also reduced in equal proportions, and the position of Q on MN is the same as the position of P on BC, connecting AQ. AB, AC, AM, AND AN are black dashed lines, MN are red dashed lines, and the area of triangle ABC that does not coincide with triangle AMN is blue.", "ocr_zh": "三角形ABC中，P为BC上靠近B一点，连接AP，另外做三角形AMN相似于三角形ABC并逆时针旋转，AQ也等比例缩小，Q在MN上的位置与P在BC上的位置一致，连接AQ。其中，AB、AC、AM、AN为黑色虚线，MN为红色虚线，三角形ABC中与三角形AMN不重合的面积为蓝色。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2460.jpg" }, { "index": 2461, "ocr_en": "There is a point P near B on the line BC, A is the outer point of the PC part of the line segment, which is connected to PA, Q is the outer point of the PC part of the line segment on the same side as point A, connected to AQ, and the angle PAQ is an acute angle and is denoted as the angle α.", "ocr_zh": "线段BC上靠近B的地方有一点P，A为线段PC部分的外部一点，连接PA，Q为线段PC部分的外部一点与点A在同一侧，连接AQ，角PAQ为锐角记作角α。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2461.jpg" }, { "index": 2462, "ocr_en": "There is an angular QAP, the size is α, the length of the AP is slightly longer than AQ, rotate the angular QAP clockwise around the point A (the rotation angle is less than α), and extend the two sides according to the same proportion to obtain the angle MAO, connect MQ and OP, take M as the center of the circle, MQ as the radius to make the circle M, take O as the center of the circle, and OP as the radius to make the circle O. Among them, AM, MQ, AO, OP are black dotted lines, circle M is red dashed line, and each point is represented by a red solid circle.", "ocr_zh": "存在角QAP，大小为α，AP长度略长于AQ，将角QAP绕点A顺时针旋转（旋转角度小于α），并将两边按照相同比例延长得到角MAO，连接MQ和OP，以M为圆心，MQ为半径做圆M，以O为圆心，OP为半径做圆O。其中，AM、MQ、AO、OP为黑色虚线，圆M为红色虚线，每个点用红色实心圆表示。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_2462.jpg" }, { "index": 2463, "ocr_en": "In triangle ABC, P is a point close to B on BC and connects to AP, and in addition, the triangle AMN is similar to triangle ABC (reduced in equal proportions) and rotated counterclockwise, AQ is also reduced in equal proportions, and the position of Q on MN is the same as the position of P on BC, connecting AQ. Among them, AB, AC, AM, AN, MN are black dotted lines.", "ocr_zh": "三角形ABC中，P为BC上靠近B一点，连接AP，另外做三角形AMN相似于三角形ABC（等比例缩小）并逆时针旋转，AQ也等比例缩小，Q在MN上的位置与P在BC上的位置一致，连接AQ。其中，AB、AC、AM、AN、MN为黑色虚线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2463.jpg" }, { "index": 2464, "ocr_en": "There is an angular QAP, the size is α, the length of the AP is slightly longer than AQ, rotate the angular QAP clockwise around the point A (the rotation angle is less than α), and extend the two sides according to the same proportion to obtain the angle MAO, connect MQ and OP, take M as the center of the circle, MQ as the radius to make the circle M, take O as the center of the circle, and OP as the radius to make the circle O. AM, MQ, AO, OP and circle M are black dotted lines.", "ocr_zh": "存在角QAP，大小为α，AP长度略长于AQ，将角QAP绕点A顺时针旋转（旋转角度小于α），并将两边按照相同比例延长得到角MAO，连接MQ和OP，以M为圆心，MQ为半径做圆M，以O为圆心，OP为半径做圆O。其中，AM、MQ、AO、OP以及圆M为黑色虚线。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_2464.jpg" }, { "index": 2465, "ocr_en": "In the isosceles right triangle ABC, AC is equal to BC, angle C is the right angle, O is the midpoint of hypotenuse AB, P is on AC, Q is on BC, OP, OQ and PQ are connected, OP is perpendicular to OQ and perpendicular to OQ is O, and point M is the midpoint of PQ, extending OQ.", "ocr_zh": "等腰直角三角形ABC中，AC等于BC，角C为直角，O为斜边AB的中点，P在AC上，Q在BC上，连接OP、OQ和PQ，满足OP垂直于OQ垂足为O，点M为PQ中点，延长OQ。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2465.jpg" }, { "index": 2466, "ocr_en": "In the isosceles right triangle ABC, AC is equal to BC, angle C is the right angle, O is the midpoint of hypotenuse AB, P is on AC, Q is on BC, OP, OQ and PQ are connected, OP is perpendicular to OQ and perpendicular to OQ is O, and point M is the midpoint of PQ, extending OQ. OM and ON are perpendicular to PQ and AC, and the vertical feet are respectively point M and point N, connecting MN. Connect to OC. Among them, ON, MN, OC are black dotted lines, OM is red dashed lines, and OP and ray OQ are red solid lines.", "ocr_zh": "等腰直角三角形ABC中，AC等于BC，角C为直角，O为斜边AB的中点，P在AC上，Q在BC上，连接OP、OQ和PQ，满足OP垂直于OQ垂足为O，点M为PQ中点，延长OQ。分别做OM、ON垂直于PQ、AC，垂足分别为点M和点N，连接MN。连接OC。其中，ON、MN、OC为黑色虚线，OM为红色虚线，OP、射线OQ为红色实线。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_2466.jpg" }, { "index": 2467, "ocr_en": "In the right triangle ABC, the angle ACB is a right angle, AC is slightly longer than BC, AB is the diameter, the midpoint of AB is the center of the circle and the outside of the triangle is a semicircle, P is a point on the arc, and the midpoint connecting CP and CP is M", "ocr_zh": "直角三角形ABC中，角ACB为直角，AC略长于BC，以AB为直径，AB中点为圆心在三角形外部做半圆，P为圆弧上一点，连接CP，CP的中点为M", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2467.jpg" }, { "index": 2468, "ocr_en": "In the square ABCD, E is a point close to B on BC, F is a point near A on AB, connect EF, and make an equilateral triangle EFG (inside the square) with EF as the side, and connect CG.", "ocr_zh": "正方形ABCD中，E为BC上靠近B一点，F为AB上靠近A一点，连接EF，并以EF为边做等边三角形EFG（在正方形内部），连接CG。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2468.jpg" }, { "index": 2469, "ocr_en": "In the rectangular ABCD, AD is the long side, P is a point on BC close to B, connected to AP, and Q is a point inside the rectangle, connected to AQ and DQ.", "ocr_zh": "长方形ABCD中，AD为长边，P为BC上靠近B的一点，连接AP，Q为长方形内部一点，连接AQ、DQ。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2469.jpg" }, { "index": 2470, "ocr_en": "In the acute triangle ABC, the angle bisector of angle ABC and angle ACB intersects with AC and AB at points E and F respectively, the intersection of the two angle bisectors is D, G is the midpoint of CD, M is the point on BC, the dotted line connects MG, and MG is rotated 90 degrees counterclockwise around G to obtain a solid line segment NG, which connects DN.", "ocr_zh": "锐角三角形ABC中，角ABC和角ACB的角平分线分别与AC、AB交于点E和点F，两条角平分线的交点为D，G为CD的中点，M为BC上一点，虚线连接MG，将MG绕G逆时针旋转90度得到实线段NG，连接DN。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_2470.jpg" }, { "index": 2471, "ocr_en": "In the right triangle ABC, the angle BAC is a right angle, the angle bisector of angle ABC and angle ACB intersects with AC and AB at the point E and the point F respectively, the intersection of the two angle bisectors is D, and M is the point on the AF, connecting DM.", "ocr_zh": "直角三角形ABC中，角BAC为直角，角ABC和角ACB的角平分线分别与AC、AB交于点E和点F，两条角平分线的交点为D，M为AF上一点，连接DM。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2471.jpg" }, { "index": 2472, "ocr_en": "In the acute triangle ABC, the angular bisector of the angle ABC and the angle ACB intersect with AC and AB at the points E and F respectively, and the intersection of the two angular bisectors is D.", "ocr_zh": "锐角三角形ABC中，角ABC和角ACB的角平分线分别与AC、AB交于点E和点F，两条角平分线的交点为D。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2472.jpg" }, { "index": 2473, "ocr_en": "In the Cartesian coordinate system xOy, A and B are two points on the positive semi-axis of the x-axis, OA is shorter than OB, C is one point on the positive semi-axis of the y-axis, and the opening of the upward parabola passes through the three points of A, B and C. Make any circle with B as the center of the circle (radius less than AB), circle B and x-axis intersect E (left) and F (right) two points, make a point P on the circle, connect AP and AP as the waist to make an isosceles right-angle triangle APD, the angle PAD is 90 degrees, and connect DF.", "ocr_zh": "直角坐标系xOy中，A、B依次为x轴正半轴上两点，OA短于OB，C为y轴正半轴上一点，一开口向上抛物线过A、B、C三点。以B为圆心做任意圆（半径小于AB），圆B与x轴交于E（左侧）、F（右侧）两点，圆上做一点P，连接AP并以AP为腰做等腰直角三角形APD，角PAD为90度，连接DF。", "class": "Analytic Geometry", "difficulty": 3, "image_path": "fig_2473.jpg" }, { "index": 2474, "ocr_en": "In the Cartesian coordinate system xOy, A and B are two points on the positive half axis of the x-axis, OA is shorter than OB, C is one point on the positive half-axis of the y-axis, an opening upward parabola passes through three points A, B and C, and the straight line passes through two points A and C, and M is one point below the x-axis on the parabola, connecting AM and BC.", "ocr_zh": "直角坐标系xOy中，A、B依次为x轴正半轴上两点，OA短于OB，C为y轴正半轴上一点，一开口向上抛物线过A、B、C三点，一直线经过A、C两点，M为抛物线上x轴下方一点，连接AM和BC。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_2474.jpg" }, { "index": 2475, "ocr_en": "In the rectangular ABCD, AD is the long side, E is a point close to D on AD, F is a point close to B on BC, the symmetrical graph of trapezoidal CDEF is made with line segment EF as the axis of symmetry, the symmetry point of point C is A, and the symmetry point of point D is D', wherein, the line segment EF is red, AD, DC, CF are dotted lines, and the areas of trapezoidal CDEF and trapezoidal AD'EF are blue and yellow respectively.", "ocr_zh": "长方形ABCD中，AD为长边，E为AD上靠近D的一点，F为BC上靠近B的一点，以线段EF作为对称轴，做梯形CDEF的对称图形，C点的对称点为A，D点的对称点为D'，其中，线段EF为红色，AD、DC、CF为虚线，梯形CDEF与梯形AD'EF的面积分别为蓝色与黄色。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2475.jpg" }, { "index": 2476, "ocr_en": "In the rectangular ABCD, AD is the long side, E is a point close to D on AD, F is a point close to B on BC, the line segment EF is used as the axis of symmetry, the symmetrical figure of trapezoidal CDEF is made, the symmetry point of point C is A, the symmetry point of point D is D', the dotted line connects CE and AC, AC and EF intersect at the point O, AD, DC, CF are dotted lines, the area of the triangle ACF is skin color, and the area of the triangle ABF is blue.", "ocr_zh": "长方形ABCD中，AD为长边，E为AD上靠近D的一点，F为BC上靠近B的一点，以线段EF作为对称轴，做梯形CDEF的对称图形，C点的对称点为A，D点的对称点为D'，虚线连接CE、AC，AC与EF交于点O，AD、DC、CF为虚线，三角形ACF的面积为肤色，三角形ABF的面积为蓝色。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2476.jpg" }, { "index": 2477, "ocr_en": "In the rectangular ABCD, AD is the long side, BD is the diagonal, and the symmetrical figure of the triangle BCD is made with BD as the axis of symmetry, and the symmetry point of the triangle BED, E is the symmetry point of C, and the intersection point of BE and AD is F. Among them, BD is a red line segment, BC, CD, and DF are black dotted lines, and the areas of triangle BCD and triangle BED are blue and yellow, respectively.", "ocr_zh": "长方形ABCD中，AD为长边，BD为对角线，以BD为对称轴做三角形BCD的对称图形，得到三角形BED，E为C的对称点，BE与AD交点为F。其中BD为红色线段，BC、CD、DF为黑色虚线，三角形BCD和三角形BED的面积分别为蓝色和黄色。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2477.jpg" }, { "index": 2478, "ocr_en": "In the rectangular ABCD, AD is the long side, BD is the diagonal, and the symmetrical figure of the triangle BCD is made with BD as the axis of symmetry, and the symmetry point of the triangle BED, E is the symmetry point of C, and the intersection point of BE and AD is F. Among them, BC, CD, and DF are black dotted lines, and the areas of triangle BAF and triangle BDF are blue and skin color, respectively.", "ocr_zh": "长方形ABCD中，AD为长边，BD为对角线，以BD为对称轴做三角形BCD的对称图形，得到三角形BED，E为C的对称点，BE与AD交点为F。其中BC、CD、DF为黑色虚线，三角形BAF和三角形BDF的面积分别为蓝色和肤色。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2478.jpg" }, { "index": 2479, "ocr_en": "In the rectangular ABCD, AD is the long side, E is a point on AB, connect DE, and use DE as the axis of symmetry to make a triangle The symmetrical graph of ADE makes the symmetry point A' of A fall on BC, where AD and AE are dashed lines, and the area of triangle A'EB is blue.", "ocr_zh": "长方形ABCD中，AD为长边，E为AB上一点，连接DE，以DE为对称轴做三角形ADE的对称图形使得A的对称点A'落在BC上，其中，AD、AE为虚线，三角形A'EB面积为蓝色。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2479.jpg" }, { "index": 2480, "ocr_en": "In the rectangular ABCD, AD is the long side, BD is the diagonal, E is the point on AB, connect DE, and use DE as the axis of symmetry to make a triangle The symmetrical graph of ADE makes the symmetry point A' of A fall on the diagonal BD, where AD and AE are dashed lines, and the area of triangle A'EB is blue.", "ocr_zh": "长方形ABCD中，AD为长边，BD为对角线，E为AB上一点，连接DE，以DE为对称轴做三角形ADE的对称图形使得A的对称点A'落在对角线BD上，其中，AD、AE为虚线，三角形A'EB面积为蓝色。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2480.jpg" }, { "index": 2481, "ocr_en": "In the rectangular ABCD, AB is the long side, AC is the diagonal, and the symmetrical figure triangle AB'C and B' are the symmetrical triangles of ABC with AC as the axis of symmetry, and the symmetry point of B' falls outside the rectangle, and AB' and CD intersect at the point E. Among them, AB, BC, CE are dashed lines.", "ocr_zh": "长方形ABCD中，AB为长边，AC为对角线，以AC为对称轴做三角形ABC的对称图形三角形AB'C，B'为B的对称点落在长方形外，AB’与CD交于点E。其中AB、BC、CE为虚线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2481.jpg" }, { "index": 2482, "ocr_en": "In the rectangular ABC, AD is the long side, B' and E are the upper points of AD and BC, respectively, and connect B'E AND ABEB' to form a square, connecting AE and ED. F is a point on AB, and EF is used as the axis of symmetry to make a symmetrical figure of the triangle BEF, so that the symmetry point G of B falls on AE. Among them, BE, BF, B'E are all dashed lines.", "ocr_zh": "长方形ABC中，AD为长边，B'、E分别为AD、BC上一点，连接B'E，ABEB'构成正方形，连接AE、ED。F为AB上一点，以EF为对称轴做三角形BEF的对称图形，使得B的对称点G落在AE上。其中，BE、BF、B'E均为虚线。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_2482.jpg" }, { "index": 2483, "ocr_en": "In the square ABCD, BD is the diagonal, E is the point on the BD close to D, connecting AE and CE, where CE is the red dotted line.", "ocr_zh": "正方形ABCD中，BD为对角线，E为BD上靠近D的一点，连接AE、CE，其中CE为红色虚线。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2483.jpg" }, { "index": 2484, "ocr_en": "In the square ABCD, BD is the diagonal and E is the point on the BD close to D, connecting AE, where BD is the red line segment.", "ocr_zh": "正方形ABCD中，BD为对角线，E为BD上靠近D的一点，连接AE，其中BD为红色线段。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2484.jpg" }, { "index": 2485, "ocr_en": "In the square ABCD, BD is the diagonal, E is the point on the BD close to D, connecting AE and CE, where CE is the dashed line, the quadrilateral ADCE area is blue, and the quadrilateral ABCE area is light green.", "ocr_zh": "正方形ABCD中，BD为对角线，E为BD上靠近D的一点，连接AE、CE，其中CE为虚线，四边形ADCE面积为蓝色，四边形ABCE面积为浅绿色。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2485.jpg" }, { "index": 2486, "ocr_en": "In the square ABCD, E and F are used as two points on the DC, satisfying the equal length of ED and CF, connecting AE and BE, and making the perpendicular bisector (dashed line) of AB, where the area of the triangle ADE and the triangle BCF is blue.", "ocr_zh": "正方形ABCD中，E、F以此为DC上两点，满足ED和CF长度相等，连接AE、BE，做AB的垂直平分线（虚线），其中三角形ADE和三角形BCF的面积为蓝色。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2486.jpg" }, { "index": 2487, "ocr_en": "In the square ABCD, AC is the diagonal, P is the point on AC close to A, and E is a point on DC, connecting BP and EP, satisfying that BP is perpendicular to EP and perpendicular to P. Extend DC to F.", "ocr_zh": "正方形ABCD中，AC为对角线，P为AC上靠近A的一点，E为DC上一点，连接BP、EP，满足BP垂直于EP且垂足为P。延长DC至F。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2487.jpg" }, { "index": 2488, "ocr_en": "In the square ABCD, AC is the diagonal, P is the point on the AC close to A, and E is the point on the DC, connecting BP, EP, DP, and satisfying the BP perpendicular to the EP and the vertical foot is P. Extend DC to F. The DP is a dashed line, and the areas of the triangle ABP and the triangle ADP are blue and yellow, respectively.", "ocr_zh": "正方形ABCD中，AC为对角线，P为AC上靠近A的一点，E为DC上一点，连接BP、EP、DP，满足BP垂直于EP且垂足为P。延长DC至F。其中DP为虚线，三角形ABP和三角形ADP的面积分别为蓝色和黄色。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2488.jpg" }, { "index": 2489, "ocr_en": "In the square ABCD, BD is the diagonal, F is the point on the BD close to D, AF is connected and extended with CD at point E, FG is perpendicular to AE through point F, and the vertical foot is F, and FG is crossed by BC at point G, connecting AG.", "ocr_zh": "正方形ABCD中，BD为对角线，F为BD上靠近D的一点，连接AF并延长与CD交于点E，过点F做FG垂直于AE，垂足为F，FG交BC于点G，连接AG。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2489.jpg" }, { "index": 2490, "ocr_en": "In the square ABCD, BD is the diagonal and P is the point on the BD close to D, which connects the AP. Through P, PE is perpendicular to BC at point E, PF is perpendicular to CD at point F, and EF is connected.", "ocr_zh": "正方形ABCD中，BD为对角线，P为BD上靠近D的一点，连接AP。过P分别做PE垂直于BC于点E，做PF垂直于CD于点F，连接EF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2490.jpg" }, { "index": 2491, "ocr_en": "In the square ABCD, BD is the diagonal, E is the point on the BD close to D, AE is connected, CE is connected and the intersection with AD is extended at point F.", "ocr_zh": "正方形ABCD中，BD为对角线，E为BD上靠近D的一点，连接AE，连接CE并延长与AD交于点F。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2491.jpg" }, { "index": 2492, "ocr_en": "In the square ABCD, point E is a point near the midpoint of AD, point F is a point on BC near B, point H is a point near the midpoint of CD, and point G is a point near A on AB. Crossing the point F to make FM perpendicular to AD to point M, passing H to do HN perpendicular to AB to point N, FM and HN are both dashed lines, and the complementary parts of triangle HGN and triangle FEM are blue and light green, respectively.", "ocr_zh": "正方形ABCD中，点E为AD中点附近的一点，点F为BC上靠近B的一点，点H为CD中点附近的一点，点G为AB上靠近A的一点，分别连接EF、GH，满足EF垂直于GH且垂足为O。过点F做FM垂直于AD于点M，过H做HN垂直于AB于点N，FM和HN均为虚线，三角形HGN与三角形FEM补充和的部分分别为蓝色和浅绿色。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_2492.jpg" }, { "index": 2493, "ocr_en": "In the square ABCD, point E is a point near the midpoint of AD, point F is a point on BC near B, point H is a point near the midpoint of CD, and point G is a point near A on AB. Crossing the point F is perpendicular to FM to AD to point M, crossing H is perpendicular to AB to point N, FM and HN are red dotted lines.", "ocr_zh": "正方形ABCD中，点E为AD中点附近的一点，点F为BC上靠近B的一点，点H为CD中点附近的一点，点G为AB上靠近A的一点，分别连接EF、GH，满足EF垂直于GH且垂足为O。过点F做FM垂直于AD于点M，过H做HN垂直于AB于点N，FM和HN均为红色虚线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2493.jpg" }, { "index": 2494, "ocr_en": "In the square ABCD, point E is a point near the midpoint of AD, point F is a point near B on BC, point H is a point near the midpoint of CD, and point G is a point near A on AB.", "ocr_zh": "正方形ABCD中，点E为AD中点附近的一点，点F为BC上靠近B的一点，点H为CD中点附近的一点，点G为AB上靠近A的一点，分别用红色实线连接EF、GH，满足EF垂直于GH且垂足为O。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2494.jpg" }, { "index": 2495, "ocr_en": "In the square ABCD, H and I are two points on AD, J and G are two points on BC, E is a point on AB near B, F is one point near D on CD, connecting HG, JI and EF, and the three intersect at the same point and are equal in length, and EF is perpendicular to HG. The vertical foot is the intersection point. EF and HG are red and JI are yellow.", "ocr_zh": "正方形ABCD中，H、I依次为AD上两点，J、G依次为BC上两点，E为AB上靠近B的一点，F为CD上靠近D一点，连接HG、JI和EF，三者交于同一点并且三者长度相等，EF垂直于HG。垂足为交点。其中EF、HG为红色线段，JI为黄色线段。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2495.jpg" }, { "index": 2496, "ocr_en": "In the square ABCD, the point P is a point near B on BC, M is a point near B, and N is a point near D on CD, connecting AP and MN so that AP is perpendicular to MN and perpendicular to point E.", "ocr_zh": "正方形ABCD中，点P为BC上靠近B一点，M为AB上一点，N为CD上靠近D的一点，连接AP和MN，使得AP垂直于MN，垂足为点E。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2496.jpg" }, { "index": 2497, "ocr_en": "In the rectangular ABCD, AB is the long side, E is the point on AB near A, F is the point near C on CD, G is the point near D on AD, and H is the point near B on BC. EF and GH are red line segments.", "ocr_zh": "长方形ABCD中，AB为长边，E为AB上靠近A的一点，F为CD上靠近C的一点，G为AD上靠近D的一点,H为BC上靠近B的一点,连接EF、GH，满足EF垂直于GH，垂足为交点。EF、GH为红色线段。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2497.jpg" }, { "index": 2498, "ocr_en": "In the rectangular ABCD, AB is the long side, E is the point on AB near A, F is the point near C on CD, G is the point near D on AD, and H is the point near B on BC. The red dashed line EM is perpendicular to CD and the perpendicular foot is M, and the red dashed line GN is perpendicular to CB and the perpendicular foot is N. The areas of the triangle EMF and triangle GNH (except for the coincident parts) are light green and blue, respectively", "ocr_zh": "长方形ABCD中，AB为长边，E为AB上靠近A的一点，F为CD上靠近C的一点，G为AD上靠近D的一点,H为BC上靠近B的一点,连接EF、GH，满足EF垂直于GH，垂足为交点。过点E做红色虚线EM垂直于CD，垂足为M，过点G做红色虚线GN垂直于CB，垂足为N。三角形EMF和三角形GNH的面积（除了重合部分）分别为浅绿色和蓝色", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_2498.jpg" }, { "index": 2499, "ocr_en": "In the rectangular ABCD, AB is the short side, E is the point on AB near B, F is the point near D on CD, G is the point near A on AD, and H is the point near B on BC.", "ocr_zh": "长方形ABCD中，AB为短边，E为AB上靠近B的一点，F为CD上靠近D的一点，G为AD上靠近A的一点,H为BC上靠近B的一点,连接EF、GH，满足EF垂直于GH，垂足为交点M。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2499.jpg" }, { "index": 2500, "ocr_en": "In the rectangular ABCD, AB is the short side, E is the point on AB near B, F is the point near D on CD, G is the point near A on AD, and H is the point near B on BC. The red dashed line GK is perpendicular to CB and the perpendicular foot is K, the red dotted line FN is perpendicular to AB and the perpendicular foot is N. The area of the triangular ENF (except for the coincident parts) is blue.", "ocr_zh": "长方形ABCD中，AB为短边，E为AB上靠近B的一点，F为CD上靠近D的一点，G为AD上靠近A的一点,H为BC上靠近B的一点,连接EF、GH，满足EF垂直于GH，垂足为交点M。过点G做红色虚线GK垂直于CB，垂足为K，过点F做红色虚线FN垂直于AB，垂足为N。三角形ENF的面积（除了重合部分）为蓝色。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_2500.jpg" }, { "index": 2501, "ocr_en": "In the square ABCD, P is a point on BC close to B, connecting AP, M and G are the midpoints of AB and CD, respectively, connecting MG, N is a point on GD, connecting MN, MN is perpendicular to AP, and vertical foot is E. where MG is a red dotted line, the triangle ABP area (except for the part that coincides with the triangle MNG) is blue, and the triangle MNG area is yellow.", "ocr_zh": "正方形ABCD中，P为BC上靠近B的一点，连接AP，M、G分别为AB、CD的中点，连接MG，N为GD上一点，连接MN，MN垂直于AP，垂足为E。其中MG为红色虚线，三角形ABP面积（除了与三角形MNG重合部分）为蓝色，三角形MNG面积为黄色。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2501.jpg" }, { "index": 2502, "ocr_en": "In the square ABCD, E is the point on BC and F is a point on CD, connecting AE and BF, and the two intersect at the point O, and the AE is perpendicular to BF, and the vertical foot is O.", "ocr_zh": "正方形ABCD中，E为BC上一点，F为CD上一点，连接AE、BF，两者相交于点O，且满足AE垂直于BF，垂足为O。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2502.jpg" }, { "index": 2503, "ocr_en": "In the square ABCD, E is a point on CD and F is a point on AD, connecting AE and BF, and the two intersect at a point, and AE is perpendicular to BF, and the vertical foot is the intersection point.", "ocr_zh": "正方形ABCD中，E为CD上一点，F为AD上一点，连接AE、BF，两者相交于一点，且满足AE垂直于BF，垂足为交点。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2503.jpg" }, { "index": 2504, "ocr_en": "In the rectangular ABCD, AB is the long side, G is a point on CD, F is a point on AB, and the symmetrical figure of trapezoidal AFGD is made with FG as the axis of symmetry, the symmetry point of A is E falling on BC, the symmetry point of D is H falling outside the rectangle, EH and CD intersect at the point P, connecting AE, and AE and FG intersect at the point O.", "ocr_zh": "长方形ABCD中，AB为长边，G为CD上一点，F为AB上一点，以FG为对称轴做梯形AFGD的对称图形，A的对称点为E落在BC上，D的对称点为H落在长方形外，EH与CD交于点P，连接AE，AE与FG交于点O。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2504.jpg" }, { "index": 2505, "ocr_en": "In the square ABCD, BD is the diagonal, N is the point on the BD close to D, M is the point near B on the BC, connect AM, do NP perpendicular to AM at the point P, and extend the N part to both ends to intersect with AB at the point E and CD at the point F. Connect the MN.", "ocr_zh": "正方形ABCD中，BD为对角线，N为BD上靠近D的一点，M为BC上靠近B的一点，连接AM，过N做NP垂直于AM于点P，N篇向两端延长分别与AB交于点E，与CD交于点F。连接MN。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2505.jpg" }, { "index": 2506, "ocr_en": "In the square ABCD, M is a point on BC close to B, E is a point on AB, F is a point on CD, AM and EF are connected to AM and are perpendicular to EF, and the vertical foot is P.", "ocr_zh": "正方形ABCD中，M为BC上靠近B的一点，E为AB上一点，F为CD上一点，连接AM、EF满足AM垂直于EF，垂足为P。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2506.jpg" }, { "index": 2507, "ocr_en": "In the rectangular ABCD, AB is the long side, E is a point close to B on BC, and F is a point close to A on AB, connecting AE and DF, satisfying AE perpendicular to DF, and perpendicular to the intersection point.", "ocr_zh": "长方形ABCD中，AB为长边，E为BC上靠近B一点，F为AB上靠近A一点，连接AE、DF，满足AE垂直于DF，垂足为交点。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2507.jpg" }, { "index": 2508, "ocr_en": "In the convex polygon ABCD, AB is equal to AD, BC is equal to CD, angle A is an obtuse angle, and angle C is an acute angle. E, F, G, and H are the midpoints of AB, BC, CD, and DA, respectively, connecting EF, FG, GH, HE, to form a rectangular EFGH, and the dotted line connecting AC and BD.", "ocr_zh": "凸多边形ABCD中，AB等于AD，BC等于CD，角A为钝角，角C为锐角。E、F、G、H分别为AB、BC、CD、DA的中点，连接EF、FG、GH、HE，构成矩形EFGH，虚线连接AC、BD。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2508.jpg" }, { "index": 2509, "ocr_en": "In the convex polygon ABCD, AB is equal to AD, BC is equal to CD, angle A is an obtuse angle, and angle C is an acute angle.", "ocr_zh": "凸多边形ABCD中，AB等于AD，BC等于CD，角A为钝角，角C为锐角。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2509.jpg" }, { "index": 2510, "ocr_en": "In the convex polygon ABCD, AB is equal to AD, BC is equal to CD, angle A is an obtuse angle, and angle C is an acute angle. E, F, G, and H are the midpoints of AB, BC, CD, and DA, respectively, and connect EF, FG, GH, and HE to form a rectangular EFGH with an area of blue.", "ocr_zh": "凸多边形ABCD中，AB等于AD，BC等于CD，角A为钝角，角C为锐角。E、F、G、H分别为AB、BC、CD、DA的中点，连接EF、FG、GH、HE，构成矩形EFGH，面积为蓝色。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2510.jpg" }, { "index": 2511, "ocr_en": "In the convex polygon ABCD, AB is equal to AD, BC is equal to CD, angle A is an obtuse angle, and angle C is an acute angle. E, F, G, and H are the midpoints of AB, BC, CD, and DA, respectively, and are marked with a red solid circle.", "ocr_zh": "凸多边形ABCD中，AB等于AD，BC等于CD，角A为钝角，角C为锐角。E、F、G、H分别为AB、BC、CD、DA的中点，用红色实心圆标出。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2511.jpg" }, { "index": 2512, "ocr_en": "In the quadrilateral ABCD, AC and BD are connected, and AC is perpendicular to BD at the point O. Take the midpoint E of AB, the midpoint F of BC, the midpoint G of CD, and the midpoint H of AD; connect the dots EF, FG, GH, HE.", "ocr_zh": "在四边形ABCD中，连接AC，BD，且AC垂直于BD于点O；取AB中点E，取BC中点F，取CD中点G，取AD中点H；连接点EF，FG，GH，HE", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2512.jpg" }, { "index": 2513, "ocr_en": "In the quadrilateral ABCD, AC and BD are connected, and AC is equal to BD; AC, BD intersect at point O; take the midpoint F of AB, the midpoint G of BC, the midpoint H of CD, and the midpoint E of AD. Connect EF, FG, GH, HE.", "ocr_zh": "在四边形ABCD中，连接AC，BD，且AC等于BD；AC，BD相交于点O；取AB中点F，取BC中点G，取CD中点H，取AD中点E；连接EF，FG，GH，HE", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2513.jpg" }, { "index": 2514, "ocr_en": "In the quadrilateral ABCD, take the midpoint E of AB, the midpoint F of BC, the midpoint G of CD, and the midpoint H of AD. Connect EF, FG, GH, HE.", "ocr_zh": "在四边形ABCD中，取AB中点E，取BC中点F，取CD中点G，取AD中点H；连接EF，FG，GH，HE", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2514.jpg" }, { "index": 2515, "ocr_en": "In the quadrilateral ABCD, do the auxiliary lines AC, BD; take the midpoint E of AB, the midpoint F of BC, the midpoint G of CD, and the midpoint H of AD; Connect EF, FG, GH, HE.", "ocr_zh": "在四边形ABCD中，做辅助线AC，BD；取AB中点E，取BC中点F，取CD中点G，取AD中点H；连接EF，FG，GH，HE", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2515.jpg" }, { "index": 2516, "ocr_en": "In the rectangular ABCD, take the midpoint F of AB, the midpoint G of BC, the midpoint H of CD, and the midpoint E of AD. Connect EF, FG, GH, HE.", "ocr_zh": "在矩形ABCD中，取AB中点F，取BC中点G，取CD中点H，取AD中点E；连接EF，FG，GH，HE", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2516.jpg" }, { "index": 2517, "ocr_en": "In the quadrilateral ABCD, E, F, G, and H are the midpoints of AB, BC, CD, and DA, respectively. Connect EF, FG, GH, HE.", "ocr_zh": "在四边形ABCD中，E、F、G、H分别为AB、BC、CD、DA中点；连接EF，FG，GH，HE", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2517.jpg" }, { "index": 2518, "ocr_en": "In triangle ABC, O is a point within triangle ABC, connecting OA, OB, and OC.  D and E are the midpoints of sides AB and AC respectively;  F and G are the midpoints of OB and OC respectively;  Connect D, F, G and E in sequence ", "ocr_zh": "在三角形ABC中，O是三角形ABC内一点，连接 OA，OB，OC；D，E分别是边AB，AC的中点；F，G分别是OB，OC的中点；依次连接D、F、G，E", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2518.jpg" }, { "index": 2519, "ocr_en": "In quadrilateral ABCD, E, F, G and H are the midpoints of AB, BC, CD and DA respectively.  Connect E, F, G and H in sequence ", "ocr_zh": "在四边形ABCD中，E，F，G，H分别为AB，BC，CD，DA的中点；依次连接E，F，G，H", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2519.jpg" }, { "index": 2520, "ocr_en": "In quadrilateral ABCD, E, F, G and H are the midpoints of AB, BC, CD and DA respectively.  Connect E, F, G and H in sequence;  P is a point within quadrilateral ABCD, and it satisfies PA equal to PB, PC equal to PD, and Angle APB equal to Angle CPD ", "ocr_zh": "在四边形ABCD中，E、F、G、H分别是AB、BC、CD、DA的中点； 将E、F、G、H依次连接； P是四边形ABCD内的一个点，满足PA等于PB，PC等于PD，角APB等于角CPD ", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2520.jpg" }, { "index": 2521, "ocr_en": "AB and AC are two line segments with A as the vertex. D and E are the midpoints of AB and AC respectively, and DE is connected. Connect the auxiliary lines BC. Lines DE are parallel and equal to half of BC  ", "ocr_zh": "AB、AC为以A为顶点的两条线段，D、E分别为AB、AC的中点，并连接DE；连接红色辅助线BC，DE平行且等于二分之一BC", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2521.jpg" }, { "index": 2522, "ocr_en": "AB and AC are two line segments with A as the vertex. D and E are the midpoints of AB and AC respectively, and DE is connected.  Connect the auxiliary line BC ", "ocr_zh": "AB、AC为以A为公共点的两条线段，D、E分别为AB、AC的中点，并连接DE；连接辅助线BC", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2522.jpg" }, { "index": 2523, "ocr_en": "In triangle ABC, D and E are the midpoints of AB and AC respectively ", "ocr_zh": "在三角形ABC中，D、E分别为AB、AC的中点", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2523.jpg" }, { "index": 2524, "ocr_en": "In triangle ABC, side BC is red. D and E are the midpoints of AB and AC respectively, and they are connected to the red auxiliary line DE. DE is parallel and equal to half BC ", "ocr_zh": "在三角形ABC中，边BC为红色；D、E分别为AB、AC的中点，连接红色辅助线DE；DE平行且等于二分之一BC", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2524.jpg" }, { "index": 2525, "ocr_en": "In triangle ABC, D and E are the midpoints of AB and AC respectively. Connect the red auxiliary line DE", "ocr_zh": "在三角形ABC中，D、E分别为AB、AC的中点，连接红色辅助线DE", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2525.jpg" }, { "index": 2526, "ocr_en": "AB and AC are two line segments with A as the vertex, and D and E are the midpoints of AB and AC respectively, connecting DE ", "ocr_zh": "AB、AC为以A为顶点的两条线段，D、E分别为AB、AC的中点，连接DE", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2526.jpg" }, { "index": 2527, "ocr_en": "In triangle ABC, point E is on side AC and point F is on side AB. Connect CF and BE, and BE and CF are the Angle bisectors of the interior angles of the triangle.  AD is perpendicular to BE at point D, AG is perpendicular to FC at point G. Extending AD and AG respectively intersects BC at points M and N. ", "ocr_zh": "在三角形ABC中，点E在边AC上，点F在边AB上，连接CF、BE，且BE、CF为三角形内角的角平分线；AD垂直BE于点D，AG垂直FC于点G，延长AD、AG分别交BC于点M、N；", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2527.jpg" }, { "index": 2528, "ocr_en": "In triangle ABC, D is the midpoint of AB, and the red line segment CD is connected.  Draw the reverse extension line CE of AC through point C, where CE is the auxiliary line.  CE is equal to AC. Connect the red auxiliary line BE.  CD is parallel to BE, and CD is equal to half of BE ", "ocr_zh": "在三角形ABC中，D为AB中点，连接红色线段CD；过C点做AC的反向延长线CE，其中CE为辅助线；CE等于AC，连接红色辅助线BE；CD平行于BE，CD等于二分之一BE", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2528.jpg" }, { "index": 2529, "ocr_en": "In triangle ABC, D is the midpoint of AB, and connect CD.  Draw the reverse extension line CE of AC through point C, where CE is the auxiliary line.  CE is equal to AC. Connect the auxiliary line BE ", "ocr_zh": "在三角形ABC中，D为AB中点，连接CD；过C点做AC的反向延长线CE，其中CE为红色辅助线；CE等于AC，连接红色辅助线BE；", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2529.jpg" }, { "index": 2530, "ocr_en": "Triangle ABC is an acute triangle. Two equilateral triangles are drawn outward with sides AB and AC respectively.  D, E and F are the midpoints of MB, BC and CN respectively;  Connect DE and FE ", "ocr_zh": "三角形ABC为锐角三角形，分别以 AB，AC为边向外侧作两个等边三角形；D，E，F分别是 MB，BC，CN 的中点；连接 DE，FE", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2530.jpg" }, { "index": 2531, "ocr_en": "In triangle ABC, AD is the median line, AE is the Angle bisector, and both D and E are on the line segment BC.  CF is perpendicular to AE at point F, and point F is on AE.  Extend CF to intersect AB and point G. FG are auxiliary lines ", "ocr_zh": "在三角形ABC中，AD为中线，AE为角平分线，且D、E均在线段BC上；CF垂直AE于点F，F点在AE上；延长CF交AB与点G，FG为辅助线", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2531.jpg" }, { "index": 2532, "ocr_en": "Triangle ABC is an acute triangle. Two equilateral triangles are drawn outward with sides AB and AC respectively.  D, E and F are the midpoints of MB, BC and CN respectively;  Connect DE, FE;  Connect the auxiliary lines BN and CM ", "ocr_zh": "三角形ABC为锐角三角形，分别以 AB，AC为边向外侧作两个等边三角形；D，E，F分别是 MB，BC，CN 的中点；连接 DE，FE；连接辅助线BN、CM", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2532.jpg" }, { "index": 2533, "ocr_en": "In triangle ABC, AD is the median line, AE is the Angle bisector, and both D and E are on the line segment BC.  CF is perpendicular to AE at point F, and point F is on AE. ", "ocr_zh": "在三角形ABC中，AD为中线，AE为角平分线，且D、E均在线段BC上；CF垂直AE于点F，F点在AE上；", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2533.jpg" }, { "index": 2534, "ocr_en": "In the square ABCD, E and F are the midpoints of sides AB and BC respectively.  Connect EC, DF;  G and H are the midpoints of EC and DF respectively. Connect GH ", "ocr_zh": "在正方形ABCD中，E，F分别是边AB，BC的中点；连接EC，DF；G，H分别是EC，DF的中点，连接GH", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2534.jpg" }, { "index": 2535, "ocr_en": "In triangle ABC, AB is perpendicular to BC at point B.  AB equal to BC;  BD is perpendicular to AC at point D, and point D is on BC.  CE bisects Angle ACB, intersecting AB at point E and BD at point F. ", "ocr_zh": "在三角形ABC中，AB垂直BC于点B；AB等于BC；BD垂直AC于点D，点D在BC上；CE平分角ACB，交AB于点E，交BD于点F.", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2535.jpg" }, { "index": 2536, "ocr_en": "In the parallelogram ABCD, the diagonals AC and BD intersect at point O. AE is perpendicular to BD at point E ", "ocr_zh": "在平行四边形ABCD中，对角线AC、BD交于点O；AE垂直BD于点E", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2536.jpg" }, { "index": 2537, "ocr_en": "In rectangle ABCD, E is a point on side BC, and P is a point on side CD (not coinciding with points C and D), connecting AE and EP.  M and N are the midpoints of AE and PE respectively, and connect MN ", "ocr_zh": "在矩形ABCD中，E是BC边上一点，P是CD边上的一点(不与点C，D重合)，连接AE、EP；M，N分别是AE，PE的中点，连接MN", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2537.jpg" }, { "index": 2538, "ocr_en": "In triangle ABC, M is the midpoint of side BC. Points D and E are inside Angle BAC. Triangles ABD and ACE are right triangles with AB and AC as hypotenuses respectively, and Angle BAD is equal to Angle CAE. Connect MD, ME.", "ocr_zh": "在三角形ABC中，M是BC边的中点；点D，E在角BAC的内部，三角形ABD和三角形ACE分别是以AB，AC为斜边的直角三角形，且角BAD等于角CAE；连接MD，ME.", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2538.jpg" }, { "index": 2539, "ocr_en": "In triangle ABC, M is the midpoint of side BC. Points D and E are outside Angle BAC. Triangles ABD and ACE are right triangles with AB and AC as hypotenuses respectively, and Angle BAD is equal to Angle CAE. Connect MD, ME ", "ocr_zh": "在三角形ABC中，M是BC边的中点；点D，E在角BAC的外部，三角形ABD和三角形ACE分别是以AB，AC为斜边的直角三角形，且角BAD等于角CAE；连接MD，ME", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2539.jpg" }, { "index": 2540, "ocr_en": "In quadrilateral ABCD, AC and BD are connected, and AC is perpendicular to BD. E and F are the midpoints of AD and BC respectively, and connect EF.", "ocr_zh": "在四边形ABCD中，连接AC、BD，且AC垂直于BD；E、F分别为AD、BC中点，连接EF；", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2540.jpg" }, { "index": 2541, "ocr_en": "In quadrilateral ABCD, AC and BD are connected, and AC is perpendicular to BD. E and F are the midpoints of AD and BC respectively, and connect EF. Take the midpoint G of DC and connect the auxiliary lines EG and FG. Triangle EGF is a right triangle", "ocr_zh": "在四边形ABCD中，连接AC、BD，且AC垂直于BD；E、F分别为AD、BC中点，连接EF；取DC中点G，连接辅助线EG、FG；三角形EGF为直角三角形", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2541.jpg" }, { "index": 2542, "ocr_en": "In quadrilateral ABCD, AC and BD are connected, and AC is perpendicular to BD. E and F are the midpoints of AD and BC respectively, and connect EF. Take the midpoint G of DC and connect the auxiliary lines EG and FG", "ocr_zh": "在四边形ABCD中，连接AC、BD，且AC垂直于BD；E、F分别为AD、BC中点，连接EF；取DC中点G，连接辅助线EG、FG", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2542.jpg" }, { "index": 2543, "ocr_en": "In quadrilateral ABCD, connect AC and BD, and AC is equal to BD. E and F are the midpoints of AD and BC respectively, and connect EF. AC and BD are solid red lines", "ocr_zh": "在四边形ABCD中，连接AC、BD，且AC等于BD；E、F分别为AD、BC中点，连接EF；AC、BD为红色实线", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2543.jpg" }, { "index": 2544, "ocr_en": "In quadrilateral ABCD, connect AC and BD, and AC is equal to BD. E and F are the midpoints of AD and BC respectively, and connect EF. Take the midpoint G of DC and connect the red auxiliary lines EG and FG", "ocr_zh": "在四边形ABCD中，连接AC、BD，且AC等于BD；E、F分别为AD、BC中点，连接EF；取DC中点G，连接红色辅助线EG、FG", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2544.jpg" }, { "index": 2545, "ocr_en": "In quadrilateral ABCD, connect AC and BD, and AC is equal to BD. E and F are the midpoints of AD and BC respectively, and connect EF. Take the midpoint G of DC and connect the red auxiliary lines EG and FG. Triangle EGF is an isosceles triangle", "ocr_zh": "在四边形ABCD中，连接AC、BD，且AC等于BD；E、F分别为AD、BC中点，连接EF；取DC中点G，连接红色辅助线EG、FG;三角形EGF为等腰三角形", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2545.jpg" }, { "index": 2546, "ocr_en": "In quadrilateral ABCD, AB is equal to CD.  E and F are the midpoints of AD and BC respectively, and connect EF.  Connect the auxiliary line BD, take the midpoint G of BD, and connect GE and CF ", "ocr_zh": "在四边形ABCD中，AB等于CD；E、F分别为AD、BC的中点，连接EF；连接辅助线BD，取BD中点G，连接GE、CF", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2546.jpg" }, { "index": 2547, "ocr_en": "In quadrilateral ABCD, E and F are the midpoints of AD and BC respectively, and connect EF. Take the midpoint G of DC and connect the red auxiliary lines EG and FG", "ocr_zh": "在四边形ABCD中，E、F分别为AD、BC的中点，连接EF；取DC中点G，连接红色辅助线EG、FG", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2547.jpg" }, { "index": 2548, "ocr_en": "In the quadrilateral abcd, the ac is perpendicular to the bda; The ego-f is the midpoint of adder BC, which is connected to efon. Take the point of the dc point gg, and connect the red auxiliary line egis.", "ocr_zh": "在四边形ABCD中，AC垂直于BD；E、F分别为AD、BC的中点，连接EF；取DC中点G，连接红色辅助线EG、FG；", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2548.jpg" }, { "index": 2549, "ocr_en": "In quadrilateral ABCD, AB is equal to CD.  E and F are the midpoints of AD and BC respectively, and connect EF.  Connect the auxiliary line BD, take the midpoint G of BD, and connect GE and CF.  Triangle EGF is an isosceles triangle ", "ocr_zh": "在四边形ABCD中，AB等于CD；E、F分别为AD、BC的中点，连接EF；连接辅助线BD，取BD中点G，连接GE、CF；三角形EGF是等腰三角形", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2549.jpg" }, { "index": 2550, "ocr_en": "In quadrilateral ABCD, AB is equal to CD.  E and F are the midpoints of AD and BC respectively, and connect EF.  Connect the auxiliary line BD, take the midpoint G of BD, and connect GE and CF ", "ocr_zh": "在四边形ABCD中，AB等于CD；E、F分别为AD、BC的中点，连接EF；连接辅助线BD，取BD中点G，连接GE、CF", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2550.jpg" }, { "index": 2551, "ocr_en": "In quadrilateral ABCD, E and F are respectively the midpoints of CD and AB in quadrilateral ABCD, connecting EF and BD", "ocr_zh": "在四边形ABCD中，E、F分别是四边形ABCD中CD、AB的中点，连接EF、BD", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2551.jpg" }, { "index": 2552, "ocr_en": "In quadrilateral ABCD, AC and BD are connected, and AC is perpendicular to BD. E and F are the midpoints of AB and CD respectively, and connect EF.", "ocr_zh": "在四边形 ABCD中，连接AC、BD，AC垂直于BD；E、F分别为AB、CD的中点，连接EF；", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2552.jpg" }, { "index": 2553, "ocr_en": "In quadrilateral ABCD, E and F are respectively the midpoints of CD and AB in quadrilateral ABCD, connecting EF and BD. Take the midpoint G of BD, connect the red auxiliary lines EG and FG, and draw GH perpendicular to EF at point H", "ocr_zh": "在四边形ABCD中，E、F分别是四边形ABCD中CD、AB的中点，连接EF、BD；取BD中点G，连接红色辅助线EG、FG，做GH垂直EF于点H", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2553.jpg" }, { "index": 2554, "ocr_en": "In quadrilateral ABCD, AC and BD are connected, and AC is perpendicular to BD. E and F are the midpoints of sides AB and CD respectively, and connect EF", "ocr_zh": "在四边形 ABCD中，连接AC、BD，AC垂直BD；E、F分别为边AB、CD的中点，连接EF", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2554.jpg" }, { "index": 2555, "ocr_en": "In quadrilateral ABCD, AC and BD are connected, and AC is perpendicular to BD. E and F are the midpoints of sides AB and CD respectively, and connect EF. Take the midpoint P of AD and connect the auxiliary lines PE and PF", "ocr_zh": "在四边形 ABCD中，连接AC、BD，AC垂直BD；E、F分别为边AB、CD的中点，连接EF；取AD中点P，连接辅助线PE、PF", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2555.jpg" }, { "index": 2556, "ocr_en": "In the quadrional abcd, AD is equal to the BCC; The e,f is the middle point of ab and CD. Extend the extension of AD and ef to point m, and extend the extension of ef and BC to point n", "ocr_zh": "在四边形ABCD中，AD等于BC；E，F分别是AB和CD的中点；延长AD与EF的延长线交于点M，延长EF与BC的延长线交于点N", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2556.jpg" }, { "index": 2557, "ocr_en": "In quadrilateral ABCD, the diagonals AC and BD are connected, and AC is equal to BD.  E and F are the midpoints of AD and BC respectively;  Connect EF to intersect AC and BD at points M and N respectively ", "ocr_zh": "在四边形ABCD中，连接对角线AC、BD，且有AC等于BD；E，F分别是AD，BC的中点；连接EF分别交AC，BD于点M，N", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2557.jpg" }, { "index": 2558, "ocr_en": "In the right triangle ABC, Angle C is equal to 90 degrees and AC is equal to BC.  Points E and F are respectively on CA and CB, and CE is equal to CF. The length of BF is greater than that of CF, and the length of AE is greater than that of EC.  Connect AF and BE, where M and N are the midpoints of sides AF and BE respectively ", "ocr_zh": "在直角三角形ABC中，角C等于90度，AC等于BC；点E，F分别在 CA，CB上，且CE等于CF，BF长度大于CF，AE长度大于EC；连接AF、BE，M、N分别为边AF、BE的中点", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2558.jpg" }, { "index": 2559, "ocr_en": "In triangle ABC, AC is larger than AB.  Point D is on AC, and AB equal to CD.  E and F are the midpoints of BC and AD respectively. Connect EF and extend it, which intersects with the extension line of BA at point G. Connect GD ", "ocr_zh": "在三角形ABC中，AC长大于AB；点D在AC上，AB等于CD；E，F分别是BC，AD的中点，连接EF并延长，与BA的延长线交于点G，连接 GD", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2559.jpg" }, { "index": 2560, "ocr_en": "In the quadrilateral ADBC, connect AB and CD, and AB intersects CD at point O.  AB equal to CD. E and F are the midpoints of BC and AD respectively. Connect EF and intersect DC and AB at the points M and N respectively ", "ocr_zh": "在四边形ADBC中，连接AB、CD，AB与CD交于点O；AB等于CD，E，F分别是BC，AD的中点，连接EF，分别交DC，AB于点M，N", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2560.jpg" }, { "index": 2561, "ocr_en": "In quadrilateral ABCD, AB equal to CD;  E and F are the midpoints of BC and AD respectively. Connect EF and extend it, intersecting the extended lines of BA and CD at points M and N respectively.  Connect the auxiliary line BD, where H is a point on the auxiliary line BD.  Connect the auxiliary lines HF and HE, with Angle HEF being Angle 1 and Angle HEF being Angle 2 ", "ocr_zh": "在四边形ABCD中，AB等于CD；E，F分别是BC，AD的中点，连接EF并延长，分别与BA，CD的延长线交于点M，N；连接辅助线BD，H为辅助线BD上一点；连接辅助线HF、HE，角HEF为角1，角HEF为角2", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2561.jpg" }, { "index": 2562, "ocr_en": "Line AB is tangent to circle 0 at point C, and point D is on the circle. Connect CD ", "ocr_zh": "直线AB与圆0相切于点C，点 D在圆上，连接CD", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2562.jpg" }, { "index": 2563, "ocr_en": "In the right triangle ABC, Angle ACB equal to 90°;  0 is a point on side BC. A circle with 0 as its center and 0B as its radius intersects AB at point D. Connect CD, and CD equal to AC ", "ocr_zh": "在直角三角形ABC中，角ACB等于90°；0是BC边上一点，以0为圆心，0B为半径的圆与AB相交于点D，连接CD，且CD等于AC", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2563.jpg" }, { "index": 2564, "ocr_en": "AB is the diameter of circle 0. CD is tangent to circle 0 at point C and intersects the extension of AB at point D. Connect AC and BC ", "ocr_zh": "AB为圆0的直径，CD与圆0相切于点C，交AB的延长线于点D，连接AC，BC", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2564.jpg" }, { "index": 2565, "ocr_en": "Rotate point B counterclockwise around point A to point C ", "ocr_zh": "将点B绕点A逆时针旋转至点C", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2565.jpg" }, { "index": 2566, "ocr_en": "Rotate point B counterclockwise around point A to point C, and connect AB and AC.  Draw an auxiliary line CD through point C. CD is perpendicular to AB at point D. With C as the center and CD as the radius, draw an auxiliary line circle ", "ocr_zh": "将点B绕点A逆时针旋转至点C，连接AB、AC；过点C做辅助线CD，CD垂直AB于点D，以C为圆心，CD为半径做辅助线圆", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2566.jpg" }, { "index": 2567, "ocr_en": "AB is the diameter of circle O, CD is a chord of circle O, and AB is perpendicular to CD.  Connect AC, OD;  Connect DB, draw CE perpendicularly to DB through point C, and intersect the extension of DB at point E.  Extend DO and intersect AC at point F ", "ocr_zh": "AB是圆O的直径，CD是圆O的一条弦，AB垂直于CD；连接AC，OD；连接DB，过点C做CE垂直于DB，交DB延长线于点E；延长DO，交AC于点F", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2567.jpg" }, { "index": 2568, "ocr_en": "In triangle ABC, 0 is a point on AC. With 0 as the center and the length of 0C as the radius, draw a circle that is tangent to BC at point C.  Draw AD perpendicularly to BO through point A, intersecting the extension of BO at point D, and Angle AOD is equal to Angle BAD ", "ocr_zh": "在三角形ABC中，0为AC上一点，以0为圆心，0C长为半径作圆，与BC相切于点C；过点A作AD垂直BO，交BO的延长线于点D，且角AOD等于角BAD", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2568.jpg" }, { "index": 2569, "ocr_en": "AC is a chord of circle O ", "ocr_zh": "AC为圆O的一条弦", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2569.jpg" }, { "index": 2570, "ocr_en": "In triangle AOC, AC is greater than OC and OA is greater than OA. A circle is drawn with 0 as the center, passing exactly through point A and intersecting AC at point B ", "ocr_zh": "在三角形AOC中AC大于OC大于OA，以0为圆心作圆，恰好过点A，且与AC交于点B", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2570.jpg" }, { "index": 2571, "ocr_en": "In triangle AOC, AC is greater than OC and OA is greater than OA. A circle is drawn with 0 as the center, passing exactly through point A and intersecting AC at point B.  Draw a perpendicular line OD from point 0 to string AB, intersecting AB at point D. ", "ocr_zh": "在三角形AOC中AC大于OC大于OA，以0为圆心作圆，恰好过点A，且与AC交于点B；过点0作弦AB的垂线OD，交AB于点D.", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2571.jpg" }, { "index": 2572, "ocr_en": "In circle O, AB is the diameter;  C is a point on circle O. Draw CD through C, which is perpendicular to AB, intersecting circle O at point D and AB at point G.  Connect AD  Draw CE perpendicularly to AD through C, intersecting AD at point E and AB at point F ", "ocr_zh": "在圆O中，AB为直径；C为圆O上一点，过C做CD垂直于AB，交圆O于点D，交AB于点G；连接AD；过C做CE垂直AD，交AD于点 E，交AB于点F ", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2572.jpg" }, { "index": 2573, "ocr_en": "In circle O, AB is the diameter;  C is a point on circle O. Draw CD through C, which is perpendicular to AB, intersecting circle O at point D and AB at point G.  Connect AD  Draw CE perpendicularly to AD through C, intersecting AD at point E and AB at point F.  Connect the auxiliary lines BC and CO ", "ocr_zh": "在圆O中，AB为直径；C为圆O上一点，过C做CD垂直于AB，交圆O于点D，交AB于点G；连接AD；过C做CE垂直AD，交AD于点 E，交AB于点F ；连接辅助线BC、CO", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2573.jpg" }, { "index": 2574, "ocr_en": "The above figure: There is a point B within the line segment AC, and AB is less than the length of AC. D is the midpoint of line segment AB, and E is the midpoint of line segment BC. As shown in the figure below: Three line segments OA, OB, and OC are drawn starting from vertex O, among which OB is located inside OA and OC.  Line segment OD is inside Angle AOB, and OD is the bisector of Angle AOB.  OE is located inside Angle BOC, and OE is the bisector of Angle BOC ", "ocr_zh": "上图：在线段AC内有一点B，AB小于AC长；D为线段AB的中点，E为线段BC的中点 下图：以顶点O出发做三条线段OA、OB、OC，其中OB位于OA、OC的内部；线段OD在角AOB内部，OD为角AOB的平分线；OE位于角BOC内部，OE为角BOC的平分线", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_2574.jpg" }, { "index": 2575, "ocr_en": "The line segment AB in the left figure is equal to a.  The line segment BC in the right figure is equal to b ", "ocr_zh": "左图线段AB等于a；右图线段BC等于b", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2575.jpg" }, { "index": 2576, "ocr_en": "Point B is on line segment AC, and AB is greater than AC ", "ocr_zh": "点B在线段AC上，且AB大于AC", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2576.jpg" }, { "index": 2577, "ocr_en": "There is a point C on the line segment AB, and the length of AC is greater than that of BC ", "ocr_zh": "在线段AB上有一点C，AC长度大于BC", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2577.jpg" }, { "index": 2578, "ocr_en": "Line segment AB and line segment BA are the same line segment ", "ocr_zh": "线段AB和线段BA是同一条线段", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2578.jpg" }, { "index": 2579, "ocr_en": "Point B is on line segment AC, and AB is greater than AC ", "ocr_zh": "点B在线段AC上，且AB大于AC", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2579.jpg" }, { "index": 2580, "ocr_en": "Line segment AB and line segment BA are the same line segment Angle, which can be regarded as the figure formed by a ray rotating around its endpoint from one position to another.  The ray at the starting position is called the initial side of the Angle, and the ray at the ending position is called the terminal side of the Angle. ", "ocr_zh": "角可以看作是由一条射线绕着它的端点从一个位置旋转到另一个位置所成的图形；处于起始位置的射线叫做角的始边，处于终止位置的射线叫做角的终边.", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2580.jpg" }, { "index": 2581, "ocr_en": "Angle AOC is equal to Angle AOB plus Angle BOC, and Angle BOC is less than Angle AOB ", "ocr_zh": "角AOC等于角AOB加角BOC，角BOC小于角AOB", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2581.jpg" }, { "index": 2582, "ocr_en": "Given the acute Angle AOB ", "ocr_zh": "已知锐角AOB", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2582.jpg" }, { "index": 2583, "ocr_en": "In the acute Angle AOB, OM is the Angle bisector.  Outside the acute Angle AOB, ON is the Angle bisector ", "ocr_zh": "在锐角AOB内，OM为角平分线；在锐角AOB外，ON为角平分线", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2583.jpg" }, { "index": 2584, "ocr_en": "Given the acute Angle BOC ", "ocr_zh": "已知锐角BOC", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2584.jpg" }, { "index": 2585, "ocr_en": "Take the ray OC within Angle AOB, and Angle BOC is greater than Angle AOC ", "ocr_zh": "在角AOB内取射线OC，且角BOC大于角AOC", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2585.jpg" }, { "index": 2586, "ocr_en": "Three angles 1, 2 and 3 of the same size are arranged side by side. ", "ocr_zh": "三个相同大小的角1、角2、角3并列排布；", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2586.jpg" }, { "index": 2587, "ocr_en": "It is known that point C is a point on the extension line of line segment BA.  Points M and N are the midpoints of AC and BC respectively ", "ocr_zh": "已知点C是线段BA延长线上一点；点M、N分别是 AC、BC的中点", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2587.jpg" }, { "index": 2588, "ocr_en": "It is known that point C is on line segment AB;  Points M and N are the midpoints of AC and BC respectively.  The line is orange. ", "ocr_zh": "已知点C在线段AB上；点M、N分别是AC、BC的中点；线为橙色", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2588.jpg" }, { "index": 2589, "ocr_en": "It is known that point C is a point on the extension line of line segment AB.  Points M and N are the midpoints of AC and BC respectively.  The line is orange. ", "ocr_zh": "已知点C是线段AB延长线上一点；点M，N分别是AC，BC的中点；线为橙色", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_2589.jpg" }, { "index": 2590, "ocr_en": "Point C is a point on line segment AB, and AC AC. Points O and H are inside triangle ABC. Connect AO and OH. Construct an equilateral triangle DOH with OH as one side. Connect AD. Draw auxiliary line OE perpendicular to AB at E. Draw auxiliary line AH.", "ocr_zh": "三角形ABC，AB大于AC，O，H为三角形ABC内一点，连接AO，OH，以OH为边向上作等边三角形DOH，连接AD，过O作辅助线OE垂直于AB于E，作辅助线AH", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3850.jpg" }, { "index": 3851, "ocr_en": "In triangle ABC, AB < AC. Points O and H are inside the triangle. Connect AO and extend it to intersect BC at D. Draw HE parallel to AD, intersecting BC at E. Draw auxiliary line AH. Take the midpoint of AH as N and the midpoint of DE as M. Draw auxiliary lines MN and OM.", "ocr_zh": "三角形ABC，AB小于AC，O，H为三角形内一点，连接AO并延长，交BC于D，过H作HE平行于AD，交BC于E，作辅助线AH，取AH中点N，DE中点M，作辅助线MN，OM", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3851.jpg" }, { "index": 3852, "ocr_en": "In triangle ABC, point H is inside triangle ABC. Connect BH. Draw DH perpendicular to BH, intersecting AB at D. Connect CH. Draw HE perpendicular to CH, intersecting AC at E. Connect DE. Draw CF perpendicular to BC, intersecting the extension of DE at F. Connect FH. Auxiliary lines: Extend CF and HE to intersect at G. Connect AH, intersecting DE at M.", "ocr_zh": "三角形ABC，H为三角形ABC内一点，连接BH，过H作DH垂直于BH，交AB于D，连接CH，过H作HE垂直于CH，交AC于E，连接DE，过C作CF垂直于BC，交DE延长线于F，连接FH。辅助线：延长CF，HE交于G，连接AH，交DE于M", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_3852.jpg" }, { "index": 3853, "ocr_en": "In triangle ABC, point M is on BC. Points P and Q are on AB and AC, respectively. Connect PQ and AM, which intersect at N. Draw auxiliary lines MP and MQ.", "ocr_zh": "三角形ABC，M是BC上一点，P，Q分别是AB，AC上一点，连接PQ，AM交于N，作辅助线MP，MQ", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3853.jpg" }, { "index": 3854, "ocr_en": "In triangle ABC, point M is on BC. Draw the dashed line AM. Points P1 and P2 are on AB, with P1 closer to A. Point G is on AM. Connect P1G and extend it to intersect AC at Q1. Connect P2G and extend it to intersect AC at Q2. Draw the auxiliary line BX1 parallel to AM, intersecting the extension of Q2P2 (dashed line) at X2 and the extension of Q1P1 (dashed line) at X1. Draw the auxiliary line CY2 parallel to AM, intersecting the extension of P2Q2 (dashed line) at Y2 and the extension of P1Q1 (dashed line) at Y1.", "ocr_zh": "三角形ABC，M是BC上一点，作虚线AM，P1，P2是AB上两点，P1靠近A，G是AM上一点，连接P1G并延长交AC于Q1，连接P2G并延长交AC于Q2，过B作辅助线BX1平行于AM交Q2P2延长线（虚线）于X2，交Q1P1延长线（虚线）于X1，过C作辅助线CY2平行于AM交P2Q2延长线（虚线）于Y2，交P1Q1延长线（虚线）于Y1", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_3854.jpg" }, { "index": 3855, "ocr_en": "In triangle ABC, point O is inside the triangle. Connect AO and CO. Take points M and N on AB and BC, respectively, and connect MO and NO such that angle AOC is twice angle MON. Connect MN. Let angle MON be \\(\\alpha\\) and angle AOC be \\(2\\alpha\\). Draw the auxiliary line BO. Take points K and L inside angle AOC and draw the auxiliary lines AK, KL, LC, OK, and OL.", "ocr_zh": "三角形ABC，O为三角形ABC内一点，连接AO，CO，在AB，BC上分别取M，N并连接MO，NO，使角AOC是角MON的2倍，连接MN，角MON是alpha，角AOC是2alpha，作辅助线BO，在角AOC内部取K，L，作辅助线AK，KL，LC，OK，OL", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_3855.jpg" }, { "index": 3856, "ocr_en": "Isosceles triangle ABC with AB equal to AC. P is a point on BC. Draw PR parallel to AB, intersecting AC at R. Draw PQ parallel to AC, intersecting AB at Q. Draw the auxiliary line RQ. Find the symmetric point P' of P with respect to RQ. Connect BP', QP', RP', and CP'. Extend RQ to intersect BP' (dashed line). Draw the auxiliary line PP'.", "ocr_zh": "等腰三角形ABC，AB等于AC，P是BC上一点，过P作PR平行于AB交AC于R，过P作PQ平行于AC交AB于Q，作辅助线RQ，取P关于RQ的对称点P'，连接BP',QP',RP',CP'，延长RQ与BP'相交（虚线），作辅助线PP'", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_3856.jpg" }, { "index": 3857, "ocr_en": "In triangle ABC, construct the angle bisectors BI and CI of angles ABC and ACB, respectively, which intersect at I. Draw IG parallel to BC, with G inside triangle ABC. Draw the auxiliary lines AI, BG, and GC. Draw the auxiliary line AG and extend it to intersect BC at M.", "ocr_zh": "三角形ABC，作角ABC和角ACB的角平分线BI和CI交于I，过I作IG平行于BC，G在三角形ABC内，作辅助线AI，BG，GC，作辅助线AG并延长，交BC于M", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3857.jpg" }, { "index": 3858, "ocr_en": "Left figure: Rectangle ABCD (clockwise), with point K outside the side BC of the rectangle. Connect KA and KD. Draw BE perpendicular to DK at E, intersecting the extension of DK at E. Draw CF perpendicular to AK at F, intersecting the extension of AK at F. Extend BE and CF to intersect at M. Connect MK. Auxiliary lines: Construct triangles BPC and AKD to be congruent. Connect PM and PK.\n\nRight figure: Rectangle ABCD (clockwise), with point K outside the side BC of the rectangle. Connect KA and KD. Draw BE perpendicular to DK at E, intersecting DK at E. Draw CF perpendicular to AK at F, intersecting AK at F. Extend BE and CF to intersect at M. Connect MK. Auxiliary lines: Construct triangles BPC and AKD to be congruent. Connect PM and PK.Left figure: Rectangle ABCD (clockwise), with point K outside the side BC of the rectangle. Connect KA and KD. Draw BE perpendicular to DK at E, intersecting the extension of DK at E. Draw CF perpendicular to AK at F, intersecting the extension of AK at F. Extend BE and CF to intersect at M. Connect MK. Auxiliary lines: Construct triangles BPC and AKD to be congruent. Connect PM and PK.Right figure: Rectangle ABCD (clockwise), with point K outside the side BC of the rectangle. Connect KA and KD. Draw BE perpendicular to DK at E, intersecting DK at E. Draw CF perpendicular to AK at F, intersecting AK at F. Extend BE and CF to intersect at M. Connect MK. Auxiliary lines: Construct triangles BPC and AKD to be congruent. Connect PM and PK.", "ocr_zh": "左图：矩形ABCD（顺时针），K是矩形BC外侧一点，连接KA，KD，过B作BE垂直于DK于E，交DK延长线于E，过C作CF垂直于AK于F，交AK延长线于F，延长BE，CF交于M，连接MK，辅助线：作三角形BPC和三角形AKD全等，连接PM，PK，PM是虚线；右图：矩形ABCD（顺时针），K是矩形BC外侧一点，连接KA，KD，过B作BE垂直于DK于E，交DK于E，过C作CF垂直于AK于F，交AK于F，延长BE，CF交于M，连接MK，辅助线：作三角形BPC和三角形AKD全等，连接PM，PK，PK是虚线。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_3858.jpg" }, { "index": 3859, "ocr_en": "Triangle ABC, construct its circumcircle O. Draw the angle bisectors (dashed lines) of angle BAC and angle BCA, AI and CI, intersecting at I. Draw a dashed line BG perpendicular to AC at G. Draw a dashed line AE perpendicular to BC at E. BG and AE intersect at H. Connect OI and IH. Auxiliary lines: Extend BO to intersect circle O at F. Connect AF and CF. Extend AO to intersect circle O at D. Connect BD.", "ocr_zh": "三角形ABC，作其外接圆O，作角BAC和角BCA平分线（虚线）AI，CI交于I，过B作虚线BG垂直于AC于G，过A作虚线AE垂直于BC于E，BG，AE交于H，连接OI，IH。辅助线： 延长BO交圆O于F，连接AF，CF，延长AO交圆O于D，连接BD", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_3859.jpg" }, { "index": 3860, "ocr_en": "Triangle ABC, construct its circumcircle O. Draw AD perpendicular to BC (the foot of the perpendicular is not labeled), intersecting circle O at D. Draw a dashed line CH perpendicular to AB (the foot of the perpendicular is not labeled), intersecting AD at H. Connect AO and extend it to intersect circle O at E. Connect HE and intersect BC at the midpoint M of BC. Connect OM. Draw EF perpendicular to BC (the foot of the perpendicular is not labeled) intersecting the extension of DM at F. Draw auxiliary lines DE, FH, CH. Draw auxiliary line BO and extend it to intersect circle O at B1. Draw auxiliary lines AB1 and CB1.", "ocr_zh": "三角形ABC，作其外接圆O，过A作AD垂直于BC（垂足未标出），交圆O于D，过C作虚线CH垂直于AB（垂足未标出），交AD于H，连接AO并延长，交圆O于E，连接HE交BC于BC中点M，连接OM，过E作EF垂直于BC（垂足未标出）交DM延长线于F，作辅助线DE，FH，CH，作辅助线BO并延长，交圆O于B1，作辅助线AB1，CB1", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_3860.jpg" }, { "index": 3861, "ocr_en": "Triangle ABC, H is the orthocenter. Draw auxiliary line BH and extend it to intersect AC at E. Draw auxiliary line CH and extend it to intersect AB at F. Draw auxiliary line AH and extend it to intersect BC at D. Connect DE, EF, FD. Auxiliary lines: Construct the reflection of D about AC, denoted as D1, and connect DD1, D1E. Construct the reflection of D about AB, denoted as D2, and connect DD2, D2F. Take a point Q on AF and a point P on AE, and connect D2Q, QP, PD1, DP, DQ.", "ocr_zh": "三角形ABC，H是垂心，作辅助线BH并延长交AC于E，作辅助线CH并延长交AB于F，作辅助线AH并延长交BC于D，连接DE，EF，FD。辅助线：作D关于AC的对称点D1，连接DD1，D1E，作D关于AB对称点D2，连接DD2，D2F，取AF上一点Q，AE上一点P，连接D2Q，QP，PD1，DP，DQ", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_3861.jpg" }, { "index": 3862, "ocr_en": "Triangle ABC, draw the dashed line AD perpendicular to BC at D, draw the dashed line BE perpendicular to AC at E, and draw the dashed line CF perpendicular to AB at F. Connect DE, EF, and DF. Take points D', E', and F' on BD, AE, and AF respectively. Connect D'E', D'F', and E'F'. Auxiliary lines: Construct the symmetric point D1' of D' with respect to AC. Connect D'D1' intersecting AC at E0. Connect D1'E'. Construct the symmetric point D2' of D' with respect to AB. Connect D'D2' intersecting AB at F0. Connect D2'F'. Connect D1'D2', E0F0, and AD'.", "ocr_zh": "三角形ABC，过A作虚线AD垂直于BC于D，过B作虚线BE垂直于AC于E，过C作虚线CF垂直于AB于F，连接DE，EF，DF，取BD，AE，AF上的D',E',F'，连接D'E'，D'F'，E'F'。辅助线：作D'关于AC的对称点D1'，连接D'D1'交AC于E0，连接D1'E'，作D'关于AB对称点D2'，连接DD2'交AB于F0，连接D2'F',连接D1'D2',E0F0,AD'", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_3862.jpg" }, { "index": 3863, "ocr_en": "Triangle ABC, with points D, E, and F on sides BC, AC, and AB respectively. Connect DE, EF, and DF.", "ocr_zh": "三角形ABC，D,E,F分别是BC，AC,AB上的点，连接DE，EF，DF", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3863.jpg" }, { "index": 3864, "ocr_en": "Triangle ABC, with points D and E on sides AB and AC respectively. Connect DE.", "ocr_zh": "三角形ABC，D，E分别是AB，AC上的点，连接DE ", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3864.jpg" }, { "index": 3865, "ocr_en": "Triangle ABC, with points E and F on sides AC and AB respectively. Connect EF and BE.", "ocr_zh": "三角形ABC，E，F分别是AC，AB上的点，连接EF，BE", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3865.jpg" }, { "index": 3866, "ocr_en": "Triangle ABC, with points E and F on sides AB and AC respectively. Connect EF and EC. Draw auxiliary line BF.", "ocr_zh": "三角形ABC，E，F分别是AB，AC上的点，连接EF，EC，作辅助线BF", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3866.jpg" }, { "index": 3867, "ocr_en": "Triangle ABC, with points A1, B1, C1 being the midpoints of sides BC, CA, AB respectively. Points K, L, M are on segments AB1, CA1, BC1 respectively. Connect A1B1, A1C1, B1C1. Connect MK intersecting A1C1 at M2 and B1C1 at K1. Connect KL intersecting A1B1 at L1 and C1B1 at K2. Connect LM intersecting A1B1 at L2 and A1C1 at M1. Draw auxiliary lines K1M1, M1L1, L1K1. Triangles C1M2K1, B1L1K2, A1M1L2 are shaded.", "ocr_zh": "（文字噪声）三角形ABC，点A1、B1、C1分别是三条边BC、CA、AB的中点，点K、L、M分别在线段AB1、CA1、BC1上，连接A1B1,A1C1,B1C1,连接MK交A1C1于M2，交B1C1于K1，连接KL交A1B1于L1，交C1B1于K2，连接LM交A1B1于L2，交A1C1于M1，作辅助线K1M1,M1L1,L1K1,三角形C1M2K1，B1L1K2,A1M1L2有阴影", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_3867.jpg" }, { "index": 3868, "ocr_en": "Left figure: Triangle ABD, draw AC perpendicular to BD at C.Right figure: Right triangle ABC, with angle C being the right angle. D is a point on BC. Connect AD.", "ocr_zh": "左图：三角形ABD，过A作AC垂直于BD于C；右图：直角三角形ABC，角C是直角，D是BC上一点，连接AD", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3868.jpg" }, { "index": 3869, "ocr_en": "Triangle ABC, draw CD perpendicular to AB at D.", "ocr_zh": "三角形ABC，过C作CD垂直于AB于D", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3869.jpg" }, { "index": 3870, "ocr_en": "Right triangle ABC, with angle BCA being the right angle. Take the midpoint M of AB and connect CM. Draw the angle bisector CT of angle BAC, intersecting AB at T. Draw CH perpendicular to AB at H. Draw the circumcircle M of triangle ABC (dashed line). Auxiliary lines: Extend CT to intersect circle M at D. Connect DA, DM, DB.", "ocr_zh": "直角三角形ABC，角BCA是直角，取AB中点M，连接CM，作角BAC角平分线CT交AB于T，过C作CH垂直于AB于H，作三角形ABC的外接圆M（虚线），辅助线：延长CT交圆M于D，连接DA，DM，DB", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3870.jpg" }, { "index": 3871, "ocr_en": "In right triangle ABC, angle BCA is a right angle. Auxiliary lines: Draw CM perpendicular to AB at M. Draw the incircle O of triangle ABC (not labeled in the figure), which is tangent to AB, BC, AC at points D, E, F respectively. Connect OD, OE, OF. The lengths of AF and AD are x, the lengths of BD and BE are y, and the lengths of CE and CF are z.", "ocr_zh": "直角三角形ABC，角BCA是直角，辅助线：过C作CM垂直于AB于M，作三角形ABC内切圆O（图中未标出），与AB，BC，AC分别相切于D，E，F，连接OD，OE，OF。AF和AD长为x,BD和BE长为y，CE和CF长为z", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_3871.jpg" }, { "index": 3872, "ocr_en": "Right triangle ABC, with angle C being the right angle. Point D is on CB, and point E is on AB, such that AD is perpendicular to DE and AD equals DE. Draw auxiliary line EF perpendicular to BC at F.", "ocr_zh": "直角三角形ABC，角C是直角，D，E分别是CB，AB上的点且AD垂直于DE，AD等于DE，作辅助线EF垂直于BC于F", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3872.jpg" }, { "index": 3873, "ocr_en": "Triangle ABC, draw a dashed line AE perpendicular to BC at E. Take a point D on CE and connect AD.", "ocr_zh": "三角形ABC，过A作虚线AE垂直于BC于E，取CE上一点D，连接AD", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3873.jpg" }, { "index": 3874, "ocr_en": "Right triangle ABC with angle ABC being a right angle. D is the midpoint of AC. Connect BD. Draw BE perpendicular to BD, intersecting the extension of CA at E. The angle ABE is equal to angle DBC, which is equal to angle C.", "ocr_zh": "直角三角形ABC，角ABC是直角，D是AC中点，连接BD，过B作BE垂直于BD，交CA延长线于E，角ABE等于角DBC等于角C", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3874.jpg" }, { "index": 3875, "ocr_en": "In triangle ABC, point D is on side BC. Connect AD. Auxiliary lines: Extend BC to E such that CE equals AC. Connect AE. Draw AM perpendicular to BC, intersecting BC at M. Take the midpoint F of BD. Connect AF.", "ocr_zh": "三角形ABC，D是边BC上一点，连接AD，辅助线：延长BC至E，使CE等于AC，连接AE，过A作AM垂直于BC于M，取BD中点F，连接AF", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3875.jpg" }, { "index": 3876, "ocr_en": "In right triangle ABC, angle C is a right angle. Take a point D on AC close to C, and connect BD. Draw the auxiliary line BE to bisect angle ABD, intersecting AD at E. The length of AE is x.", "ocr_zh": "直角三角形ABC，角C是直角，取AC上靠近C一点D，连接BD，作辅助线BE平分角ABD，交AD于E，AE长为x ", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3876.jpg" }, { "index": 3877, "ocr_en": "In right triangle ABC, with ∠ACB being the right angle, point D lies on side AB. Connect CD. Draw auxiliary line DE perpendicular to BC, intersecting BC at point E.", "ocr_zh": "直角三角形ABC，角ACB是直角，D是AB上一点，连接CD，过D作辅助线DE垂直于BC于E", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3877.jpg" }, { "index": 3878, "ocr_en": "In right triangle ABC, with ∠ACB being the right angle, point D lies on the extension of side BA. Connect CD. Draw auxiliary line DE perpendicular to BC, intersecting the extension of BC (dashed line) at point E.", "ocr_zh": "直角三角形ABC，角ACB是直角，D是BA延长线上一点，连接CD，过D作辅助线DE垂直于BC交BC延长线（虚线）于E", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3878.jpg" }, { "index": 3879, "ocr_en": "Left figure (1): Isosceles triangle OAB with OA = OB. Draw the dashed line OM perpendicular to AB at M. Take a point P on AM and connect OP.Right figure (2): Isosceles triangle OAB with OA = OB. Draw the dashed line OM perpendicular to AB at M. Take a point P on the extension of AB and connect OP.", "ocr_zh": "左图图(1)：等腰三角形OAB，OA等于OB，过O作虚线OM垂直于AB于M，取AM上一点P，连接OP；右图图(2)：等腰三角形OAB，OA等于OB，过O作虚线OM垂直于AB于M，取AB延长线上一点P，连接OP", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3879.jpg" }, { "index": 3880, "ocr_en": "In isosceles triangle ABC, where AB = AC, take a point D on AB and a point F on BC close to C. Connect DF and extend it to intersect the extension of AC at point E. Draw the auxiliary line DG parallel to AC, intersecting BC at point G.", "ocr_zh": "等腰三角形ABC，AB等于AC，取AB上一点D，BC上靠近C一点F，连接DF并延长，交AC延长线于E，过D作辅助线DG平行于AC，交BC于G", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3880.jpg" }, { "index": 3881, "ocr_en": "In isosceles triangle PAB, take a point D on PB close to B. Connect AD and extend it to point C. Connect BC. Draw the auxiliary line extending BP to Q such that PQ equals PB. Draw the auxiliary line AQ.", "ocr_zh": "等腰三角形PAB，取PB上靠近B一点D，连接AD并延长至C，连接BC，做辅助线延长BP至Q，使PQ等于PB，作辅助线AQ", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3881.jpg" }, { "index": 3882, "ocr_en": "In isosceles triangle ABC, where AB = AC, points M and N are on AC with N closer to A and ∠ABN = ∠MBC. BM = NM. Connect BN and BM. Draw the auxiliary line ND perpendicular to BC, intersecting BC at point D.", "ocr_zh": "等腰三角形ABC，AB等于AC，M，N在AC上，N靠近A且角ABN等于角MBC，BM等于NM，连接BN，BM，过N作辅助线ND垂直于BC于D", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3882.jpg" }, { "index": 3883, "ocr_en": "In isosceles triangle ABC, where AB = AC, draw a line l parallel to BC through point A. Draw the angle bisector BE of ∠ABC, intersecting AC at D and line l at E. Draw the angle bisector CG of ∠ACB, intersecting AB at F and line l at G.", "ocr_zh": "等腰三角形ABC，AB等于AC，过A作直线l平行于BC，作角ABC平分线BE交AC于D，交l于E，作角ACB平分线CG交AB于F，交l于G", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3883.jpg" }, { "index": 3884, "ocr_en": "Right triangle ABC, with angle ACB being a right angle. Draw CD perpendicular to AB at D through C. Draw AE to bisect angle BAC, intersecting CD at H and CB at E. Draw CF to bisect angle BCD, intersecting AE at G and BA at F. Connect FH.", "ocr_zh": "直角三角形ABC，角ACB是直角，过C作CD垂直于AB于D，过A作AE平分角BAC交CD于H，交CB于E，过C作CF平分角BCD，交AE于G，交BA于F，连接FH", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3884.jpg" }, { "index": 3885, "ocr_en": "Three non-congruent isosceles right triangles ADC, DPE, and BEC, with right angles at ∠ADC, ∠DPE, and ∠CEB, respectively, and points A, P, and B are collinear. Auxiliary lines: Connect AP and BP. Extend DP to F such that PF equals PD. Connect FE and FB.", "ocr_zh": "三个互相不全等的等腰直角三角形ADC，DPE，BEC，直角分别是角ADC，角DPE，角CEB，且A，P，B共线。辅助线：连接AP，BP，延长DP至F使PF等于PD，连接FE，FB", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_3885.jpg" }, { "index": 3886, "ocr_en": "In triangle ABC, take a point D on BC such that the distance from D to AB is equal to AB. Draw DH perpendicular to AB at H. Extend HD to intersect AC at E. Auxiliary lines: Draw AF perpendicular to AB, and make AF equal to BH. Connect BF and DF.", "ocr_zh": "三角形ABC，取BC上一点D使D到AB的距离等于AB，过D作DH垂直于AB于H，延长HD，AC交于E。辅助线：过A作AF垂直于AB，且AF等于BH，连接BF，DF", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3886.jpg" }, { "index": 3887, "ocr_en": "In isosceles triangle ABC, AB equals AC. Extend AB to D such that AD equals BC. Extend CA to E such that CE equals BC. Connect DE, and we have DE equals AD. Auxiliary lines: Draw DF parallel to BC through D, and draw CF parallel to AD through C. Let DF and CF intersect at F. Connect EF.", "ocr_zh": "等腰三角形ABC，AB等于AC，延长AB至D使AD等于BC，延长CA至E使CE等于BC，连接DE，有DE等于AD。辅助线：过D作DF平行于BC，过C作CF平行于AD，DF，CF交于F，连接EF", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3887.jpg" }, { "index": 3888, "ocr_en": "Triangle ABC. Auxiliary lines: Extend CB to C' such that BC' equals AB. Extend BA to B' such that AB' equals AC. Connect AC' and B'C. The lengths of AB and BC' are c, the lengths of BC and B'C are a, and the lengths of AC, AC', and AB' are b.", "ocr_zh": "三角形ABC。辅助线：延长CB至C'，使BC'等于AB，延长BA至B'，使AB'等于AC，连接AC',B'C，AB和BC'长为c，BC和B'C长为a，AC，AC',AB'长为b", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3888.jpg" }, { "index": 3889, "ocr_en": "Left figure: Triangle ABC, where AB is greater than AC. Draw AT to bisect angle BAC, intersecting BC at T. Take point S on BT such that BS equals TC. Auxiliary line: Take point T' on BC such that AT equals AT', and connect AT'.Right figure: Triangle ABC, where AB is greater than AC. Draw AT to bisect angle BAC, intersecting BC at T. Take point S on BT such that BS equals TC. Auxiliary line: Take point T' on the extension of BC such that AT equals AT', and connect AT'.", "ocr_zh": "左图：三角形ABC，AB大于AC，过A作AT平分角BAC交BC于T，在BT上取S使BS等于TC，辅助线：在BC上取T'，使AT等于AT'，连接AT';右图：三角形ABC，AB大于AC，过A作AT平分角BAC交BC于T，在BT上取S使BS等于TC，辅助线：在BC延长线上取T'，使AT等于AT'，连接AT'", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3889.jpg" }, { "index": 3890, "ocr_en": "Isosceles triangle ABC with AB equal to AC. D is a point inside the triangle on the perpendicular bisector of AB. Connect AD, BD, and CD. Auxiliary lines: Construct an equilateral triangle ACE with AC as one side, inside triangle ABC. Connect EB and ED.", "ocr_zh": "等腰三角形ABC，AB等于AC，D是三角形内在AB中垂线上一点，连接AD，BD，CD，辅助线：以AC为边，向三角形ABC内作等边三角形ACE，连接EB，ED", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3890.jpg" }, { "index": 3891, "ocr_en": "Equilateral triangle ABC, take the midpoints D and E of BC and AC respectively, connect AD and BE, angle CBE equals angle DAC.", "ocr_zh": "等边三角形ABC，取BC，AC中点D，E，连接AD，BC，角CBE等于角DAC", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3891.jpg" }, { "index": 3892, "ocr_en": "Isosceles triangle ABC, where AB equals AC. Point D is inside the triangle on the perpendicular bisector of AB. Connect AD, BD, and CD. Auxiliary line: Take the point E, which is the reflection of D across the perpendicular bisector of BC. Connect EA, ED, and EC.", "ocr_zh": "等腰三角形ABC，AB等于AC，D是三角形内在AB中垂线上一点，连接AD，BD，CD，辅助线：取D关于BC中垂线的对称点E，连接EA，ED，EC", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3892.jpg" }, { "index": 3893, "ocr_en": "Convex quadrilateral ABCD (clockwise), connect AC and BD. Auxiliary line: Construct an equilateral triangle CBE upward with CB as one side, and connect DE.", "ocr_zh": "凸四边形ABCD（顺时针），连接AC，BD，辅助线：以CB为边向上作正三角形CBE，连接DE", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3893.jpg" }, { "index": 3894, "ocr_en": "Square ABCD, take points E and F on BC and CD respectively such that triangle AEF is an equilateral triangle. Connect AE, EF, and AF.", "ocr_zh": "正方形ABCD，在BC，CD上分别取E，F使三角形AEF是正三角形，连接AE，EF，AF", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3894.jpg" }, { "index": 3895, "ocr_en": "Line segment AB, point C is on AB closer to A. Construct equilateral triangles ACD and CBE on the same side of AB. Connect AE intersecting CD at G, connect BD intersecting CE at H, and connect GH.", "ocr_zh": "线段AB，C是AB上靠近A的一点，作等边三角形ACD和CBE（在AB的同侧），连接AE交CD于G，连接BD交CE于H，连接GH", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3895.jpg" }, { "index": 3896, "ocr_en": "Hexagon ABCDEF has all interior angles equal. Auxiliary lines: Extend BA and EF to meet at R, extend AB and DC to meet at P, and extend CD and FE to meet at Q.", "ocr_zh": "六边形ABCDEF所有内角都相等，辅助线：延长BA，EF交于R，延长AB，DC交于P，延长CD，FE交于Q", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3896.jpg" }, { "index": 3897, "ocr_en": "Convex quadrilateral PQRS, E and F are the midpoints of SP and RQ respectively, connect EF. Auxiliary lines: Extend PQ and SR to meet at T, extend EF and PQ to meet at L, EL and RT to meet at N, connect PQ, take the midpoint M of PR, connect EM and FM. PQ equals a, SR equals b, angle STP equals theta, angle SNE is angle 2.", "ocr_zh": "凸四边形PQRS，E，F分别为SP，RQ中点，连接EF。辅助线：延长PQ，SR交于T，延长EF，PQ交于L，EL，RT交于N，连接PQ，取PR中点M，连接EM，FM，PQ等于a，SR等于b，角STP等于theta，角SNE是角2", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_3897.jpg" }, { "index": 3898, "ocr_en": "Equilateral triangle A1B1C1 and equilateral triangle A2B2C2. Connect A1A2, B1B2, C1C2 (dashed lines). Take the midpoints A, B, C of A1A2, B1B2, C1C2 respectively, and connect AB, BC, AC. Extend A1B1 and A2B2 to meet at X (omitted in the diagram), extend A1C1 and A2C2 to meet at Y (omitted in the diagram), A1C1 and A2B2 intersect at Z.", "ocr_zh": "等边三角形A1B1C1和等边三角形A2B2C2,连接A1A2,B1B2,C1C2（虚线），取A1A2,B1B2,C1C2中点A,B,C ,连接AB，BC，AC，延长A1B1，A2B2交于X（图中省略相交），延长A1C1，A2C1交于Y（图中省略相交），A1C1和A2B2交于Z", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_3898.jpg" }, { "index": 3899, "ocr_en": "Equilateral triangles ABC, CDE, and EHK (counterclockwise). Points A, D, and K are collinear, with D being the midpoint of AK. Connect DA, DK, DC, DE, DB, and DH. Draw the auxiliary line EB and extend it to intersect AD at P.", "ocr_zh": "等边三角形ABC，CDE，EHK（逆时针），A，D，K共线且D是AK中点，连接DA，DK，DC，DE，DB，DH，作辅助线EB并延长，交AD于P", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3899.jpg" }, { "index": 3900, "ocr_en": "Equilateral triangle ABC, take points A1 (closer to B) and A2 on BC, B1 (closer to C) and B2 on CA, C1 (closer to A) and C2 on AB, such that the convex hexagon A1A2B1B2C1C2 has equal side lengths. Connect A2B1, B2C1, C2A1, A1B2, B1C2, C1A2. Auxiliary line: Construct an equilateral triangle PA1A2 inside triangle ABC, and connect PB1, PB2, PC1, PC2, A1B1, B1C1, C1A1.", "ocr_zh": "正三角形ABC，在BC上取A1（靠近B）和A2,CA上取B1（靠近C）和B2，AB上取C1（靠近A）和C2，使凸六边形A1A2B1B2C1C2边长均相等，连接A2B1,B2C1,C2A1,A1B2,B1C2,C1A2.辅助线：在三角形ABC内作正三角形PA1A2，连接PB1,PB2,PC1,PC2,A1B1,B1C1,C1A1", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_3900.jpg" }, { "index": 3901, "ocr_en": "Convex quadrilateral ABCD, connect AC and BD intersecting at O. AB equals a, BC equals b, CD equals c, DA equals d, AC equals e, BD equals f.", "ocr_zh": "凸四边形ABCD，连接AC，BD交于O，AB等于a,BC等于b,CD等于c,DA等于d，AC等于e,BD等于f", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3901.jpg" }, { "index": 3902, "ocr_en": "Figure 1: Convex quadrilateral ABCD, draw AE to bisect angle BAD, intersecting BC at E. Draw CF to bisect angle ACD, intersecting AE at F. Angle CFE is theta.Figure 2: Concave quadrilateral ABCD, with angle C concave inward. Draw the angle bisectors of angle BAD and angle BCD, intersecting at F. AF intersects BC at E. Angle AFC is theta.Figure 3: Folded quadrilateral ABCD (AB and CD intersect). Draw the angle bisectors of angle BAD and angle BCD, intersecting at F. AF intersects CD at E, and CF intersects AB at G. Angle AFC is theta.", "ocr_zh": "图1：凸四边形ABCD，过A作AE平分角BAD交BC于E，过C作CF平分角ACD交AE于F，角CFE是theta；图2：凹四边形ABCD，角C向内凹，作角BAD和角BCD的角平分线AF，CF交于F，AF交BC于E，角AFC是theta；图3：折四边形ABCD（AB，CD相交），作角BAD和角BCD的角平分线AF，CF交于F，AF交CD于E，CF交AB于G，角AFC是theta", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3902.jpg" }, { "index": 3903, "ocr_en": "Convex quadrilateral ABCD (counterclockwise), connect AC. Auxiliary lines: Draw CH1 perpendicular to AB at H1, and draw CH2 perpendicular to AD at H2, intersecting the extension of AD at H2. Connect BD.", "ocr_zh": "凸四边形ABCD（逆时针），连接AC。辅助线：过C作CH1垂直于AB于H1，过C作CH2垂直于AD于H2交AD延长线于H2，连接BD", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3903.jpg" }, { "index": 3904, "ocr_en": "In the plane, there is a convex quadrilateral ABCD, where the length of AB is greater than that of AD, and the length of BC is equal to that of CD. Connect AC. On segment AB, near point B, there is a point D₁. Points D and D₁ are symmetric with respect to AC (that is, the length of AD is equal to that of AD₁). Connect CD₁ with a dashed line.", "ocr_zh": "在平面内有一个凸四边形 ABCD，其中 AB 的长度大于 AD 的长度，BC 的长度与 CD 的长度相等，连接 AC。在线段 AB 上靠近点 B 的地方有一点 D₁，点 D 与点 D₁关于 AC 对称（即 AD 的长度与 AD₁的长度相等），用虚线连接 CD₁。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3904.jpg" }, { "index": 3905, "ocr_en": "In the complete quadrilateral ABCDEF, connect AC, BD and EF with dashed lines.", "ocr_zh": "在完全四边形ABCDEF中，虚线连接AC,BD和EF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3905.jpg" }, { "index": 3906, "ocr_en": "In quadrilateral ABCD, connect AC and BD, which intersect at point M. BA is perpendicular to AC, and BD is perpendicular to CD, with the feet of the perpendiculars being A and D respectively. Draw a perpendicular line from point A to segment BM, intersecting segment BM at point H (where H is the foot of the perpendicular), and segment AH is a dashed line.", "ocr_zh": "在四边形 ABCD 中，连接 AC 和 BD，二者相交于点 M。BA 与 AC 垂直，BD 与 CD 垂直，垂足分别为 A 和 D。过点 A 作线段 BM 的垂线，交线段 BM 于点 H（H 即为垂足），且线段 AH 为虚线。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3906.jpg" }, { "index": 3907, "ocr_en": "In the convex quadrilateral ABCD, connect AC. The lengths of side CD and side BC are equal, and both ∠BAD and ∠BCD are obtuse angles.", "ocr_zh": "在凸四边形 ABCD 中，连接 AC，边CD和BC的长度相等，角BAD和角BCD均为钝角。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3907.jpg" }, { "index": 3908, "ocr_en": "In quadrilateral ABCD, connect AC and BD, where the lengths of segments AD, DC, and CB are equal.", "ocr_zh": "在四边形 ABCD 中，连接 AC 和 BD，其中线段 AD、DC、CB 的长度相等。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3908.jpg" }, { "index": 3909, "ocr_en": "In quadrilateral ABCD, connect AC and BD, which intersect at point O. Here, AD is parallel to BC, CA is the angle bisector of ∠BCD, and the length of segment CD is equal to that of AO, while the length of segment BC is equal to that of OD.", "ocr_zh": "在四边形 ABCD 中，连接 AC 和 BD，二者相交于点 O，其中AD和BC平行，CA是角BCD的角平分线，线段CD和AO相等，线段BC和OD相等。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3909.jpg" }, { "index": 3910, "ocr_en": "In the plane, there is a quadrilateral ABCD. Connect BD and connect AC with a dashed line. The length of segment AD is greater than that of AB, and the lengths of side BA and side BC are equal. Taking AD as a side, construct an equilateral triangle ADB' outside quadrilateral ABCD, with AB' and DB' as dashed lines. Let P be a point inside triangle ABD; connect AP, CP, and DP, and connect PB' and CB' with dashed lines.", "ocr_zh": "平面内存在一四边形 ABCD，连接 BD，用虚线连接 AC。其中，线段 AD 的长度大于 AB 的长度，边 BA 与边 BC 的长度相等。以 AD 为边，在四边形 ABCD 的外侧作一个正三角形 ADB'，且 AB' 和 DB' 为虚线。P 为三角形 ABD 内部一点，连接 AP、CP 和 DP，用虚线连接 PB' 和 CB'。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3910.jpg" }, { "index": 3911, "ocr_en": "In quadrilateral ABCD, the lengths of sides AB and AD are equal. Side BC is a dashed line, and the dashed line BC is extended to point E. Connect DE, BD, and AC with dashed lines. The lengths of segments CE and CD are equal.", "ocr_zh": "在四边形 ABCD 中，边 AB 和 AD 的长度相等。边 BC 为虚线，且将虚线 BC 延长至点 E，用虚线连接 DE、BD 和 AC。其中，线段 CE 和线段 CD 的长度相等。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3911.jpg" }, { "index": 3912, "ocr_en": "In quadrilateral ABCD, connect AC and BD, and the two segments intersect at point O. E is the midpoint of side AB; there is a point F on side BC near point B, a point G on side CD near point C, and a point H on side AD near point D. Connect EF, FG, GH, and EH, and connect BH with a dashed line.", "ocr_zh": "在四边形 ABCD 中，连接 AC 和 BD，两条线段相交于点 O。AB 边的中点为 E；BC 边上靠近点 B 的位置有一点 F；CD 边上靠近点 C 的位置有一点 G；AD 边上靠近点 D 的位置有一点 H。连接 EF、FG、GH 和 EH，用虚线连接 BH。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3912.jpg" }, { "index": 3913, "ocr_en": "In the quadrilateral ABCD, the lengths of sides AD and CD are equal, both being 1, and DC is perpendicular to BC, DA is perpendicular to AB, with the feet of the perpendiculars being point C and point A respectively. Extend BC and AD, and their extensions intersect at point P.", "ocr_zh": "在四边形ABCD中，边AD和CD的长度相等，均为1，且DC与BC垂直，DA与AB垂直，垂足分别为C点和A点，延长BC和AD，二者的延长线相交于点P。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3913.jpg" }, { "index": 3914, "ocr_en": "In the complete quadrilateral ABCDEF, connect CE and BF, and extend BF and CE, which intersect at a point G. Connect AD, which intersects segment BF at point M; extend AD, which intersects segment CE at point N.", "ocr_zh": "在完全四边形 ABCDEF 中，连接 CE 和 BF，并延长 BF 和 CE，二者相交于一点 G。连接 AD，交线段 BF 于点 M；延长 AD，交线段 CE 于点 N。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3914.jpg" }, { "index": 3915, "ocr_en": "In the complete quadrilateral ABCDEF, connect AD, BF, and CE. The midpoints of segments AD, BF, CE, CD, BD, and BC are M, N, P, Q, R, and S respectively. Connect MN, PN, NR, RS, MR, QR, SQ, and PQ with dashed lines.", "ocr_zh": "在完全四边形 ABCDEF 中，连接 AD、BF 和 CE。其中，线段 AD、BF、CE、CD、BD、BC 各自的中点分别为 M、N、P、Q、R、S，用虚线连接 MN、PN、NR、RS、MR、QR、SQ、PQ。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3915.jpg" }, { "index": 3916, "ocr_en": "There are seven points in the plane, which are A, G, F, E, D, C, B in clockwise order, with A at the upper left. No three of these seven points are collinear. Connect AC, CE, EG, GB, BD, DF, FA to form a heptagram.", "ocr_zh": "在平面内有七个点，按照顺时针的顺序依次为 A、G、F、E、D、C、B，其中 A 在左上方，这七个点中任意三点都不共线，连接 AC、CE、EG、GB、BD、DF、FA，构成一个七角星。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3916.jpg" }, { "index": 3917, "ocr_en": "In quadrilateral ABCD, the length of BC is greater than that of CD, and the length of CD is greater than that of DA. Let O be the midpoint of side AB, and connect DO and CO.", "ocr_zh": "在四边形 ABCD 中，BC 的长度大于 CD 的长度，CD 的长度大于 DA 的长度。AB 边的中点为 O，连接 DO 和 CO。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3917.jpg" }, { "index": 3918, "ocr_en": "In triangle ABC, the lengths of sides AB and AC are equal. There is a point D on side AC; draw a line through D parallel to AB, intersecting side BC at point E. Connect DE and BD. Let F be the midpoint of side BD, and connect AF. There is a point O₁ inside triangle CED; outside triangle ABC and to the upper-left of O₁, there is a point O₂, which is also to the lower-left of A and to the upper-right of B. Extend AF with a dashed line to point G (where G is also outside triangle ABC) such that the length of segment FG is equal to that of AF; connect BG, CG, EG, O₁G, O₁E, O₁C, O₁A, O₁D, O₁O₂, and O₂D with dashed lines; connect O₁F and O₂F.", "ocr_zh": "在三角形 ABC 中，AB 边与 AC 边的长度相等。AC 边上有一点 D，过点 D 作 AB 的平行线，与边 BC 相交于点 E，连接 DE 和 BD。BD 边上的中点为 F，连接 AF。在三角形 CED 内部有一点 O₁；在三角形 ABC 外部、O₁的左上方有一点 O₂，且 O₂同时位于 A 的左下方、B 的右上方。用虚线延长 AF 至点 G（G 也在三角形 ABC 外部），使得线段 FG 与 AF 的长度相等；用虚线连接 BG、CG、EG、O₁G、O₁E、O₁C、O₁A、O₁D、O₁O₂、O₂D；连接 O₁F 和 O₂F。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3918.jpg" }, { "index": 3919, "ocr_en": "In the parallelogram ABCD, connect BD. Draw a perpendicular line from point A to side BC, intersecting side BC at point F (F is the foot of the perpendicular and is close to point B), and AF intersects segment BD at point E. Let M be the midpoint of segment DE, and connect AM with a dashed line.", "ocr_zh": "在平行四边形 ABCD 中，连接 BD。过点 A 作 BC 边的垂线，与 BC 边相交于点 F（F 为垂足，且 F 靠近点 B），AF 与线段 BD 相交于点 E。线段 DE 的中点为 M，用虚线连接 AM。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3919.jpg" }, { "index": 3920, "ocr_en": "In the parallelogram ABCD, connect AC and BD, and the two segments intersect at point O. Extend AB to point E, connect OE, and OE intersects side BC at point F. Draw a line through point E parallel to BC, which intersects the extension of OB at point G. Connect BG and EG with dashed lines.", "ocr_zh": "在平行四边形 ABCD 中，连接 AC 和 BD，两条线段相交于点 O。延长 AB 至点 E，连接 OE，OE 与 BC 边相交于点 F。过点 E 作 BC 的平行线，该平行线与 OB 的延长线相交于点 G，用虚线连接 BG 和 EG。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3920.jpg" }, { "index": 3921, "ocr_en": "In the parallelogram ABCD, there is a point P inside it. Connect AC and BD, and the two segments intersect at point O, with point P located to the upper-right of point O. Connect PA, PD, and PO with dashed lines, and connect PC and PB. Let N be the midpoint of segment PC, and M be the midpoint of segment PB. Connect DM and AN, and these two segments intersect at point Q.", "ocr_zh": "在平行四边形 ABCD 内部有一点 P，连接 AC 和 BD，两条线段相交于点 O，且点 P 位于点 O 的右上方。用虚线连接 PA、PD、PO，连接 PC 和 PB。线段 PC 的中点为 N，线段 PB 的中点为 M，连接 DM 和 AN，这两条线段相交于点 Q。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3921.jpg" }, { "index": 3922, "ocr_en": "In the parallelogram ABCD, connect BD. On segment BD, there are points P₁, P₂, Pₙ₋₂, Pₙ₋₁ in order from B to D, where P₁ and P₂ are close to point B, and Pₙ₋₂ and Pₙ₋₁ are close to point D. Connect AP₂ and extend it to intersect side BC at point E; connect APₙ₋₂ and extend it to intersect side CD at point F, then connect EF.", "ocr_zh": "在平行四边形 ABCD 中，连接 BD。在线段 BD 上，从 B 到 D 依次有点 P₁、P₂、Pₙ₋₂、Pₙ₋₁，其中 P₁、P₂靠近点 B，Pₙ₋₂、Pₙ₋₁靠近点 D。连接 AP₂并延长，与 BC 边相交于点 E；连接 APₙ₋₂并延长，与 CD 边相交于点 F，然后连接 EF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3922.jpg" }, { "index": 3923, "ocr_en": "In the plane, there are three figures, denoted as (1), (2), and (3) from left to right, with the labels placed below their respective figures.(1) In parallelogram AB'CD, connect AC and BD, which intersect at point O. There is a point B on segment OB', and connect AB and CB;(2) In parallelogram ABCD, connect AC and BD, which intersect at point O;(3) In quadrilateral ABCD, connect AC and BD, which intersect at point O. The two pairs of opposite sides of the quadrilateral are not parallel, O is the midpoint of segment BD, and the measures of ∠ADC and ∠ABC are equal.", "ocr_zh": "平面内存在三个图形，从左至右分别记为（1），（2），（3），标记在各自的图形下方。（1）：平行四边形AB'CD中连接AC和BD，二者相交于点O，在线段OB'上存在一点B，连接AB和CB；（2）：在平行四边形ABCD中连接AC和BD，二者相交于点O；（3）：在四边形ABCD中连接AC和BD，二者相交于点O，且四边形的两组对边均不平行，O为线段BD的中点，角ADC和角ABC大小相等。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3923.jpg" }, { "index": 3924, "ocr_en": "In quadrilateral ABCD, connect AC and BD, which intersect at point P. There is a point E on side AD near point D; connect EP and extend it to intersect side BC at point F, with the lengths of segments PE and PF being equal. Extend PA to point M with a dashed line, and extend PC to point N with a dashed line, such that the length of segment MA is equal to that of AE, and the length of segment CN is equal to that of CF. Then connect ME, MF, NE, and NF with dashed lines.", "ocr_zh": "在四边形 ABCD 中，连接 AC 和 BD，二者相交于点 P。在 AD 边上靠近 D 点的地方有一点 E，连接 EP 并延长，与 BC 边相交于点 F，且线段 PE 和线段 PF 的长度相等。用虚线延长 PA 至点 M，用虚线延长 PC 至点 N，使得线段 MA 与线段 AE 的长度相等，线段 CN 与线段 CF 的长度相等，再用虚线连接 ME、MF、NE、NF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3924.jpg" }, { "index": 3925, "ocr_en": "In the convex quadrilateral ABCD, connect AC and BD, which intersect at point M. Let E be the midpoint of side AB, and F be the midpoint of side BC; connect EF. Connect ED and FD, which intersect segment AC at points P and Q respectively. Here, P is on the left of M, Q is on the right of M, and the lengths of segments AP, PQ, and QC are all equal. Connect BP and BQ with dashed lines.", "ocr_zh": "在凸四边形 ABCD 中，连接 AC 和 BD，二者相交于点 M。AB 边上的中点为 E，BC 边上的中点为 F，连接 EF。连接 ED 和 FD，它们分别与线段 AC 相交于点 P 和 Q，其中 P 在 M 的左侧，Q 在 M 的右侧，且线段 AP、PQ、QC 的长度均相等，用虚线连接 BP 和 BQ。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3925.jpg" }, { "index": 3926, "ocr_en": "In quadrilateral ABCD, connect AC and BD. Draw a line through point D parallel to AC, a line through point A parallel to BD, a line through point C parallel to BD, and a line through point B parallel to AC. Among them, the line through D intersects the line through C at point M and intersects the line through A at point N; the line through B intersects the line through A at point P and intersects the line through C at point Q, thus forming parallelogram PQMN, with sides PN, PQ, QM, and MN all as dashed lines.", "ocr_zh": "在四边形 ABCD 中，连接 AC 和 BD。过点 D 作 AC 的平行线，过点 A 作 BD 的平行线，过点 C 作 BD 的平行线，过点 B 作 AC 的平行线。其中，过点 D 的平行线与过点 C 的平行线相交于点 M，与过点 A 的平行线相交于点 N；过点 B 的平行线与过点 A 的平行线相交于点 P，与过点 C 的平行线相交于点 Q，由此得到平行四边形 PQMN，且边 PN、PQ、QM、MN 均为虚线。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3926.jpg" }, { "index": 3927, "ocr_en": "In the plane, there are two figures, denoted as (1) and (2) from left to right, with the labels placed below their respective figures.(1) In the convex quadrilateral PMQN, connect PQ and MN, which intersect at point H. PQ is perpendicular to MN, and the foot of the perpendicular is H;(2) There is a point Q inside triangle PMN. Connect PQ and extend it with a dashed line to intersect side MN at point H. Connect QM and QN. Line PQ is perpendicular to MN, with the foot of the perpendicular being H.", "ocr_zh": "平面内存在两个图形，从左至右分别记为（1），（2），标记在各自的图形下方。（1）：在凸四边形PMQN中连接PQ和MN，二者相交于点H，PQ与MN垂直且垂足为H；（2）：在三角形PMN内存在一点Q，连接PQ并虚线延长，与MN边相交于点H，连接QM和QN，直线PQ与MN垂直，垂足为H。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3927.jpg" }, { "index": 3928, "ocr_en": "In pentagon ABCDE, M is the midpoint of side AB, N is the midpoint of side CD, P is the midpoint of side BC, and Q is the midpoint of side DE. Connect MN and PQ, and connect BE with a dashed line. K is the midpoint of segment MN, L is the midpoint of segment PQ, with K located to the upper-left of L; R is the midpoint of segment BE. Connect KL, and connect MR, QR, QN, NP, PR, and NR with dashed lines.", "ocr_zh": "在五边形 ABCDE 中，AB 边的中点为 M，CD 边的中点为 N，BC 边的中点为 P，DE 边的中点为 Q。连接 MN 和 PQ，用虚线连接 BE。线段 MN 的中点为 K，线段 PQ 的中点为 L，且 K 位于 L 的左上方；线段 BE 的中点为 R。连接 KL，用虚线连接 MR、QR、QN、NP、PR、NR。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3928.jpg" }, { "index": 3929, "ocr_en": "In the plane, there are two figures, denoted as (1) and (2) from left to right, with the labels placed below their respective figures.(1) In the convex quadrilateral PMQN, connect PQ and MN, where PQ and MN are perpendicular to each other. The midpoints of segments PM, PN, QM, QN, MN, and PQ are S, I, J, T, L, and R respectively. Connect SI, IT, JT, SJ, SL, LT, TR, SR, IL, IR, JR, and JL with dashed lines;(2) In triangle PMN, there is a point Q inside it. Connect PQ, where PQ and MN are perpendicular to each other, and connect QM and QN. The midpoints of segments PM, PN, MN, PQ, MQ, and NQ are S, I, L, R, J, and T respectively. Connect RS, SJ, JL, TL, TI, IR, SI, IL, LS, JT, TR, and JR with dashed lines.", "ocr_zh": "平面内存在两个图形，从左至右分别记为（1），（2），标记在各自的图形下方。（1）：在凸四边形 PMQN 中连接 PQ 和 MN，且PQ和MN互相垂直，线段 PM、PN、QM、QN、MN、PQ 上各自的中点分别为S、I、J、T、L、R，虚线连接 SI、IT、JT、SJ、SL、LT、TR、SR、IL、IR、JR、JL；（2）：在三角形 PMN 内存在一点 Q，连接 PQ，且PQ和MN互相垂直，连接 QM 和 QN，线段 PM、PN、MN、PQ、MQ、NQ 各自的中点分别为 S、I、L、R、J、T，虚线连接 RS、SJ、JL、TL、TI、IR、SI、IL、LS、JT、TR、JR。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_3929.jpg" }, { "index": 3930, "ocr_en": "In parallelogram ABCD, there are points E and P in sequence from A to B on side AB (following the order of AB); there are points N and T in sequence on side BC (following the order of BC). Draw lines through points N and T parallel to AB, which intersect side AD at points M and S in sequence; draw lines through points E and P parallel to BC, which intersect side DC at points F and Q in sequence. Segment MN intersects EF at point G and intersects PQ at point H; segment ST intersects EF at point L and intersects PQ at point K.", "ocr_zh": "在平行四边形 ABCD 中，边 AB 上从 A 到 B 依次存在点 E 和 P（按 AB 的顺序）；边 BC 上依次存在点 N 和 T（按 BC 的顺序）。过点 N 和 T 分别作 AB 的平行线，与边 AD 依次相交于点 M 和 S；过点 E 和 P 分别作 BC 的平行线，与边 DC 依次相交于点 F 和 Q。线段 MN 与 EF 相交于点 G，与 PQ 相交于点 H；线段 ST 与 EF 相交于点 L，与 PQ 相交于点 K。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3930.jpg" }, { "index": 3931, "ocr_en": "Inside triangle ABC, there is a point Q. Draw a line through Q parallel to AC, intersecting side AB at point I and side BC at point H; draw a line through Q parallel to AB, intersecting side AC at point G and side BC at point F; draw a line through Q parallel to BC, intersecting side AB at point E and side AC at point D. Denote the areas of parallelograms CDQH, AIQG, and QEBF as S₂, S₃, and S₁ respectively.", "ocr_zh": "在三角形 ABC 内部有一点 Q。过点 Q 作 AC 的平行线，与边 AB 相交于点 I，与边 BC 相交于点 H；过点 Q 作 AB 的平行线，与边 AC 相交于点 G，与边 BC 相交于点 F；过点 Q 作 BC 的平行线，与边 AB 相交于点 E，与边 AC 相交于点 D。记平行四边形 CDQH、AIQG、QEBF 的面积分别为 S₂、S₃、S₁。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3931.jpg" }, { "index": 3932, "ocr_en": "In triangle ABC, there is a point P inside it. Draw a line through P parallel to AC, intersecting side AB at point I and side BC at point H; draw a line through P parallel to AB, intersecting side AC at point D and side BC at point E; draw a line through P parallel to BC, intersecting side AB at point F and side AC at point G, and the lengths of segments DE, FG, and HI are all equal.", "ocr_zh": "在三角形 ABC 内部有一点 P。过点 P 作 AC 的平行线，与边 AB 相交于点 I，与边 BC 相交于点 H；过点 P 作 AB 的平行线，与边 AC 相交于点 D，与边 BC 相交于点 E；过点 P 作 BC 的平行线，与边 AB 相交于点 F，与边 AC 相交于点 G，且线段 DE、FG 和 HI 的长度均相等。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3932.jpg" }, { "index": 3933, "ocr_en": "In rectangle ABCD, the length of side AB is greater than that of side BC. There is a point F to the upper-left of point A and to the lower-left of point D; there is a point E to the upper-right of point B and to the lower-right of point C. Connect EF, DF, AF, CE, and BE; extend FA and EB with dashed lines, and these two extensions intersect at point G. The length of segment AF is equal to that of CE, the length of segment DF is equal to that of BE, and ∠AFD, ∠BEC, and ∠AGB are all right angles.", "ocr_zh": "在矩形 ABCD 中，AB 边的长度大于 BC 边的长度，在 A 点的左上方、D 点的左下方有一点 F，在 B 点的右上方、C 点的右下方有一点 E。连接 EF、DF、AF、CE、BE，用虚线延长 FA 和 EB，两条延长线相交于点 G，且线段AF和CE长度相等，线段DF和BE长度相等，角AFD，角BEC和角AGB均为直角。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3933.jpg" }, { "index": 3934, "ocr_en": "In rectangle ABCD, the length of side AB is less than that of side AD. Connect AC with a dashed line, and let O be the midpoint of segment AC. Draw a perpendicular line to AC through point O (with O as the foot of the perpendicular), which intersects side AD at point F and side BC at point E. Connect CF with a dashed line. It is obvious that the lengths of CE and CF are equal, meaning points E and F are symmetric with respect to diagonal AC.", "ocr_zh": "在矩形 ABCD 中，AB 边的长度小于 AD 边的长度。用虚线连接 AC，线段 AC 的中点为 O。过点 O 作 AC 的垂线（垂足为点 O），该垂线与 AD 边相交于点 F，与 BC 边相交于点 E，用虚线连接 CF。显然，CE 和 CF 的长度相等，即点 E 和点 F 关于对角线 AC 对称。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3934.jpg" }, { "index": 3935, "ocr_en": "The four sides of rectangle ABCD are all dashed lines. There is a point E on side AB near point A, a point H on side AD near point A, a point G on side CD near point C, and a point F on side BC near point C. Connect EG, HE, EF, FG, and GH; there are points P and Q in sequence on segment EG (following the order of EG). Point P is symmetric to point A with respect to EH, point C is symmetric to point Q with respect to FG, point B is symmetric to point Q with respect to EF, and point D is symmetric to point P with respect to HG. Connect HP and QF.", "ocr_zh": "矩形 ABCD 的四条边均为虚线。边 AB 上靠近点 A 的位置有一点 E，AD 边上靠近点 A 的位置有一点 H，CD 边上靠近点 C 的位置有一点 G，BC 边上靠近点 C 的位置有一点 F。连接 EG、HE、EF、FG、GH，在线段 EG 上依次有点 P 和 Q（按 EG 的顺序排列）。其中，点 P 与点 A 关于 EH 对称，点 C 与点 Q 关于 FG 对称，点 B 与点 Q 关于 EF 对称，点 D 与点 P 关于 HG 对称，连接 HP 和 QF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3935.jpg" }, { "index": 3936, "ocr_en": "In triangle ABC, construct an inscribed rhombus EFGH, where point E lies on side AB, point F lies on side AC, and points H and G both lie on side BC. Point H is located to the lower-right of E, point G is located to the lower-right of F, with H close to point B and G close to point C. The lengths of segments AF, EF, CG, and EB are all equal.", "ocr_zh": "在三角形 ABC 内作一个内接菱形 EFGH，其中点 E 在边 AB 上，点 F 在边 AC 上，点 H 和点 G 均在边 BC 上，且 H 位于 E 的右下方，G 位于 F 的右下方，同时 H 靠近点 B，G 靠近点 C。线段 AF、EF、CG、EB 的长度均相等。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3936.jpg" }, { "index": 3937, "ocr_en": "In rectangle ABCD, extend CD to point Q, and extend BC to point P. Connect QA, QP, and AP, where segment AP intersects side CD at point E. Draw a line through E parallel to BC, intersecting side AB at point F, and connect EF with a dashed line. The measures of ∠PAD and ∠QAD are equal.", "ocr_zh": "在矩形ABCD中，延长CD至点Q，延长BC至点P，连接QA,QP,AP，其中线段AP交CD边于点E，过点E做BC的平行线，交AB边于点F，虚线连接EF，角PAD和角QAD大小相等。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3937.jpg" }, { "index": 3938, "ocr_en": "In the plane, there is a convex quadrilateral ABCD, with AC connected. AC is perpendicular to BC, with the foot of the perpendicular at point C; AD is perpendicular to CD and they are equal in length, with the foot of the perpendicular at point D. E is the midpoint of side AB, and F is the midpoint of side BC. Connect DE and CE, where segment DE intersects segment AC at point M. Connect EF with a dashed line. Among them, ∠DAC is 45 degrees, and ∠CAB is 30 degrees.", "ocr_zh": "平面内存在一凸四边形 ABCD，连接 AC，其中 AC 与 BC 垂直，垂足为 C 点；AD 与 CD 垂直且相等，垂足为 D 点。AB 边上的中点为 E，BC 边上的中点为 F，连接 DE 和 CE，线段 DE 与线段 AC 相交于点 M，用虚线连接 EF。其中，角 DAC 为 45 度，角 CAB 为 30 度。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3938.jpg" }, { "index": 3939, "ocr_en": "In the convex quadrilateral ABCD, connect AC and BD. There is an inscribed rhombus EFGH in the quadrilateral, with point E on side AB, point F on side BC, point G on side CD, and point H on side AD. Moreover, EF is parallel to AC, and FG is parallel to BD.", "ocr_zh": "在凸四边形 ABCD 中，连接 AC 和 BD。四边形内接有一个菱形 EFGH，其中点 E 在 AB 边上，点 F 在 BC 边上，点 G 在 CD 边上，点 H 在 AD 边上，且 EF 与 AC 平行，FG 与 BD 平行。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3939.jpg" }, { "index": 3940, "ocr_en": "In the plane, there is a triangle BEF. There is a point D on side EF, a point A on side BE near point E, and a point C on side BF near point B, with point C located to the upper-right of point A. Connect AC, CD, and DA; connect AF and CE, and the two segments intersect at point M, which is inside triangle ACD. Denote line EF as l, where the lengths of segments AB, AD, BC, and CD are all equal.", "ocr_zh": "平面内有一个三角形 BEF，边 EF 上有一点 D，边 BE 上靠近点 E 的位置有一点 A，边 BF 上靠近点 B 的位置有一点 C，且点 C 在点 A 的右上方。连接 AC、CD、DA，连接 AF 和 CE，这两条线段相交于点 M，点 M 在三角形 ACD 的内部。记直线 EF 为 l，其中线段 AB、AD、BC、CD 的长度均相等。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3940.jpg" }, { "index": 3941, "ocr_en": "In the plane, there are two figures, denoted as (1) and (2) from left to right, with the labels placed below their respective figures.(1) In the convex quadrilateral ABCD, connect AC and BD, which intersect at point O. There is a point N on segment OD near D such that the lengths of OB and ON are equal, denoted as b; there is a point M on segment OC near C such that the lengths of OA and OM are equal, denoted as a. Connect AN, MN, and BM with dashed lines, where ABMN is a parallelogram.(2) In rhombus ABCD, connect AC and BD, which intersect at point O. Denote the lengths of both OA and OC as a, and the lengths of both OB and OD as b.", "ocr_zh": "平面内存在两个图形，从左至右分别记为（1），（2），标记在各自的图形下方。（1）：在凸四边形ABCD中连接AC和BD，二者相交于点O，在线段OD上靠近D的位置上存在一点N，使得OB和ON长度相等，记为b，在线段OC上靠近C的位置上存在一点M，使得OA和OM长度相等，记为a，虚线连接AN,MN,BM，ABMN为平行四边形。；（2）：在菱形ABCD中连接AC和BD，二者相交于点O，记OA和OC的长度均为a，记OB和OD的长度均为b。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3941.jpg" }, { "index": 3942, "ocr_en": "In the plane, there is a square ABCD. There is a point M on side BC near point B, and a point N on side CD near point D. Extend CD to point M' such that the lengths of BM and DM' are equal, then connect AM, MN, AN, and AM'.", "ocr_zh": "平面内有一个正方形 ABCD，边 BC 上靠近点 B 的位置有一点 M，边 CD 上靠近点 D 的位置有一点 N。延长 CD 至点 M'，使得 BM 与 DM' 的长度相等，连接 AM、MN、AN、AM'。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3942.jpg" }, { "index": 3943, "ocr_en": "In square ABCD, there is a point P on side CD. Extend CB to point E such that the lengths of CB and BE are equal, and connect EP, which intersects side AB at point F.", "ocr_zh": "在正方形 ABCD 的 CD 边上有一点 P，延长 CB 至点 E，使得 CB 与 BE 的长度相等，连接 EP，该线段与 AB 边相交于点 F。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3943.jpg" }, { "index": 3944, "ocr_en": "In square ABCD, there is a point N on side DC, a point M near point A on side AD, and a point E on side AB. Connect EN with a dashed line, connect MB and MN. Segment EN intersects segment MB at point F, and there is a point O on segment MB. Extend MN and BC with dashed lines, and the two extensions intersect at point T. Connect OT with a dashed line.", "ocr_zh": "在正方形 ABCD 中，边 DC 上的中点为 N，边 AD 上靠近点 A 的位置有一点 M，边 AB 上的中点为 E。用虚线连接 EN，连接 MB 和 MN，线段 EN 与线段 MB 相交于点 F，线段 MB 上的中点为 O。用虚线延长 MN 和 BC，两条延长线相交于点 T，用虚线连接 OT。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3944.jpg" }, { "index": 3945, "ocr_en": "Inside square ABCD, there is a point P such that triangle BPC is an equilateral triangle. Connect BP, CP, DP, and BD. Draw a perpendicular line from P to side CD, intersecting side CD at point E (where E is the foot of the perpendicular), and PE is a dashed line.", "ocr_zh": "在正方形 ABCD 内有一点 P，使得三角形 BPC 是正三角形。连接 BP、CP、DP、BD，过点 P 作 CD 边的垂线，交 CD 边于点 E（E 为垂足），且 PE 为虚线。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3945.jpg" }, { "index": 3946, "ocr_en": "On the sides AB, BC, CD, and DA of square ABCD, there are points E, F, G, and H respectively, with point E close to point A, point F close to point B, point G close to point C, and point H close to point D. Connect EF, FG, GH, HE, EG, and HF. Both ∠BEG and ∠CFH are acute angles. Draw perpendicular lines from points E, F, G, and H to their opposite sides respectively, intersecting the opposite sides at points K, L, I, and J (these four points are the corresponding feet of the perpendiculars). Connect EK, IG, LF, and HJ with dashed lines. Segment EK intersects LF at point P and intersects HJ at point T; segment IG intersects LF at point Q and intersects HJ at point R (note: points I, J, K, and L are all set points, and these points are not marked with letters in the figure).", "ocr_zh": "在正方形 ABCD 的边 AB、BC、CD、DA 上分别有一点 E、F、G、H，其中点 E 靠近点 A，点 F 靠近点 B，点 G 靠近点 C，点 H 靠近点 D。连接 EF、FG、GH、HE、EG、HF，角BEG 和角CFH 均为锐角。分别过点 E、F、G、H 作对边的垂线，交对边于点 K、L、I、J（这四个点即为对应的垂足），用虚线连接 EK、IG、LF、HJ。线段 EK 与 LF 相交于点 P，与 HJ 相交于点 T；线段 IG 与 LF 相交于点 Q，与 HJ 相交于点 R（注意：点 I、J、K、L 均为设定的点，图中并未用字母标注这些点）。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3946.jpg" }, { "index": 3947, "ocr_en": "Inside square ABCD, connect diagonals AC and BD, and these two diagonals intersect at point O.", "ocr_zh": "在正方形 ABCD 内，连接对角线 AC 和 BD，这两条对角线相交于点 O。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3947.jpg" }, { "index": 3948, "ocr_en": "In the plane, there is a square ABCD. Extend BC to point E, connect AE, and connect AC with a dashed line. Construct a square AEFG with AE as a side, where points A and G are on the same side of BE. Draw a perpendicular line from point F to BC, intersecting the extension of BE at point M. Connect BD, which intersects segment AC at point O. Extend BD to intersect segment AF at point H and the extension of MF at point N. Note that segments HN, ME, MN, and AC are all dashed lines.", "ocr_zh": "平面内有一个正方形 ABCD，延长 BC 至点 E，连接 AE，用虚线连接 AC。以 AE 为边作一个正方形 AEFG，且点 A、G 在 BE 的同侧。过点 F 作 BC 的垂线，与 BE 的延长线相交于点 M，连接 BD，BD 与线段 AC 相交于点 O。延长 BD，与线段 AF 相交于点 H，与 MF 的延长线相交于点 N。注意，线段 HN、ME、MN、AC 均为虚线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3948.jpg" }, { "index": 3949, "ocr_en": "In the plane, there is a square ABCD. Rotating it 60 degrees counterclockwise around point A results in a new square AB'C'D'. Segment B'C' intersects segment CD at point K, with point B' inside square ABCD. Draw a line through point B' parallel to AD, intersecting segment CK at point N and side AB at point M, where MN is a dashed line. Connect AK with a dashed line, and the measure of ∠B'AB is 60 degrees.", "ocr_zh": "平面内存在一正方形ABCD，绕A点逆时针旋转60度，得到一个新正方形AB'C'D'，线段B'C'与线段CD相交于点K，B'在正方形ABCD内部，过该点做AD的平行线，交线段CK于点N，交AB边于点M，且MN为虚线，虚线连接AK，角B'AB的大小为60度。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3949.jpg" }, { "index": 3950, "ocr_en": "In the plane, there is a triangle ABC. Construct a square ABDE outside the triangle with side AB, and construct a square ACFG outside the triangle with side AC. Connect BG and CE, and these two segments intersect at point K; connect AK and DF, and connect EG, CD, BF, AF, and CG with dashed lines. Let M be the midpoint of segment EG; connect AM with a dashed line and extend it to point P such that the lengths of AM and PM are equal. Connect PE and PG with dashed lines, and extend MA with a dashed line, where the extension intersects side BC at point H.", "ocr_zh": "平面内有一个三角形 ABC，以边 AB 为一边向三角形外部作一个正方形 ABDE，以边 AC 为一边向三角形外部作一个正方形 ACFG。连接 BG 和 CE，这两条线段相交于点 K，连接 AK 和 DF，用虚线连接 EG、CD、BF、AF、CG。线段 EG 的中点为 M，用虚线连接 AM 并将其延长至点 P，使得 AM 与 PM 的长度相等，用虚线连接 PE 和 PG，用虚线延长 MA，该延长线与 BC 边相交于点 H。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_3950.jpg" }, { "index": 3951, "ocr_en": "In the plane, there is a triangle ABC. Construct a square ABMN inside the triangle with AB as a side, where point C lies inside square ABMN; construct a square ACPQ inside triangle ABC with AC as a side, where point B lies outside square ACPQ. Connect BQ and CN.", "ocr_zh": "平面内有一个三角形 ABC，以 AB 边为一边向三角形内部作一个正方形 ABMN，点 C 在正方形 ABMN 的内部；以 AC 边为一边，向三角形 ABC 内部作一个正方形 ACPQ，点 B 在正方形 ACPQ 的外部。连接 BQ 和 CN。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3951.jpg" }, { "index": 3952, "ocr_en": "In the plane, there is a triangle ABC. Construct a square ABDE outside the triangle with side AB, and construct a square ACFG outside the triangle with side AC. Connect EG, AD, and AF; let O₁, M, O₂, and N be the midpoints of segments AD, BC, AF, and EG respectively. Connect BE, BG, CE, CG, and DF with dashed lines; segment BG intersects segment CE at point K, and let L be the midpoint of segment DF. Connect O₁N, NO₂, O₂M, and MO₁.", "ocr_zh": "平面内存在一个三角形 ABC，以边 AB 为一边向三角形外作一个正方形 ABDE，以边 AC 为一边向三角形外作一个正方形 ACFG。连接 EG、AD、AF，线段 AD、BC、AF、EG 各自的中点分别为 O₁、M、O₂、N。用虚线连接 BE、BG、CE、CG、DF，线段 BG 与线段 CE 相交于点 K，线段 DF 的中点为 L，连接 O₁N、NO₂、O₂M、MO₁。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_3952.jpg" }, { "index": 3953, "ocr_en": "In the plane, there are two non-congruent right triangles ABC and ADE, where ∠ABC and ∠ADE are right angles. The lengths of sides AD and DE are equal, the lengths of sides AB and BC are equal, and the length of AB is greater than that of AD. Triangle ADE is outside triangle ABC. Connect BD and CE; there is a point M on CE, and connect BM and DM.", "ocr_zh": "平面内存在两个不全等的直角三角形ABC和ADE，其中角ABC和角ADE为直角且边AD和DE长度相等，边AB和BC长度相等，AB的长度大于AD，三角形ADE在三角形ABC外部，连接BD和CE，在CE上存在一点M，连接BM和DM。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3953.jpg" }, { "index": 3954, "ocr_en": "In the plane, there is an obtuse triangle ABC with ∠ACB being the obtuse angle. Construct squares ACID and CBEJ outside the triangle with AC and BC as sides respectively, connect DE, and let M be the midpoint of segment DE.", "ocr_zh": "平面内有一个钝角三角形 ABC，其中角ACB 为钝角。分别以 AC、BC 为边，在三角形外部作正方形 ACID 和正方形 CBEJ，连接 DE，线段 DE 的中点为 M。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3954.jpg" }, { "index": 3955, "ocr_en": "On square ABCD, there is a point G on side AB and a point F on side BC. Draw a line through G parallel to BC, intersecting side CD at point H; draw a line through F parallel to AB, intersecting side AD at point E. Here, point G is close to point A, point F is close to point B, point H is close to point D, and point E is close to point A. Connect GH, EF, AH, and AF, with segment EF intersecting segment GH at point P.", "ocr_zh": "正方形 ABCD 的 AB 边上有一点 G，BC 边上有一点 F。过点 G 作 BC 的平行线，与 CD 边相交于点 H；过点 F 作 AB 的平行线，与 AD 边相交于点 E。其中，点 G 靠近点 A，点 F 靠近点 B，点 H 靠近点 D，点 E 靠近点 A。连接 GH、EF、AH、AF，线段 EF 与线段 GH 相交于点 P。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3955.jpg" }, { "index": 3956, "ocr_en": "On the AB and BC sides of square ABCD, there are points E and F respectively, with E close to point A and F close to point B. Connect AF and CE; the two segments intersect at point G, and connect BG.", "ocr_zh": "在正方形 ABCD 的 AB 边和 BC 边上分别存在一点 E 和 F，其中 E 靠近 A 点，F 靠近 B 点。连接 AF 和 CE，两条线段相交于点 G，连接 BG。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3956.jpg" }, { "index": 3957, "ocr_en": "In square ABCD, there is a point E on side AB and a point F on side BC; connect EF. Draw a perpendicular line from point D to segment EF, intersecting segment EF at point G (where G is the foot of the perpendicular), and the length of segment DG is equal to that of DA.", "ocr_zh": "在正方形 ABCD 中，AB 边上有一点 E，BC 边上有一点 F，连接 EF。过点 D 作线段 EF 的垂线，交线段 EF 于点 G（G 为垂足），且线段 DG 与 DA 的长度相等。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3957.jpg" }, { "index": 3958, "ocr_en": "In the plane, there is a triangle AFG. Construct squares AFED and AGCB outside the triangle with AF and AG as sides respectively. Connect DB. Draw a perpendicular line from point A to side FG, intersecting FG at point H (where H is the foot of the perpendicular). Connect HA and extend it to point K such that the length of segment AK is equal to that of FG. Draw a perpendicular line from point A to BD, intersecting segment BD at point S (where S is the foot of the perpendicular). Connect SA and extend it to point R such that the length of segment AR is equal to that of BD. Connect KD, KB, GD, BF, FR, and GR with dashed lines; connect KE, KC, CR, and ER.", "ocr_zh": "平面内有一个三角形 AFG，分别以 AF、AG 为边，在三角形外部作正方形 AFED 和正方形 AGCB。连接 DB，过点 A 作边 FG 的垂线，交 FG 于点 H（H 为垂足），连接 HA 并延长至点 K，使线段 AK 与 FG 的长度相等；过点 A 作 BD 的垂线，交线段 BD 于点 S（S 为垂足），连接 SA 并延长至点 R，使线段 AR 与 BD 的长度相等。用虚线连接 KD、KB、GD、BF、FR、GR，连接 KE、KC、CR、ER。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_3958.jpg" }, { "index": 3959, "ocr_en": "In the plane, there are two squares ABCD and AB₁C₁D₁, and the two squares are in the same direction. Both points B₁ and C₁ are inside square ABCD. Connect DD₁; connect BB₁ and extend it, and the extension intersects segment DD₁ at point X. Connect CC₁ and C₁X.", "ocr_zh": "平面内存在两个正方形 ABCD 和 AB₁C₁D₁，且两个正方形同向。点 B₁和点 C₁均在正方形 ABCD 内部，连接 DD₁；连接 BB₁并延长，延长线与线段 DD₁相交于点 X，连接 CC₁和 C₁X。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3959.jpg" }, { "index": 3960, "ocr_en": "In the plane, there is a right triangle ADG with ∠DAG as the right angle, and the length of side AD is greater than that of side AG. Construct square ABCD and square DGFE outside the triangle with AD and DG as sides respectively, where points B, A, and G are collinear. Connect CE.", "ocr_zh": "平面内存在一直角三角形 ADG，角DAG为直角且边AD的长度大于边AG的长度，分别以 AD、DG 为边，在三角形外部作正方形 ABCD 和正方形 DGFE，且点 B、A、G 三点共线，连接 CE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3960.jpg" }, { "index": 3961, "ocr_en": "On the AB side of square ABCD, there is a point K, and on the AD side, there is a point N. Connect BD, CN, and CK. Segment CN intersects segment BD at point M, segment CK intersects segment BD at point L, and connect AM.", "ocr_zh": "在正方形 ABCD 的 AB 边上存在一点 K，在 AD 边上存在一点 N。连接 BD、CN、CK，线段 CN 与线段 BD 相交于点 M，线段 CK 与线段 BD 相交于点 L，连接 AM。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3961.jpg" }, { "index": 3962, "ocr_en": "There is a triangle ABC in a plane. With side AC of the triangle as the base, points D and E outside triangle ABC are the other two vertices, and a square ACDE is constructed. With side BC of the triangle as the base, points F and G outside triangle ABC are the other two vertices, and a square BCGF is constructed. Connect EF, and point P is the midpoint of line segment EF.", "ocr_zh": "平面内有三角形ABC。以三角形的边AC为底边，三角形ABC外的点D、E为另外两个顶点，作正方形ACDE；以三角形的边BC为底边，三角形ABC外的点F、G为另外两个顶点，作正方形BCGF。连接EF，点P是线段EF的中点。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3962.jpg" }, { "index": 3963, "ocr_en": "In Figure (1) on the left, there is a triangle EDF in the plane. Using the side DE of the triangle as the base, and points A and B outside the triangle DEF as the other two vertices, construct a square ABDE. Using the side DF of the triangle as the base, and points C and I outside the triangle DEF as the other two vertices, construct a square DFIC. Using the side EF of the triangle as the base, and points H and G outside the triangle DEF as the other two vertices, construct a square EFGH, and connect AH, BC, and GI. The area of the square ABDE region is labeled as 17, the area of the square DFIC region is labeled as 10, and the area of the square EFGH region is labeled as 13;In Figure 2 on the right, there is a rectangle PDRQ in the plane. Point E is a point on line segment PQ, with line segment PE labeled as 1 and line segment EQ labeled as 2. Point F is a point on line segment QR, with line segment QF labeled as 3 and line segment FR labeled as 1. Connect EF, FD, and ED.", "ocr_zh": "在左边的图（1）中，平面内有三角形EDF。以三角形的边DE为底边，三角形DEF外的点A、B为另外两个顶点，作正方形ABDE。以三角形的边DF为底边，三角形DEF外的点C、I为另外两个顶点，作正方形DFIC。以三角形的边EF为底边，三角形DEF外的点H、G为另外两个顶点，作正方形EFGH，连接AH、BC、GI。正方形ABDE区域的面积标注为17，正方形DFIC区域的面积标注为10，正方形EFGH区域的面积标注为13；在右边的图（2）中，平面内有矩形PDRQ。点E是线段PQ上的一点，线段PE长度标注为1，线段EQ长度标注为2.点F是线段QR上的一点，线段QF长度标注为3，线段FR长度标注为1.连接EF、FD、ED。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3963.jpg" }, { "index": 3964, "ocr_en": "In a plane, there is an isosceles trapezoid ABCD, where segment AB is the lower base and segment DC is the upper base. Connect AC and BD, and segment AC intersects BD at point E. Extend AD and BC with dashed lines, and the two dashed lines intersect at point F. Connect EF and extend it downward, and segment EF intersects segment CD at point M. The extension of segment EF intersects AB at point N.", "ocr_zh": "平面内有等腰梯形ABCD，线段AB为下底，线段DC为上底，连接AC、BD，线段AC与BD相交于点E。以虚线延长AD、BC，两条虚线相交于点F。连接EF并向下延长，线段EF与线段CD相交于点M，线段EF的延长线与AB相交于点N。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3964.jpg" }, { "index": 3965, "ocr_en": "There is a trapezoid ADCB in the plane, with line segment AD as the upper base and line segment BC as the lower base. Draw point A and auxiliary line AH perpendicular to BC, with point H as the foot of the perpendicular. E is a point on line segment HC, and draw auxiliary line AE.", "ocr_zh": "平面内有梯形ADCB，线段AD为上底，线段BC为下底。作点A作辅助线AH垂直于BC，点H为垂足。E是线段HC上的一点，作辅助线AE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3965.jpg" }, { "index": 3966, "ocr_en": "In a plane, there is a trapezoid ADCB, where line segment AD is the upper base and line segment BC is the lower base. Extend line segment AD to a point E on the left side of the trapezoid, extend line segment BC to a point J on the right side of the trapezoid, and connect EJ. The intersection points of line segment EJ with AB, BD, AC, and DC are F, G, H, and I, respectively.", "ocr_zh": "在平面内有梯形ADCB，线段AD为上底，线段BC为下底。延长线段AD至梯形左侧的一点E，延长线段BC至梯形右侧的一点J，连接EJ，线段EJ与AB、BD、AC、DC的交点分别是F、G、H、I。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3966.jpg" }, { "index": 3967, "ocr_en": "There is a trapezoid ABCD in the plane, with segment AB as the lower base and segment CD as the upper base. Point G is the midpoint of segment DC, point H is the midpoint of segment AB, and GH is connected. Draw an auxiliary line CE through point C perpendicular to segment AB, with the foot of the perpendicular being E. Draw an auxiliary line CC1 through point C parallel to segment AD, with the intersection point of segment CC1 and segment AB being C1.", "ocr_zh": "平面内有梯形ABCD，线段AB为下底，线段CD为上底，点G是线段DC的中点，点H是线段AB的中点，连接GH。过点C作辅助线CE垂直于线段AB，垂足为E。过点C作辅助线CC1平行于线段AD，线段CC1与线段AB的交点为C1。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3967.jpg" }, { "index": 3968, "ocr_en": "In a plane, there is a trapezoid ABCD, where line segment AB is the lower base and line segment CD is the upper base. Connect AC, and point E is the midpoint of line segment AC. Connect BE and extend it, and the extension of line segment BE intersects line segment AD at point F. Extend line segment AD with a dashed line to line segment DC. Draw a dashed line CG parallel to BF through point C, and the intersection of line segment CG and the extension of line segment AD is point G.", "ocr_zh": "在平面内有梯形ABCD，线段AB为下底，线段CD为上底，连接AC，点E为线段AC中点。连接BE并延长，线段BE的延长线与线段AD相交于点F。以虚线延长线段AD至线段DC之上。过点C作虚线CG平行于BF，线段CG与线段AD延长线的交点为点G。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3968.jpg" }, { "index": 3969, "ocr_en": "In a plane, there is a trapezoid ADCB, where segment AD is the upper base and segment BC is the lower base. Point E is a point on segment AD, and segment CE is drawn. Extend segments CE and BA; the two extensions intersect at point F. Extend segment CD, and draw segment ME parallel to segment AB through point E; the intersection of segment ME and the extension of segment CD is point M. Draw segment MB; the intersection of segment MB and segment AD is point N, and the intersection of segment CF and segment MB is point P; draw segment FN.", "ocr_zh": "在平面内有梯形ADCB，线段AD为上底，线段BC为下底。点E是线段AD上的一点，连接CE。延长线段CE、BA，两条延长线相交于点F。延长线段CD，过点E作线段ME平行于线段AB，线段ME与CD延长线的交点为M。连接MB，线段MB与线段AD的交点为N，线段CF和线段MB的交点为P，连接FN。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3969.jpg" }, { "index": 3970, "ocr_en": "There is a trapezoid ABCD in a plane, with line segment AB as the upper base and line segment CD as the lower base. Draw an auxiliary line AF perpendicular to line segment CD through point A, with the foot of the perpendicular being F. Draw an auxiliary line BE perpendicular to line segment CD through point B, with the foot of the perpendicular being E.", "ocr_zh": "平面内有梯形ABCD，线段AB为上底，线段CD为下底。过点A作辅助线AF垂直于线段CD，垂足为F。过点B作辅助线BE垂直于线段CD，垂足为E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3970.jpg" }, { "index": 3971, "ocr_en": "In a plane, there is a trapezoid ADCB, where line segment AD is the upper base, line segment BC is the lower base, and BD and AC are connected. Draw an auxiliary line AF perpendicular to line segment BC through point A, with the foot of the perpendicular being F. Extend line segment CB with a dashed line, draw a dashed line AE parallel to line segment BD through point A, and the intersection of the extended dashed line of line segment CB and the dashed line segment AE is E.", "ocr_zh": "平面内有梯形ADCB，线段AD为上底，线段BC为下底，连接BD、AC。过点A作辅助线AF垂直于线段BC，垂足为F。以虚线延长线段CB，过点A作虚线AE与线段BD平行，线段CB的虚线延长线与虚线线段AE的交点为E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3971.jpg" }, { "index": 3972, "ocr_en": "In a plane, there is a right-angled trapezoid ABCD. Segment AD is the lower base, segment BC is the upper base, and segment AB is one of the heights of the right-angled trapezoid. The foot of the perpendicular from segment AB to segment CB is B, and the foot of the perpendicular from segment AB to segment AD is A. Segment CD is the hypotenuse. Connect AC and BD. Segment AC intersects BD at point E. Draw segment EF perpendicular to segment AB through point E, with the foot of the perpendicular being F. The midpoint of segment AB is O. Connect EO.", "ocr_zh": "平面内有直角梯形ABCD，线段AD为下底，线段BC为上底，线段AB是直角梯形的一条高，与线段CB的垂足为B，与线段AD的垂足为A，线段CD为斜边。连接AC、BD，线段AC与BD相交于点E，过点E线段EF垂直于线段AB，垂足为F。线段AB的中点为O，连接EO。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3972.jpg" }, { "index": 3973, "ocr_en": "There is a trapezoid ABCD in a plane, with AB as the upper base, CD as the lower base, and AC connecting the two bases. Draw an auxiliary line AF perpendicular to line segment CD through point A, with the foot of the perpendicular being F. Point E is a point outside trapezoid ABCD. Connect AE and CE, with line segment AE perpendicular to line segment CE, with the foot of the perpendicular being E. The lengths of line segments AD, CB, and CE are all labeled as x.", "ocr_zh": "平面内有梯形ABCD，AB为上底，CD为下底，连接AC。过点A作辅助线AF垂直于线段CD，垂足为F。点E是梯形ABCD外的一点，连接AE、CE，线段AE与线段CE垂直，垂足为E。线段AD、CB、CE的长度都标注为x。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3973.jpg" }, { "index": 3974, "ocr_en": "There is a right-angled trapezoid ABCD in the plane, with the upper base being CD, the lower base being AB, and AC connecting the two bases. Draw a line segment DE through point D perpendicular to line segment AC, with the foot of the perpendicular being E, and connect BE.", "ocr_zh": "平面内有直角梯形ABCD，上底为CD，下底为AB，连接AC。过点D作线段DE垂直于线段AC，垂足为E，连接BE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3974.jpg" }, { "index": 3975, "ocr_en": "There is an isosceles trapezoid ABCD in the plane, with the upper base AB and the lower base CD. Point M is the midpoint of line segment AD, and auxiliary lines MB and MC are drawn.", "ocr_zh": "平面内有等腰梯形ABCD，上底为AB，下底为CD。点M为线段AD的中点，作辅助线MB、MC。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3975.jpg" }, { "index": 3976, "ocr_en": "In a plane, there is a right-angled trapezoid ABCD, where AB is the lower base, CD is the upper base, and segment AD is one of the heights of the right-angled trapezoid. The foot of the perpendicular from segment AD to segment CD is D, and the foot of the perpendicular from segment AD to segment AB is A. Segment CB is the hypotenuse. Draw an auxiliary line AC. Draw an auxiliary line AF through point A perpendicular to segment CB, with the foot of the perpendicular being F. Draw an auxiliary line CE through point C perpendicular to segment AB, with the foot of the perpendicular being E. ", "ocr_zh": "平面内有直角梯形ABCD，AB为下底，CD为上底，线段AD是直角梯形的一条高，与线段CD的垂足为D，与线段AB的垂足为A，线段CB为斜边。作辅助线AC，过点A作辅助线AF垂直于线段CB，垂足为F。过点C作辅助线CE垂直于线段AB，垂足为E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3976.jpg" }, { "index": 3977, "ocr_en": "There is a regular hexagon ABCDEF in the plane. Point H is the midpoint of line segment DE, and point G is a point on line segment BC close to B. Connect HG and GA, draw auxiliary line BE, and draw a dotted line parallel to line segment BE through point H, intersecting line segment BC at point K.", "ocr_zh": "平面内有正六边形ABCDEF，点H是线段DE的中点，点G是线段BC是靠近B的一点，连接HG、GA，作辅助线BE，过点H以虚线作线段BE的平行线并与线段BC相交于点K。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3977.jpg" }, { "index": 3978, "ocr_en": "In a plane, there is a right-angled trapezoid AFEC, where CE is the lower base, AF is the upper base, and segment FE is one of the heights of the right-angled trapezoid. The foot of the perpendicular from segment FE to segment CE is E, and the foot of the perpendicular from segment FE to segment AF is F. Segment AC is the hypotenuse. Point D is the midpoint of segment AC. Connect DF and DE, where segment DE is perpendicular to segment AC. Draw auxiliary line DG parallel to segment CE, where the intersection of dashed segment DG and segment FE is point G. Draw segment AB perpendicular to segment CE through point A, where the foot of the perpendicular is point B.", "ocr_zh": "平面内有直角梯形AFEC，CE为下底，AF为上底，线段FE是直角梯形的一条高，与线段CE的垂足为E，与线段AF的垂足为F，线段AC为斜边。点D是线段AC的中点，连接DF、DE，线段DE与线段AC垂直。点D作辅助线DG平行于线段CE，虚线线段DG与线段FE的交点为G。过点A作线段AB垂直于线段CE，垂足为B。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3978.jpg" }, { "index": 3979, "ocr_en": "In a plane, there is a trapezoid ABCE, where segment CD is the upper base, segment AB is the lower base, and segments AD and BC are two equal-length legs. Connect segment BD and extend it to the left of trapezoid ABCD to a point E outside the trapezoid, such that point D is the midpoint of segment BE. Extend segment AB to the left of trapezoid ABCD, and draw segment EF perpendicular to the extension of segment AB through point E, with the foot of the perpendicular being F.", "ocr_zh": "平面内有梯形ABCE，线段CD为上底，线段AB为下底。连接线段BD并向梯形ABCD左侧延长至梯形外的一点E，使点D为线段BE的中点。向梯形ABCD左侧延长线段AB，过点E作线段EF垂直于线段AB的延长线段，垂足为F。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3979.jpg" }, { "index": 3980, "ocr_en": "There is a regular nonagon ABCDEFGHI in the plane. Connect AE and draw auxiliary lines AD and AC. Extend auxiliary line AC with a dashed line to a point K outside the regular nonagon ABCDEFGHI. Draw auxiliary line KD. The triangle KCD formed by the dashed lines KD and KC and line segment CD is an equilateral triangle.", "ocr_zh": "平面内有正九边形ABCDEFGHI，连接AE，作辅助线AD、AC，以虚线延长辅助线AC至正九边形ABCDEFGHI外的一点K，作辅助线KD，虚线KD、KC和线段CD构成的三角形KCD为正三角形。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3980.jpg" }, { "index": 3981, "ocr_en": "There is a trapezoid ABCD in a plane, with line segment AB as the upper base and line segment CD as the lower base. Connect AC and BD, and the intersection point of line segment AC and line segment BD is O. Points F and E are the midpoints of line segments BD and AC, respectively, and connect FE.", "ocr_zh": "平面内有梯形ABCD，线段AB为上底，线段CD为下底，连接AC、BD，线段AC与线段BD的交点为O。点F、E分别是线段BD、AC的中点，连接FE.", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3981.jpg" }, { "index": 3982, "ocr_en": "There is a trapezoid ABCD in a plane, with line segment AB as the lower base and line segment CD as the upper base. Connect AC and BD, and the intersection point of line segment AC and line segment BD is O. Point E is a point on line segment AO, and connect ED and EB. Point F is a point on line segment BO, and connect FC and FA.", "ocr_zh": "平面内有梯形ABCD，线段AB为下底，线段CD为上底，连接AC、BD,线段AC与线段BD的交点为O。点E是线段AO上的一点，连接ED、EB。点F是线段BO上的一点，连接FC、FA。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3982.jpg" }, { "index": 3983, "ocr_en": "There is a trapezoid ABCD in a plane, with line segment AB as the lower base and line segment CD as the upper base. Connect AC and BD. Points M and N are the midpoints of line segments AC and BD, respectively. Connect MN, MD, and NC.", "ocr_zh": "平面内有梯形ABCD，线段AB为下底，线段CD为上底，连接AC、BD。点M、N分别是线段AC、BD的中点，连接MN、MD、NC。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3983.jpg" }, { "index": 3984, "ocr_en": "In a plane, there is a convex quadrilateral ADCB, where angle D and angle A are obtuse angles, and angle B and angle C are acute angles. Connect AC and BD, and angle ABE is equal to angle ACE. Take a point E in the region below the line segment connecting the intersection point of line segments AC and BD to point B and above line segment BC. Draw auxiliary lines EB, ED, and EA so that angle BAE and angle CAD are equal.", "ocr_zh": "平面内有凸四边形ADCB，其中角D、角A为钝角，角B、角C为锐角。连接AC、BD，角ABE等于角ACE。在线段AC与线段BD的交点到点B所连成的线段之下、线段BC之上的区域中取一点E，作辅助线EB、ED、EA，使角BAE和角CAD相等。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3984.jpg" }, { "index": 3985, "ocr_en": "There is a quadrilateral ABCD in a plane, where angle A and angle C are right angles, angle B is an obtuse angle, and angle D is an acute angle. Draw an auxiliary line BD and draw a circumcircle of quadrilateral ABCD.", "ocr_zh": "平面内有四边形ABCD，其中角A、角C为直角，角B为钝角，角D为锐角。作辅助线BD，作四边形ABCD的外接圆。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3985.jpg" }, { "index": 3986, "ocr_en": "There is a circle with diameter AD in the plane. Points B and C are both on the same side of arc AD. Connect AB, BC, and CD. Angles ABC and BCD are both obtuse angles. Draw auxiliary line AC, extend segments AB and CD, and intersect at point P.", "ocr_zh": "平面内有以线段AD为直径的圆，点B、C都是圆弧AD同一侧上的点，连接AB、BC、CD，角ABC和角BCD都是钝角。作辅助线AC，延长线段AB、CD并相交于点P。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3986.jpg" }, { "index": 3987, "ocr_en": "There is a quadrilateral ABCD in a plane, where angle A and angle D are acute angles, and angle B and angle C are obtuse angles. Connect AC and BD. The intersection point of line segments AC and BD is E, and E is the midpoint of line segment BD. Draw the circumcircle of quadrilateral ABCD.", "ocr_zh": "平面内有四边形ABCD，其中角A、角D为锐角，角B、角C为钝角，连接AC、BD，线段AC与BD的交点为E，E为线段BD的中点。作四边形ABCD的外接圆。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3987.jpg" }, { "index": 3988, "ocr_en": "There is a parallelogram ABCD in a plane. Connect AC and BD. Point E is a point on the line segment connecting the intersection point of AC and BD to point D. Connect EC. Angle ECD is equal to angle ACB. Construct the circumcircle of triangle ABD. Extend AC and intersect the circumcircle of triangle ABD at point F. Connect EF, DF, and BF.", "ocr_zh": "平面内有平行四边形ABCD，连接AC、BD，点E是在线段AC与BD的交点到点D所连成的线段上的一点，连接EC，角ECD与角ACB大小相同。作三角形ABD的外接圆，延长AC并与三角形ABD的外接圆相交于点F，连接EF、DF、BF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3988.jpg" }, { "index": 3989, "ocr_en": "In a plane, there are triangle ABC and equilateral triangle ABD. Line segment AB is the common side of equilateral triangle ABD and triangle ABC. Other than that, triangle ABD and triangle ABC do not overlap. Draw a dotted line to form the circumcircle of triangle ABC. Point P is a point located inside triangle ABD and on arc AB. Connect PC and draw auxiliary lines PA and PB. Line segment PA is equal to line segment PB.", "ocr_zh": "平面内有三角形ABC和正三角形ABD，线段AB为正三角形ABD与三角形ABC的公用边，除此之外三角形ABD与三角形ABC无重叠区域。以虚线作三角形ABC的外接圆，点P是位于三角形ABD内、圆弧AB上的一点，连接PC，作辅助线PA、PB，线段PA与线段PB相等。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3989.jpg" }, { "index": 3990, "ocr_en": "In a plane, there is a triangle ABC. D is the midpoint of side AC, and E is the midpoint of side AB. Connect BD and CE. Line segment BD intersects CE at point G. Connect AG and extend it with a dotted line. The extended line segment AG intersects line segment BC at point F. Draw a circle through points A, E, and D. Draw auxiliary line ED. The intersection point of line segment ED and line segment AG is M.", "ocr_zh": "平面内有三角形ABC，D为AC边上的中点，E为AB边上的中点，连接BD、CE，线段BD与CE相交于点G，连接AG并以虚线延长，AG的延长线段与线段BC相交于点F。过点A、E、D作圆，作辅助线ED，线段ED与线段AG的交点为M。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3990.jpg" }, { "index": 3991, "ocr_en": "In a plane, there is an isosceles triangle ABC, where segments AB and AC are equal sides, and BC is the base. Draw the circumcircle of triangle ABC. Point D is a point on the circle. Connect BD, AD, and CD. Segment BD is perpendicular to segment AC, with the foot of the perpendicular being E. Extend segment AB to a point F outside the circle, and connect FD. Segment FD is perpendicular to segment BD, with the foot of the perpendicular being D. Draw the angle bisector FN of angle BFD. Segment FN intersects segment AD at point M and segment BD at point N. Connect point D, the intersection of FN and AC with a dashed line and extend it. The extended dashed line intersects AB at point Q. Draw a dashed line segment AP parallel to segment FN through point A. The intersection of segment AP and segment BD is point P.", "ocr_zh": "平面内有等腰三角形ABC，线段AB、AC为相等的腰，BC为底边，作三角形ABC的外接圆。点D是圆上的一点，连接BD、AD、CD，线段BD与线段AC垂直，垂足为E。延长线段AB至圆外的一点F，连接FD，线段FD与线段BD垂直，垂足为D。作角BFD的角平分线FN，线段FN与线段AD相交于点M、与线段BD相交于点N。以虚线连接点D、FN与AC的交点并延长，虚线延长线与AB交于点Q。过点A作虚线线段AP平行于线段FN，线段AP与线段BD的交点为P。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3991.jpg" }, { "index": 3992, "ocr_en": "In a plane, there is a quadrilateral ABCD, where angles B and A are acute angles, and angles C and D are obtuse angles. Draw a circumcircle around quadrilateral ABCD, connect AC and BD, and let the intersection point of segments AC and BD be P. Draw a segment FE parallel to segment AB through point P, and let the intersection points of segments FE and AD be E, and of segments FE and BC be F. Draw segment GH parallel to segment AD through point P. The intersection of segment GH and segment AB is G, and the intersection of segment GH and segment CD is H. Connect FH.", "ocr_zh": "平面内有四边形ABCD，角B、角A是锐角，角C、角D是钝角。作四边形ABCD的外接圆，连接AC、BD，线段AC与线段BD的交点为P。过点P作线段FE平行于线段AB，线段FE与线段AD的交点为E、与线段BC的交点为F。过点P作线段GH平行于线段AD，线段GH与线段AB的交点为G，与线段CD的交点为H。连接FH。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3992.jpg" }, { "index": 3993, "ocr_en": "In a plane, there is a quadrilateral ABCD, where angles B and C are acute angles, and angles A and D are obtuse angles. Connect AC and BD, and draw the circumcircle of quadrilateral ABCD. Connect point A to the midpoint of segment CD and the midpoint of segment CB, E, with a dashed line. Connect point D to the midpoint of segment CB and the midpoint of segment AB with a dashed line. Connect point C to the midpoint of segment AD and the midpoint of segment AB with a dashed line. Connect point B to the midpoint of segment CD and the midpoint of segment AD with a dashed line. Point G1 is the centroid of triangle ABC, point G2 is the centroid of triangle BCD, point G3 is the centroid of triangle ACD, and point G4 is the centroid of triangle ABD. Connect G1G2, G1G4, G4G3, and G3G2.", "ocr_zh": "平面内有四边形ABCD，角B、角C是锐角，角A、角D是钝角，连接AC、BD，作四边形ABCD的外接圆。以虚线连接点A与线段CD的中点、线段CB的中点E。以虚线连接点D与线段CB的中点、线段AB的中点。以虚线连接点C与线段AD的中点、线段AB的中点。以虚线连接点B与线段CD的中点、线段AD的中点。点G1是三角形ABC的重心，点G2是三角形BCD的重心，点G3是三角形ACD的重心，点G4是三角形ABD的重心，连接G1G2、G1G4、G4G3、G3G2。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3993.jpg" }, { "index": 3994, "ocr_en": "There is a quadrilateral A1A2A3A4 in the plane, with angles A4 and A1 being acute angles and angles A2 and A3 being obtuse angles. Draw a circumcircle of quadrilateral A1A2A3A4 with center O. Draw an auxiliary line A3O and extend it. The extension intersects circle O at point B. Draw auxiliary lines BA1, BA4, BA3, and BA2. Point H1 is a point in the region enclosed by quadrilateral A1A2A3A4 that is close to point A3. Draw auxiliary lines H1A2, H1A3, H1A4, H1A1, and H1B. Extend A4H1 with a dashed line until it intersects with segment A2A3. Point H2 is a point in the region enclosed by quadrilateral A1A2A3A4 that is to the upper right of point H1. Draw auxiliary lines H2A1, H2A4, H2A3, and H2A2, and extend A1H2 with a dashed line until it intersects with line segment A4H1. Point P is the midpoint of line segment H1A1. Draw auxiliary line OP and extend it to point O’ such that point P is the midpoint of line segment O’O.", "ocr_zh": "平面内有四边形A1A2A3A4，角A4、A1为锐角，角A2、A3为钝角，作四边形A1A2A3A4的外接圆，圆心为O。作辅助线A3O并延长，延长线段与圆O相交于点B，作辅助线BA1、BA4、BA3、BA2。点H1是四边形A1A2A3A4所围成的区域中靠近点A3的一点，作辅助线H1A2、H1A3、H1A4、H1A1、H1B，以虚线延长A4H1至与线段A2A3相交。点H2是四边形A1A2A3A4所围成的区域中在点H1右上方的一点，作辅助线H2A1、H2A4、H2A3、H2A2，以虚线延长A1H2至与线段A4H1相交。点P是线段H1A1的中点，作辅助线OP并延长至点O’，使点P为线段O’O的中点。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3994.jpg" }, { "index": 3995, "ocr_en": "In a plane, there is a quadrilateral ABCD, where angles A and D are obtuse angles, and angles B and C are acute angles. Connect AC and BD, and draw the circumcircle of quadrilateral ABCD. Points M, M’, N, and N’ are the midpoints of arcs AB, CD, CB, and AD, respectively. Draw auxiliary lines MM’, NN’, MC, MD, M’A, and M’B. Points I₁, I₂, I₃, and I₄ are points on line segments M’B, M’A, MD, and MC, respectively. Connect I₁I₂ and I₁I₄, I₂I₃ and I₃I₄ to form quadrilateral I₁I₂I₃I₄, which is a parallelogram.", "ocr_zh": "平面内有四边形ABCD，角A、D为钝角，角B、C为锐角，连接AC、BD，作四边形ABCD的外接圆，点M、M’、N、N’分别是圆弧AB、CD、CB、AD上的中点，作辅助线MM’、NN’、MC、MD、M’A、M’B。点I1、I2、I3、I4分别是线段M’B、M’A、MD、MC上的点，连接I1I2、I1I4、I2I3、I3I4构成四边形I1I2I3I4，四边形I1I2I3I4是平行四边形。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3995.jpg" }, { "index": 3996, "ocr_en": "In a plane, there is a quadrilateral ABCD, where angles A and B are obtuse angles, and angles C and D are acute angles. Connect AC and BD; the intersection point of segments AC and BD is G. Construct the circumcircle of quadrilateral ABCD. Point I3 is a point inside triangle ABG, and connect GI3. Points I1 and I2 are two points inside triangle CDG, with I1 on the left and I2 on the right. Connect I1I2, and the line segment I1I2 is perpendicular to but does not intersect with the line segment I3G. Draw auxiliary lines I1D, I1C, I2C, and I2D. Extend segment I1I2 to both sides with dashed lines. The extended line of segment I1I2 intersects segment GD at point E and segment GC at point F.", "ocr_zh": "平面内有四边形ABCD，角A、B为钝角，角C、D为锐角，连接AC、BD，线段AC与BD的交点为G，作四边形ABCD的外接圆。点I3是三角形ABG内的一点，连接GI3。点I1、I2平面内有四边形ABCD，角A、B为钝角，角C、D为锐角，连接AC、BD，线段AC与BD的交点为G，作四边形ABCD的外接圆。点I3是三角形ABG内的一点，连接GI3。点I1、I2是三角形CDG内的两点，I1在左侧、I2在右侧，连接I1I2，线段I1I2与线段I3G垂直但不相交。作辅助线I1D、I1C、I2C、I2D，以虚线向两侧延长线段I1I2，线段I1I2的延长线与线段GD相交于点E、与线段GC相交于点F。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3996.jpg" }, { "index": 3997, "ocr_en": "In a plane, there is a quadrilateral ABCD, where angles A and C are obtuse angles, angles B and D are acute angles, and angle B is greater than angle D. Draw a circle inscribed in quadrilateral ABCD, with center O. Connect AC and BD, and the intersection point of segments AC and BD is E.", "ocr_zh": "平面内有四边形ABCD，角A、C为钝角，角B、D为锐角，角B大于角D。作四边形ABCD的内接圆，圆心为O，连接AC、BD，线段AC与BD的交点为E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_3997.jpg" }, { "index": 3998, "ocr_en": "In a plane, there is a quadrilateral ABCD, where angles B and C are acute angles, and angles A and D are obtuse angles. Draw the incircle of quadrilateral ABCD. The points of intersection of the incircle with sides AB, BC, CD, and DA are E, F, G, and H, respectively. Connect AF and DF. The line segment AF intersects the line segment EG at point M, and the line segment DF intersects the line segment EG at point N. Draw auxiliary lines EF, GF, HF, and EH. The intersection point of the line segment HF and the line segment EG is T. Draw a line segment PQ parallel to the line segment EG through point A. Extend the line segment FH and intersect it with the extended line at point Q. Extend the line segment EF and intersect it with the extended line at point P.", "ocr_zh": "平面内有四边形ABCD，角B、C为锐角，角A、D为钝角，作四边形ABCD的内接圆，内接圆与AB、BC、CD、DA的交点分别是E、F、G、H。连接AF、DF，线段AF与线段EG相交于点M，线段DF与线段EG相交于点N。作辅助线EF、GF、HF、EH，线段HF与线段EG的交点为T。过点A作线段EG的平行线PQ，延长线段FH并与该延长线相交于点Q，延长线段EF并与该延长线相交于点P。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3998.jpg" }, { "index": 3999, "ocr_en": "In a plane, there is a quadrilateral ABCD, where angles C and D are obtuse angles, angles B and A are acute angles, and angle A is greater than angle B. Draw the circumcircle of quadrilateral ABCD, with center O. Extend line segments BA and CD, and the two extended lines intersect at point P. Extend line segments AD and BC, and the two extended lines intersect at point Q. Draw auxiliary lines PO, PQ, and OQ. Draw the circumcircles of triangles PBC and CDQ. The two circumcircles intersect at point M, in addition to point C. Draw auxiliary lines CM, PM, QM, OP, and OQ.", "ocr_zh": "平面内有四边形ABCD，角C、D为钝角，角B、A为锐角，角A大于角B。作四边形ABCD的外接圆，圆心为O。延长线段BA、CD，两条延长线相交于点P。延长线段AD、BC，两条延长线相交于点Q。作辅助线PO、PQ、OQ。作三角形PBC的外接圆、三角形CDQ的外接圆，两个外接圆除了点C外的另一交点为M。作辅助线CM、PM、QM、OP、OQ。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_3999.jpg" }, { "index": 4000, "ocr_en": "There is a circle within a plane, and within the circle there is a convex quadrilateral. Extend each side of the quadrilateral outward on both sides. The intersection points of the extended sides with the circle, in clockwise order, are B1, B2, C1, C2, D1, D2, A1, and A2. Connect B1B2, A1A2, D1D2, and C1C2, and extend the line segments B1B2, A1A2, D1D2, and C1C2 in both directions until they intersect with another extended line.", "ocr_zh": "平面内有一个圆，圆内有一个凸四边形。将四边形的每条边都向两侧延长，四条边的延长线与圆的交点按顺时针顺序分别是B1、B2、C1、C2、D1、D2、A1、A2.连接B1B2、A1A2、D1D2、C1C2，并将线段B1B2、A1A2、D1D2、C1C2都向两侧延长至与另一条延长线相交为止。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4000.jpg" }, { "index": 4001, "ocr_en": "In Figure (1) on the left, there is a trapezoid ABCD in the plane, with segment AD as the upper base and segment BC as the lower base. Point E is a point on segment BC. Draw an auxiliary line DE such that segment DE is parallel to segment AB. In Figure (2) on the right, there is a trapezoid ADBC in the plane, with segment BC as the upper base and segment AD as the lower base. Connect AB and CD. Extend segment CB with a dashed line to a point E outside the trapezoid, where segment BE is equal to segment AD, and draw an auxiliary line ED.", "ocr_zh": "在左侧的图（1）中，平面内有梯形ABCD，线段AD为上底，线段BC为下底，点E是线段BC上的一点，作辅助线DE，线段DE与线段AB平行；在右侧的图（2）中，平面内有梯形ADBC，线段BC为上底，线段AD为下底，连接AB、CD。以虚线延长线段CB至梯形外的一点E，线段BE与线段AD相同，作辅助线ED。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4001.jpg" }, { "index": 4002, "ocr_en": "In Figure (1) on the left, there is a quadrilateral ABCD in the plane. Angles B and C are acute angles, while angles A and D are obtuse angles. Angle C is greater than angle B, and angle A is greater than angle D. Connect AC and BD, with the intersection point being S. Extend segment AB with a dashed line and extend segment CD with a dashed line. The two extended lines intersect at point T. Draw the angle bisector of angle ATD. The end of this angle bisector extends to the region containing triangle BSC, intersecting line segment BD at point Q and line segment AC at point P. Point K is a point on the left side of triangle TBC. Draw auxiliary line KD. where segment KD is parallel to and equal to segment AC. Draw auxiliary lines KA and KB. In the middle figure (2), there is an acute triangle BCD in the plane. Point A is a point inside triangle BCD. Connect AB, AC, and AD. Extend segment AC and intersect it with segment BD at point S. Extend segment AB with a dashed line and intersect it with segment CD at point T. Extend segment BD with a dashed line to a point Q on the right side of triangle BDC, connect QT with a dashed line and extend it, and the extension intersects segment AC at point P. Point K is a point on the left side of triangle BDC. Draw auxiliary line KD, which is parallel to and equal to segment AC. Draw auxiliary lines KA and KB. In the right-hand figure (3), there is a triangle SBC in the plane. Point A is the midpoint of segment SC. Connect AB. Point D is a point on segment SB closer to point B. Connect DA and DC. The intersection of segment DC and segment AB is point T. Point K is a point on the left side of triangle SBC. Draw auxiliary line KD, where segment KD is parallel and equal to segment AC. Draw auxiliary lines KA and KB. Point P is a point on the left side of triangle SBC. Draw auxiliary line PT, where segment PT is parallel to segment KB. Extend segment SC to the left and extend segment AB downward with a dashed line to point L.", "ocr_zh": "在左侧的图（1）中，平面内有四边形ABCD，角B、角C为锐角，角A、角D为钝角，角C大于角B，角A大于角D，连接AC、BD，交点为S。以虚线延长线段AB、以虚线延长线段CD，两条延长线相交于点T。作角ATD的角平分线，该角平分线末端到三角形BSC所在区域为止，与线段BD相交于点Q，与线段AC相交于点P。点K是三角形TBC左侧的一点，作辅助线KD，线段KD与线段AC平行且相等，作辅助线KA、KB；在中侧的图（2）中，平面内有锐角三角形BCD，点A是三角形BCD内的一点，连接AB、AC、AD，延长线段AC并与线段BD相交于点S，以虚线延长线段AB并与线段CD相交于点T。以虚线延长线段BD至三角形BDC右侧的一点Q，以虚线连接QT并延长，延长线与线段AC相交于点P。点K是三角形BDC左侧的一点，作辅助线KD，线段KD与线段AC平行且相等，作辅助线KA、KB；在右侧的图（3）中，平面内有三角形SBC，点A是线段SC的中点，连接AB，点D是线段SB上靠近点B的一点，连接DA、DC，线段DC与线段AB的交点为T。点K是三角形SBC左侧的一点，作辅助线KD，线段KD与线段AC平行且相等，作辅助线KA、KB。点P是三角形SBC左侧的一点，作辅助线PT，线段PT与线段KB平行。向左延长线段SC，以虚线向下延长线段AB至点L。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4002.jpg" }, { "index": 4003, "ocr_en": "In Figure (1) on the left, there are two circles in the plane: circle O1 with center O1 and circle O2 with center O2. The two circles intersect at points P and Q, with point Q located above point P. Points A and B are two points on circle O1 located on the upper half of the minor arc QP, and points C and D are two points on circle O2 located on the upper half of the major arc QP. Connect CD, DA, CB, and AB. Extend segments DA and CB with dashed lines, and the two extended lines intersect at point P. Points (A) and (B) are two points on circle O1 located on the lower half of the major arc QP, with point (A) above point (B). Points (C) and (D) are two points on circle O2 located on the lower half of the major arc QP, with point (D) above point (C). Connect (A)(D), (A)(B), and (C)(D), and draw auxiliary lines (A)(P), (D)(P), (B)(P), and (C)(P). If segment O1O2 exists, then segment (A)(D) is parallel to segment O1O2 (segment O1O2 is actually not connected); In Figure (2) on the middle side, there are two circles in the plane: circle O1 with center O1 and circle O2 with center O2. The two circles intersect at points P and Q, with point Q located above point P. Draw an auxiliary line O2P and extend it with a dashed line. The extension of segment O2P intersects circle O1 at point A. Extend segment AP with a solid line. The extension of segment AP intersects circle O2 at point D. Point B is a point on circle O1 located on the lower half of arc QP. Connect BP and extend it; the extension intersects circle O2 at point C. Connect CD and AB, and draw an auxiliary line AP. In the figure on the right (3), there are two circles in the plane: circle O1 with center O1 and circle O2 with center O2. The two circles intersect at points P and Q, with point Q located above point P. Draw an auxiliary line O1P and extend it with a dashed line. The extension of O1P intersects circle O2 at point D. Extend segment PD with a solid line and intersect circle O1 at point A. Point B is a point on circle O1 located on the upper half of the minor arc QP. Draw auxiliary line BP and extend it with a solid line. The extension of BP intersects circle O2 at point C. Connect AB and CD. Points (A) and (B) are two points on circle O1 located on the lower half of the major arc QP. Point (A) is above point (B). Connect (A) and (B), (A)(P), and (B)(P), and extend (A)(P) and (B)(P). The extensions of segments (A)(P) and (B)(P) intersect circle O2 at points (D) and (C), respectively. Connect (D) and (C).", "ocr_zh": "在左侧的图（1）中，平面内有以点O1为圆心的圆O1和以点O2为圆心的圆O2，两圆相交于点P、Q，点Q位于点P上方。点A、B是圆O1上位于劣弧QP上半段的两点，点C、D是圆O2上位于优弧QP上半段的两点，连接CD、DA、CB、AB。以虚线延长线段DA和线段CB，两条延长线相交于点P。点（A）、（B）是圆O1上位于优弧QP下半段的两点，点(A)在点（B）之上，点（C）、（D）是圆O2上位于优弧QP下半段的两点，点（D）在点（C）之上。连接（A）（D）、（A）（B）、（C）（D），作辅助线（A）(P)、(D)(P) 、（B）(P)、(C)(P)，若线段O1O2存在则线段（A）（D）与线段O1O2平行（线段O1O2实际不连接）；在中侧的图（2）中，平面内有以点O1为圆心的圆O1和以点O2为圆心的圆O2，两圆相交于点P、Q，点Q位于点P上方。作辅助线O2P并以虚线延长，线段O2P的延长线与圆O1相交于点A.以实线延长线段AP，线段AP的延长线与圆O2相交于点D。点B是圆O1上位于优弧QP下半段的一点，连接BP并延长，延长线与圆O2相交于点C。连接CD、AB，作辅助线AP；在右侧的图（3）中，平面内有以点O1为圆心的圆O1和以点O2为圆心的圆O2，两圆相交于点P、Q，点Q位于点P上方。作辅助线O1P以虚线延长，O1P的延长线与圆O2相交于点D，以实线延长线段PD并与圆O1相交于点A。点B是圆O1上位于劣弧QP上半段的一点，作辅助线BP并以实线延长，BP的延长线与圆O2相交于点C，连接AB、CD。点（A）、（B）是圆O1上位于优弧QP下半段的两点，点(A)在点（B）之上，连接（A）（B）、（A）（P）、（B）(P)并延长（A）（P）、（B）(P) ，线段（A）（P）、（B）(P) 的延长线分别与圆O2相交于点(D)(C)，连接(D)(C)。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4003.jpg" }, { "index": 4004, "ocr_en": "In a plane, there is a quadrilateral ABCD, where angles B and C are acute angles, and angles A and D are obtuse angles. Connect AC and BD. Point P is the midpoint of segment AD, and point Q is the midpoint of segment BC. Connect PQ, and segment PQ passes through the intersection point of segments AC and BD. Point M is the midpoint of segment AC, and point N is the midpoint of segment BD. Draw auxiliary lines MN, MP, NP, NQ, and MQ.", "ocr_zh": "平面内有四边形ABCD，角B、角C为锐角，角A、角D为钝角，连接AC、BD。点P是线段AD的中点，点Q是线段BC的中点，连接PQ，线段PQ经过线段AC和BD的交点。点M是线段AC的中点，点N是线段BD的中点，作辅助线MN、MP、NP、NQ、MQ。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4004.jpg" }, { "index": 4005, "ocr_en": "In Figure (1) on the left, there are two circles in the plane: circle O1 with center O1 and circle O2 with center O2. The radius of circle O2 is greater than that of circle O1. The two circles intersect at points P and Q, with point Q located above point P. Connect QP. Point A is a point on the major arc QP of circle O1, point C is a point on the minor arc QP of circle O1, point B is a point on the minor arc QP of circle O2, and point D is a point on the major arc QP of circle O2. Connect AB, AD, CD, and BC, and extend the line segments AB and CD with dashed lines. The two extended lines intersect at point P. Draw auxiliary lines AC, BQ, QC, AQ, and QD. In the middle figure (2), there are two circles in the plane: circle O1 with center O1 and circle O2 with center O2. The radius of circle O1 is greater than that of circle O2, and the two circles intersect at points P and Q, with point Q located above point P. Connect QP. Points A and C are two points on the superior arc QP of circle O1, with point C located above point A. Points B and D are two points on the superior arc QP of circle O2, with point B located above point D. Connect AB, AD, CD, and BC. The line segment CD intersects the line segment AB at point P. Draw auxiliary lines CQ, AQ, DQ, and BQ. In the figure on the right (3), there are two circles in the plane. The circle with the smaller radius is called the smaller circle, and the circle with the larger radius is called the larger circle. The smaller circle is to the left of the larger circle, and the two circles intersect at points P and Q, with point Q located above point P. Connect QP. Point A is a point on the small circle to the left of segment QP, and point C is a point on the small circle to the right of segment QP, with point C located above point A. Point B is a point on the lower half of the major arc QP of the large circle, and point D is a point on the upper half of the major arc QP of the large circle. Connect AB, AD, CD, and BC. The intersection point of segment AB with the large circle and the small circle is P. Draw auxiliary lines AC, BQ, QC, AQ, QD, and PC.", "ocr_zh": "在左侧的图（1）中，平面内有以点O1为圆心的圆O1和以点O2为圆心的圆O2，圆O2的半径大于圆O1，两圆相交于点P、Q，点Q位于点P上方，连接QP。点A是位于圆O1的优弧QP上半段的一点，点C是位于圆O1的劣弧QP上半段的一点，点B是位于圆O2上的劣弧QP上半段的一点，点D是位于圆O2的优弧QP上半段的一点，连接AB、AD、CD、BC并以虚线延长线段AB、CD，两条延长线相交于点P。作辅助线AC、BQ、QC、AQ、QD；在中侧的图（2）中，平面内有以点O1为圆心的圆O1和以点O2为圆心的圆O2，圆O1的半径大于圆O2，两圆相交于点P、Q，点Q位于点P上方，连接QP。点A、C是位于圆O1的优弧QP下半段的两点，点C位于点A上方。点B、D是位于圆O2的优弧QP下半段的两点，点B是位于点D上方。连接AB、AD、CD、BC，线段CD与线段AB相交于点P。作辅助线CQ、AQ、DQ、BQ；在右侧的图（3）中，平面内有两个圆，半径较小的称为小圆，半径较大的称为大圆，小圆在大圆左侧，两圆相交于点P、Q，点Q位于点P上方，连接QP。点A是位于小圆上、线段QP左侧的一点，点C是位于小圆上、线段QP右侧的一点，点C在点A的上方。点B是位于大圆的优弧QP下半段的一点，点D是位于大圆的优弧QP上半段的一点，连接AB、AD、CD、BC，线段AB与大圆和小圆的公平交点为P。作辅助线AC、BQ、QC、AQ、QD、PC。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4005.jpg" }, { "index": 4006, "ocr_en": "In quadrilateral ABCD, angles A and C are acute angles, and angles B and D are obtuse angles. Draw the angle bisector of angle B and intersect it with line segment AD at point M. Draw line segment DN through point D and intersect it with line segment BC at point N.", "ocr_zh": "在四边形ABCD中，角A、角C为锐角，角B、角D为钝角。作角B的角平分线并与线段AD相交于点M，过点D作线段DN并与线段BC相交于点N。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4006.jpg" }, { "index": 4007, "ocr_en": "In a convex quadrilateral ABCD, connect AC and BD, and the points G, H, E, and F are the midpoints of the line segments AD, CD, AC, and BD, respectively. Connect the auxiliary lines GH, GE, GF, EH, EF, FH.", "ocr_zh": "凸四边形ABCD中，连接AC与BD，点G、H、E、F分别为线段AD、CD、AC、BD的中点。连接辅助线GH、GE、GF、EH、EF、FH。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4007.jpg" }, { "index": 4008, "ocr_en": "This article consists of figure (1) and figure (2). Figure (1) on the left, extend the two sides of the convex quadrilateral ABCD AB and DC to make auxiliary lines BE and CE intersected at point E, this respectively do the triangle AED and triangle BEC external circle, the two circles intersect at point E and point M, connect the auxiliary lines CM and ME, the point B 'is a point in BE. Figure (2) on the right, concave quadrilateral ABCD, Angle B, Angle D and Angle A are acute angles, Angle C is an angle greater than 180 degrees, connect AC, extend DC to make the auxiliary line CE to cross AB at point E, respectively, to do the external circle of triangle AED and CEB, the two circles intersect at point E and point M, connecting CM and ME, point B is also called the point B '.", "ocr_zh": "本条由图（1）和图（2）组成。其中图（1）在左侧，延长凸四边形ABCD的两条边AB和DC做辅助线BE和CE交于点E，本别做三角形AED和三角形BEC的外接圆，两个圆相交于点E和点M，连接辅助线CM和ME，点B'为BE中一点。图（2）在右侧，凹四边形ABCD中，角B、角D和角A均为锐角，角C为大于180度的角，连接AC，延长DC做辅助线CE交AB于点E，分别做三角形AED和CEB的外接圆，两圆交于点E和点M，连接CM和ME，点B也叫点B'。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4008.jpg" }, { "index": 4009, "ocr_en": "Point BDEF for the auxiliary circle on the four points, where the length of the inferior arc BD is less than the length of the inferior arc EF, connect BF, DE intersection at point C, connect EB and FD and extend the intersection at point A, connect AC.", "ocr_zh": "点BDEF为辅助圆上四点，其中劣弧BD长度小于劣弧EF的长度，连接BF、DE交于点C，连接EB和FD并延长交于点A，连接AC。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4009.jpg" }, { "index": 4010, "ocr_en": "The height on the BD side of an acute triangle ABD is the imaginary segment AL, with the vertical foot L. Point C is a point on LB, connecting to AC.", "ocr_zh": "锐角三角形ABD的BD边上的高为虚线段AL，垂足为L，点C为LB上一点，连接AC。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4010.jpg" }, { "index": 4011, "ocr_en": "In triangle ABC, CF and BE are the heights on the sides of AB and AC respectively, the vertical feet are F and E respectively, CF and BE intersect at point H, connect AH and extend to intersect BC at point D, where the line segment HD is an imaginary segment.", "ocr_zh": "三角形ABC中，CF和BE分别是AB和AC边上的高，垂足分别为F和E，CF和BE交于点H，连接AH并延长交BC于点D，其中线段HD为虚线段。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4011.jpg" }, { "index": 4012, "ocr_en": "This strip consists of figures (1) and (2). Where figure (1) is on the left hand side, there is a circle with a chord A'D'. Point P is a point on the inferior arc A'D' close to point D', over point P make ray PB' perpendicular to the chord A'D', with pendant B, intersecting with the circle at point B', over point P make rays PA' and PD', over points A' and B' make tangents to the circle A'Q and B'Q intersecting at point Q, AQ' and B'Q are both auxiliaries, connecting auxiliary D'Q intersecting the circle at point C', angle A'PC' is angle α, angle C' PB' is angle β, angle B'PD' is angle γ. Point A is also point A' and point D is also point D'. Figure (2) On the right side, there is a circle with a chord CD, point C is also point C', point D is also point D', point P is a point on the inferior arc CD close to point C, point P is also point A', make the ray PC and the ray PD, make the tangent to the circle PQ across point P to intersect the extension line of DC at point A, point Q is coincident with point A, and draw the tangent to the circle AB' across point A. AB' is the imaginary line segment, and B' is on the circle, connecting PB' intersects CD at point B. Angle APC' is angle α, angle CPB is angle β, and angle B'PD' is angle γ.", "ocr_zh": "本条由图（1）和图（2）组成。其中图（1）在左侧，有一个圆，圆上有一条弦A'D'。点P为劣弧A'D'上靠近点D'的一点，过点P做射线PB'垂直于弦A'D'，垂足为B，与圆交于点B'，过P点做射线PA'和PD'，过点A'和B'做圆的切线A'Q和B'Q交于点Q，AQ'和B'Q都是辅助线，连接辅助线D'Q交圆于点C'，角A'PC'为角α，角C'PB'为角β，角B'PD'为角γ，点A也是点A'，点D也是点D'。图（2）在右侧，有一个圆，圆上有一条弦CD，点C也是点C'，点D也是点D'，点P是劣弧CD上靠近点C的一点，点P也是点A'，做射线PC和射线PD，过点P做圆的切线PQ交DC的延长线于点A，点Q与点A重合，过点A引圆的切线AB'，AB'为虚线段，B'在圆上，连接PB'交CD于点B，角APC'为角α，角CPB为角β，角B'PD'为角γ。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4012.jpg" }, { "index": 4013, "ocr_en": "This article consists of figure (1), figure (2) and figure (3). Figure (1) On the left, there is a circle tangent to two parallel lines with tangent points C and A, connecting AC, making a perpendicular bisector BD of AC intersecting with the circle at point D and point B, connecting DC, CB, BA, AD. Figure (2) In the center, there is a circle tangent to two parallel lines with tangent points C and A, connecting AC, and there is a straight line BC perpendicular to AC intersecting with the circle at point D and point B. Connect DC, CB, BA, AD. Figure (3) on the right side, there is a circle, Q is a point outside the circle, through the point Q to lead the ray QC and QA tangent to the circle, the tangent points are C and A, connecting the auxiliary line AC, the point B is a point on the superior arc AC, connecting the QB intersection of the circle at the point D, intersection of AC at the point G, connecting the DC, CB, BA, AD, extend the CD and DA to make auxiliary lines DF and DE, where the point E and point F are on QC and QA respectively.", "ocr_zh": "本条由图（1）、图（2）和图（3）组成。图（1）在左侧，有一个圆与两条平行线相切，切点分别为C和A，连接AC，做AC的垂直平分直线BD与圆交于点D和点B，连接DC、CB、BA、AD。图（2）在中间，有一个圆与两条平行线相切，切点分别为C和A，连接AC，有一条直线BC垂直于AC并与圆交于点D和点B，连接DC、CB、BA、AD。图（3）在右侧，有一个圆，Q为圆外一点，过点Q引射线QC和QA与圆相切，切点分别为C和A，连接辅助线AC，点B为优弧AC上一点，连接QB交圆于点D，交AC于点G，连接DC、CB、BA、AD，延长CD和DA做辅助线DF和DE，其中点E和点F分别在QC和QA上。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4013.jpg" }, { "index": 4014, "ocr_en": "There are five points ABTCD on the circle, connect CT, CD, DA, AB, DB, BC, connect the auxiliary lines TD, TB, and connect the auxiliary line TA to intersect BD at point M.", "ocr_zh": "圆上有ABTCD五点，连接CT、CD、DA、AB、DB、BC，连接辅助线TD、TB，连接辅助线TA与BD交于点M。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4014.jpg" }, { "index": 4015, "ocr_en": "Quadrilateral ABCD is an interior quadrilateral of a circle, connecting CA and BD, the midpoints of CA and BD are point M and point N, respectively, connecting MD, MB, NA, NC.", "ocr_zh": "四边形ABCD为圆的一个内接四边形，连接CA和BD，CA和BD的中点分别为点M和点N，连接MD、MB、NA、NC。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4015.jpg" }, { "index": 4016, "ocr_en": "There are four points ABCD on the circle, connecting AC, AD, DC, CB, BA, point T1 is a point on AC, make a line segment DT through point T1, and make a line segment BT2 through point T1.", "ocr_zh": "圆上有ABCD四点，连接AC、AD、DC、CB、BA，点T1为AC上一点，过点T1做线段DT，过点T1做线段BT2。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4016.jpg" }, { "index": 4017, "ocr_en": "This strip consists of figure (a), figure (b) and figure (c). Figure (a) in the left, circle O1 and circle O2 for two intersecting the same diameter of the circle, the two circles intersect at the point D and point A, connecting AD, point E in the circle O2, for the superior arc DA on the point near the D point, connecting auxiliary line EA, point B in the circle O2, connecting the auxiliary line BD so that the angle BDA is equal to the Angle EAD. line EA and the circle O1 and the circle O1 intersected in the point C, connecting BA, BC and CD. Figure (b) ) in the middle, circle O1 and circle O2 are two intersecting circles with the same diameter, the two circles intersect at point D and point A, connect AD, point E is on circle O2, a point on the superior arc DA close to point D, connect the auxiliary line EA, point B is on circle O2 located on the superior arc DA close to point A, connect the auxiliary line BD to intersect with AE, the line segment EA intersects with circle O1 at point C, connect BA, BC and CD. figure (c) ) on the right, circle O1 and circle O2 are two intersecting circles of the same diameter, the two circles intersect at point D and point A, connect AD, point E is on circle O2, a point on superior arc DA close to point D, connect the auxiliary line EA, point B is on circle O2 on superior arc DA close to point A, connect the auxiliary line BD and AE have no intersection inside the circle, the line segment EA and circle O1 intersect at point C, connect BA, BC and CD. .", "ocr_zh": "本条由图（a）、图（b）和图（c）组成。其中图（a）在左侧，圆O1和圆O2为两个相交的相同直径的圆，两圆相交于点D和点A，连接AD，点E在圆O2上，为优弧DA上靠近点D的一点，连接辅助线EA，点B在圆O2上，连接辅助线BD使角BDA等于角EAD。线段EA与圆O1交于点C，连接BA、BC和CD。图（b）在中间，圆O1和圆O2为两个相交的相同直径的圆，两圆相交于点D和点A，连接AD，点E在圆O2上，为优弧DA上靠近点D的一点，连接辅助线EA，点B在圆O2上位于优弧DA上靠近点A，连接辅助线BD与AE相交，线段EA与圆O1交于点C，连接BA、BC和CD。图（c）在右侧，圆O1和圆O2为两个相交的相同直径的圆，两圆相交于点D和点A，连接AD，点E在圆O2上，为优弧DA上靠近点D的一点，连接辅助线EA，点B在圆O2上位于优弧DA上靠近点A，连接辅助线BD与AE在圆内无交点，线段EA与圆O1交于点C，连接BA、BC和CD。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4017.jpg" }, { "index": 4018, "ocr_en": "Isosceles trapezoid ABCD is connected to circle 0, where AD is parallel to BC, connect AC, BD intersection at point M, over point M to do EF parallel to BC, point E and point F are on AB and DC, respectively, to both sides of the extension of the EF intersected by the circle at point S and the circle T, where the line segments SE and FT for the auxiliary line, connect auxiliary line EC and FB, over point M to do the straight line LN is perpendicular to BC, the vertical foot of the pendant is N, the point L in the AD. .", "ocr_zh": "等腰梯形ABCD内接于圆0，其中AD平行于BC，连接AC、BD交于点M，过点M做EF平行于BC，点E和点F分别在AB上和DC上，向两侧延长EF交圆于点S与圆T，其中线段SE和FT为辅助线，连接辅助线EC和FB，过点M做直线LN垂直于BC，垂足为N，点L在AD上。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4018.jpg" }, { "index": 4019, "ocr_en": "In triangle ABC, the bisector AS of angle A intersects BC at point S, P is the midpoint of BC, make PY parallel to AS through point P, and intersect the extension of CA at point Y, intersect AB at point X, connect YB, extend XY to point Z so that XY=YZ, connect ZC and intersect the extension of BY at point D, and make a line segment l through point D to BC, with l parallel to AS, and with point Q as the midpoint of XY.", "ocr_zh": "在三角形ABC中，角A的平分线AS交BC于点S，P为BC的中点，过P点做PY平行于AS，并交CA的延长线于点Y，交AB于点X，连接YB，延长XY至点Z使XY=YZ，连接ZC交BY的延长线于点D，过点D做线段l至BC上，l平行于AS，点Q为XY中点。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4019.jpg" }, { "index": 4020, "ocr_en": "In triangle ACE, point B is the midpoint of AC, point F is a point on AE, connect BE and CF to intersect at point D, point R is on line segment D of B, point S is on line segment DF, connect SB and RF, SB and RF are the perpendiculars of AC and AE respectively, with perpendiculars B and F, and connect RS, where FR and BS intersect at point T, and connect AT, and connect auxiliary line BF.", "ocr_zh": "三角形ACE中，点B是AC中点，点F为AE上一点，连接BE和CF交于点D，点R在B线段D上，点S在线段DF上，连接SB和RF，SB和RF分别为AC和AE的垂线，垂足为B和F，连接RS，其中FR和BS交于点T，连接AT，连接辅助线BF。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4020.jpg" }, { "index": 4021, "ocr_en": "Make a circle with point O as the center, a chord of circle O is AB, point A and point B are on the circle, and the midpoint of AB is M. Draw two chords CD and EF across M. Points C, D, E, and F are all on the circle, and connect ED and CF to intersect AB at point Q and point P respectively, and make an auxiliary line QN parallel to the line segment CF across the point Q, and the point N is on the EF.", "ocr_zh": "以点O为圆心做圆，圆O的一条弦为AB，点A和点B在圆上，AB的中点为M，过M引两条弦CD和EF，点C、D、E、F都在圆上，连接ED和CF分别交AB于点Q和点P，过Q点做辅助线QN平行于线段CF，点N在EF上。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4021.jpg" }, { "index": 4022, "ocr_en": "Point O as the center of the circle to make a circle, P is a point outside the circle, over the point P lead circle O tangent PA and PB, which point A and point B in the circle, connect AB, make auxiliary line PG perpendicular to AB, the foot of the plumbing for G, point F is a point on the superior arc AB, connect the PF intersection of the circle O at point E, connect the auxiliary line EG and extend the intersection of the circle O at point D, connect the PD intersection of the circle at point C, connect the CF, the point D and the point D 'coincide.", "ocr_zh": "以点O为圆心做圆，P为圆外一点，过点P引圆O的切线PA和PB，其中点A和点B在圆上，连接AB，做辅助线PG垂直AB，垂足为G，点F为优弧AB上一点，连接PF交圆O于点E，连接辅助线EG并延长交圆O于点D，连接PD交圆于点C，连接CF，点D和点D'重合。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4022.jpg" }, { "index": 4023, "ocr_en": "Take point O as the center of the circle to make a circle, over a point P outside the circle to lead two cut lines PB and PD, PB and the circle intersects at point A and point B, PD and the circle intersects at point C and point D, connects CB and AD, intersects at point Q, connects OQ and OP.", "ocr_zh": "以点O为圆心做圆，过圆外一点P引两条割线PB和PD，PB与圆交于点A和点B，PD与圆交于点C和点D，连接CB和AD，交于点Q，连接OQ和OP。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4023.jpg" }, { "index": 4024, "ocr_en": "Through a point P outside the circle to lead two cut lines PB and PD, PB and the circle intersect at point A and point B, PD and the circle intersect at point C and point D, connect CB and AD, intersect at point Q, connect PQ and extend to a point F on the circle, PF and the circle intersect at point F and point E.", "ocr_zh": "过圆外一点P引两条割线PB和PD，PB与圆交于点A和点B，PD与圆交于点C和点D，连接CB和AD，交于点Q，连接PQ并延长至圆上一点F，PF与圆交于点F和点E。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4024.jpg" }, { "index": 4025, "ocr_en": "A point C outside ray BN makes ray BC, point A is on ray BN, connect AC so that angle ACB is acute, make the angle bisector CL of angle ACB, point L is on line AB, make the angle bisector CN of the obtuse angle formed by CA and ray BC, and point N is on ray BN. There is one indicated point in angle BCL and one in angle LCA, and there are two indicated points in angle ACN and the angle to its right.", "ocr_zh": "射线BN外有一点C，做射线BC，点A在射线BN上，连接AC使角ACB为锐角，做角ACB的角平分线CL，点L在线段AB上，做由CA和射线BC构成的钝角的角平分线CN，点N在射线BN上。角BCL和角LCA中各有一个指示点，角ACN和其右边的角中也各有两个指示点。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4025.jpg" }, { "index": 4026, "ocr_en": "In triangle ABC, AP is the bisector of angle BAC, BQ is the bisector of angle ABC, point Q is on AC and point P is on BC. Extend AB to make the auxiliary line BD and connect PD so that PD is parallel to BQ.", "ocr_zh": "在三角形ABC中，AP为角BAC的平分线，BQ为角ABC的平分线，点Q在AC上，点P在BC上。延长AB做辅助线BD，连接PD，使PD平行于BQ。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4026.jpg" }, { "index": 4027, "ocr_en": "Triangle BCD, extend CB to make the auxiliary line BF, make the angle DBF of the bisector BA and the extension of CD to intersect at point A, over the point D to make DE parallel to BC, point E on AB, connect EC, over the point B to make the bisector BG of the angle CBD, BG is the auxiliary line, the point G on the CD.", "ocr_zh": "三角形BCD中，延长CB做辅助线BF，做角DBF的平分线BA与CD的延长线交于点A，过点D做DE平行于BC，点E在AB上，连接EC，过B点做角CBD的平分线BG，BG为辅助线，点G在CD上。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4027.jpg" }, { "index": 4028, "ocr_en": "Triangle ABC, the line segment AT is the bisector of angle BAC, point T on BC, make a straight line l and AT perpendicular to the intersection of the straight line l and AT, AB, AC are above BC, l and AB formed by an acute angle angle and with the AC formed by an acute angle angle are α, extend the BC to make an auxiliary line to intersect with the l, the extension of the line of BC and the l of the angle is β.", "ocr_zh": "三角形ABC中，线段AT是角BAC的平分线，点T在BC上，做直线l与AT垂直，直线l与AT、AB、AC的交点均在BC上方，l与AB形成的锐角夹角和与AC形成的锐角夹角均为α，延长BC做辅助线与l相交，BC的延长线和l的夹角为β。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4028.jpg" }, { "index": 4029, "ocr_en": "This strip consists of Fig. (1) and Fig. (2), where Fig. (1) is on the left side, triangle ABC, line segment AT is the bisector of angle BAC, point T is on BC, make straight line l perpendicular to AT through point C, the acute angle pinched between l and the acute angle formed by the extension line of AB and the acute angle pinched between l and the extension line formed by the extension line of AB and AC are both α, and the angle pinched by BCl is β. Fig. (2) is on the right side, triangle ABC, line segment AT is the angle BAC's bisector, the point T is on BC, make a straight line l perpendicular to AT, the intersection of the line l with AT, AB, and AC are below BC, the acute angle formed by l with the extension of AB and the acute angle formed with the extension of AC's auxiliary line are both α, the extension of CB intersects with l, and the angle between the extension of CB and l is β.", "ocr_zh": "本条由图（1）和图（2）构成，其中图（1）在左侧，三角形ABC中，线段AT是角BAC的平分线，点T在BC上，过点C做直线l与AT垂直，l与AB形成的锐角夹角和与AC形成的锐角夹角均为α，BCl的夹角为β。图（2）在右侧，三角形ABC中，线段AT是角BAC的平分线，点T在BC上，做直线l与AT垂直，直线l与AT、AB、AC的交点均在BC下方，l与AB的延长线形成的锐角夹角和与AC的延长辅助线形成的锐角夹角均为α，延长CB与l相交，CB的延长线和l的夹角为β。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4029.jpg" }, { "index": 4030, "ocr_en": "In triangle ABC, line segment AT is the bisector of angle BAC, point T is on BC, make a straight line l perpendicular to AT, the intersection of straight line l with AT, AB, and AC are all below BC, the acute angle formed by l with the extension of AB and with the extension of the auxiliary line of AC are both α, and the extension of CB intersects with l. The extension of CB and the angle between the extension of CB and l is β.", "ocr_zh": "三角形ABC中，线段AT是角BAC的平分线，点T在BC上，做直线l与AT垂直，直线l与AT、AB、AC的交点均在BC下方，l与AB的延长线形成的锐角夹角和与AC的延长辅助线形成的锐角夹角均为α，延长CB与l相交，CB的延长线和l的夹角为β。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4030.jpg" }, { "index": 4031, "ocr_en": "In triangle ABC, line segment AT is the bisector of angle BAC, point T is on BC, a straight line l is made perpendicular to AT through point C. The angle of the acute angle formed by l with AB and the angle of the acute angle formed with AC are both α, and the angle of the angle of BCl is β.", "ocr_zh": "三角形ABC中，线段AT是角BAC的平分线，点T在BC上，过点C做直线l与AT垂直，l与AB形成的锐角夹角和与AC形成的锐角夹角均为α，BCl的夹角为β。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4031.jpg" }, { "index": 4032, "ocr_en": "Triangle ABC, point A1 and A2 are two points on BC, connect AA1 and AA2 so that the angle CAA2 is equal to the angle BAA1, point P is a point on the AA2, point P to do PP1 perpendicular to AB pendant for P1, to do PP2 perpendicular to AC pendant for P2, connect P1P2, P1P2 and AA1 perpendicular to the pendant for D.", "ocr_zh": "三角形ABC中，点A1和A2是BC上两点，连接AA1和AA2使角CAA2等于角BAA1，点P为AA2上一点，过点P做PP1垂直于AB垂足为P1，做PP2垂直于AC垂足为P2，连接P1P2，P1P2与AA1垂直，垂足为D。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4032.jpg" }, { "index": 4033, "ocr_en": "Point P for the triangle ABC outside BC below a point, over the point P to do PP1 perpendicular to the AB pendant for P1, to do PP2 perpendicular to the AC pendant for P2, connecting P1P2 and AP, AP intersection of BC at point M, over the point A to do AD perpendicular to the P1P2, the point D on the BC.", "ocr_zh": "点P为三角形ABC外边BC下方一点，过点P做PP1垂直于AB垂足为P1，做PP2垂直于AC垂足为P2，连接P1P2和AP，AP交BC于点M，过点A做AD垂直于P1P2，点D在BC上。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4033.jpg" }, { "index": 4034, "ocr_en": "Rhombus ABCD can be divided into two equilateral triangles ABC and ADC, AC is the imaginary segment, do the rhombus of the tangent circle, the center of the circle is O, circle O and AB and BC of the tangent point are E and F, connect the auxiliary line OE and OF, do the auxiliary line OG perpendicular to the AC, the pendant foot is O, the point G in the circle O, point G to do the tangent to the circle O MN, the point M and the point N were in AB and BC, connect the auxiliary line OM and ON.", "ocr_zh": "菱形ABCD可以分成两个等边三角形ABC和ADC，AC为虚线段，做菱形的内切圆，圆心为O，圆O与AB和BC的切点分别是E和F，连接辅助线OE和OF，做辅助线OG垂直于AC，垂足为O，点G在圆O上，过点G做圆O的切线MN，点M和点N分别在AB和BC上，连接辅助线OM和ON。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4034.jpg" }, { "index": 4035, "ocr_en": "There is a circle, the center of the circle is O, the chord AC and chord BD are perpendicular to each other, the pendant is P, connect AB, BC, CD, DA, connect the auxiliary line OP and extend it to the ends to intersect the circle at the point F and the point E, make the auxiliary line ON perpendicular to the BD over the point O, the pendant is N, make the auxiliary line OM perpendicular to the AC over the point O, the pendant is M, the point G is the mid-point of CB, connect the auxiliary line GP and extend it to intersect the DA at the point H.", "ocr_zh": "有一个圆，圆心为O，弦AC和弦BD相垂直，垂足为P，连接AB、BC、CD、DA，连接辅助线OP并向两端延长交圆于点F和点E，过点O做辅助线ON垂直于BD，垂足为N，过点O做辅助线OM垂直于AC，垂足为M，点G为CB中点，连接辅助线GP并延长交DA于点H。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4035.jpg" }, { "index": 4036, "ocr_en": "There is a circle with center O, chord AC and chord BD are perpendicular to each other, and the pendant is P. Connect AB, BC, CD, and DA, connect OP, and make OE perpendicular to DA over point O, and the pendant is E, and make OF perpendicular to CB over point O, and the pendant is F, and connect PF, PE, and EF, and make the auxiliary line PH perpendicular to BC over point P, and the pendant is H, and make the auxiliary line PG perpendicular to DA, and the pendant is G.", "ocr_zh": "有一个圆，圆心为O，弦AC和弦BD相垂直，垂足为P，连接AB、BC、CD、DA，连接OP，过点O做OE垂直于DA，垂足为E，过点O做OF垂直于CB，垂足为F，连接PF、PE和EF，过点P做辅助线PH垂直于BC，垂足为H，做辅助线PG垂直于DA，垂足为G。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4036.jpg" }, { "index": 4037, "ocr_en": "There is a circle with center O, chord AC and chord BD are perpendicular to each other, the pendant foot is P. Connect AB, BC, CD, and DA, connect the line OP, points H, E, G, and F are the midpoints of DC, CB, BA, and AD, respectively, connect EF and HG, and connect the auxiliary lines FG and EG.", "ocr_zh": "有一个圆，圆心为O，弦AC和弦BD相垂直，垂足为P，连接AB、BC、CD、DA，连接线OP，点H、E、G、F分别是DC、CB、BA、AD的中点，连接EF、HG，连接辅助线FG、EG。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4037.jpg" }, { "index": 4038, "ocr_en": "Circle O is internally tangent to convex quadrilateral ABCD, connect OD, OC, OA, OB, make the outer circle of triangle ODC with dashed lines, point E is a point on the outer dashed arc DC of the quadrilateral, connect auxiliary lines EO, EC, ED.", "ocr_zh": "圆O内切于凸四边形ABCD，连接OD、OC、OA、OB，用虚线做三角形ODC的外接圆，点E为四边形外虚线弧DC上一点，连接辅助线EO、EC、ED。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4038.jpg" }, { "index": 4039, "ocr_en": "Triangle ABC, point E and point F are the midpoints of AB and AC respectively, connect EF, do the angle BAC bisector AP, point P on EF, over point P to do the line MN, point M on AE, point N on FC, connect NB and MC intersected at point H, connect auxiliary AH, do the auxiliary AO bisecting angle PAE, intersecting the bisector of the angle MHB HO at point O, over point M to do the auxiliary line MH1 parallel to EF, point H1 on AH, over point H1 to make auxiliary line H1O1 parallel to OH, point O1 on AO, connect H1P, PO1, O1N.", "ocr_zh": "三角形ABC中，点E和点F分别是AB和AC的中点，连接EF，做角BAC的平分线AP，点P在EF上，过点P做线段MN，点M在AE上，点N在FC上，连接NB和MC交于点H，连接辅助线AH，做辅助线AO平分角PAE，交角MHB的平分线HO于点O，过点M做辅助线MH1平行于EF，点H1在AH上，过点H1做辅助线H1O1平行于OH，点O1在AO上，连接H1P、PO1、O1N。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4039.jpg" }, { "index": 4040, "ocr_en": "Two separated circles with centers IC and IB, the two interior tangents of the two circles intersect at point A. The intersection of the two interior tangents and one exterior tangent is point B and point C. The tangent point on AC is D, the tangent point on AB is E. Connect ED, point M is the mid-point of BC, point N is the mid-point of ED, connect MN and extend it to intersect the two interior tangents at point F and point G. Point L is a point on GF. All lines in this figure are dashed except for the two circles and triangle ABC and its interior, which are solid lines.", "ocr_zh": "两个相离的圆，圆心分别为IC和IB，两个圆的两条内公切线交于点A，两条内公切线和一条外公切线的交点分别为点B、点C。AC上的切点为D，AB上的切点为E，连接ED，点M是BC中点，点N为ED中点，连接MN并延长交两条内公切线于点F和点G，点L为GF上一点。本图中除了两个圆和三角形ABC及其内部的线条为实线，其余线均为虚线。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4040.jpg" }, { "index": 4041, "ocr_en": "There is a circle, point P is a point outside the circle, over the point P lead to tangent PE and PF, point E and point F are in the circle, connect EF, over the point P lead to the circle of the two cut line auxiliary line PL and PT, PL intersects the circle at point S and point L, PT and the circle intersects the point K and point T, connect auxiliary line ST and KL intersects the point C, over the point C to do the PC and extend the intersection of the circle at the point A and point B.", "ocr_zh": "有一个圆，点P为圆外一点，过点P引切线PE和PF，点E和点F都在圆上，连接EF，过点P引圆的两条割线辅助线PL和PT，PL交圆于点S和点L，PT和圆交于点K和点T，连接辅助线ST和KL交于点C，过点C做PC并延长交圆于点A和点B。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4041.jpg" }, { "index": 4042, "ocr_en": "There is a circle, the center of the circle is O, point P is a point outside the circle, over the point P to lead the circle of the cut line PAB and PCD, two cut lines on both sides of the center of the circle, point A, B, C, D in the circle, connect BC and AD intersected at point Q, connect QO and OP, connect the auxiliary line PQ, over the point P to do the tangent auxiliary line PT tangent to the circle for the point T, T is in a bad arc AB, connect the auxiliary line OT, to the ends of the extension of the QO in the circle O to intersect at point E and point F, get auxiliary line QE and OF.", "ocr_zh": "有一个圆，圆心为O，点P为圆外一点，过点P引圆的割线PAB和PCD，两条割线在圆心两侧，点A、B、C、D在圆上，连接BC和AD交于点Q，连接QO和OP，连接辅助线PQ，过点P做圆的切线辅助线PT切点为T，T在劣弧AB上，连接辅助线OT，向两端延长QO于圆O交于点E和点F，得到辅助线QE和OF。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4042.jpg" }, { "index": 4043, "ocr_en": "Quadrilateral ABCD inside the circle, connect CA, point K is the midpoint of AC, connect the auxiliary line DK, point L is a point on BC near C, connect LK and extend to intersect the extension line of AB at point T, connect the auxiliary lines LD and DT.", "ocr_zh": "四边形ABCD内接于圆，连接CA，点K为AC中点，连接辅助线DK，点L为BC上靠近C的一点，连接LK并延长交AB的延长线于点T，连接辅助线LD和DT。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4043.jpg" }, { "index": 4044, "ocr_en": "In the square EFGO, circle O is made with O as the center and half of the length of the sides of the square as the radius, circle O intersects with GO at point B, and OE intersects with OE at point D. Extend BO to intersect circle O at point A. Extend DO and extend it to intersect circle O at point C. Take B as the center of the circle, and make the circular arc AE with AB as the radius. the area surrounded by the arc AD, the line segment DE, and the arc AE is shaded, and the rest of the area in the square except for the circle O is are shaded.", "ocr_zh": "正方形EFGO中，以O为圆心，以正方形边长的一半为半径做圆O，圆O与GO交于点B，和OE交于点D，延长BO交圆O于点A，延长DO并延长，交圆O于点C，以B为圆心，以AB为半径做圆弧AE。其中弧AD、线段DE和弧AE围城的区域有阴影，正方形中除了圆O的其余区域都有阴影。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4044.jpg" }, { "index": 4045, "ocr_en": "There is a circle whose center is O, AB is a chord of the circle, point A and point B are on the circle, connect OA and OB, point C is a point on the inferior arc AB, and point D is a point on the superior arc AB.", "ocr_zh": "有一个圆圆心为O，AB为圆的一条弦，点A、点B在圆上，连接OA、OB，点C为劣弧AB上一点，点D为优弧AB上一点。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4045.jpg" }, { "index": 4046, "ocr_en": "A semicircle is made with line segment AB as the diameter, point C is a point on the semicircle, connecting CA and CB, point H is the midpoint on arc BC, and point E is a point on arc AC, and the semicircle is made with the diameters of CA and CB, respectively, and the two semicircles are shaded by two points, G and F. The two regions formed by arcs CGB and arcs CHB, arcs CEA and arcs CFA have shadows in them.", "ocr_zh": "以线段AB为直径做半圆，点C为半圆上一点，连接CA和CB，点H为弧BC上的中点，点E为弧AC上一点，分别以CA和CB为直径做半圆，两个半圆上分别有两点G和点F。弧CGB和弧CHB、弧CEA和弧CFA构成的两个区域内有阴影。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4046.jpg" }, { "index": 4047, "ocr_en": "In a square ABCD, four quarter circles are made in the square centered at each of the four points ABCD, with the side lengths as radii, where the quarter circle centered at A and the quarter circle centered at D intersect at the point E. The area of the region jointly enclosed by the four quarter circles is shaded. The area of the region jointly formed by each of the edges and the quarter-circle centered at the two points of the opposite edge is z, and there are four regions. The area of the region jointly enclosed by the quarter-circle centered at each vertex and the quarter-circle centered at two adjacent vertices is y, and there are four regions in total. The area of the common area jointly enclosed by the four quarter circles is x, totaling one area.", "ocr_zh": "正方形ABCD中，分别以ABCD四个点为圆心，以边长为半径在正方形内分别做四个四分之一圆，其中以A为圆心的四分之一圆和以D为圆心的四分之一圆交于点E，四个四分之一圆共同围城的区域内有阴影。其中各条边和以对边的两个点为圆心的四分之一圆共同构成的区域面积为z，共四个区域。以由以各顶点为圆心的四分之一圆和以两个相邻顶点为圆心的四分之一圆共同构成的区域面积为y，共四个区域。四个四分之一圆共同围成的公共区域面积为x，共一个区域。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4047.jpg" }, { "index": 4048, "ocr_en": "In the square ABCD, four points of ABCD are used as the center of the circle, and the side lengths are used as the radius to make four quarter-circles in the square, in which the quarter-circle centered on A and the quarter-circle centered on D intersect at the point E. The four quarter-circles together encircle the area with a shadow.", "ocr_zh": "正方形ABCD中，分别以ABCD四个点为圆心，以边长为半径在正方形内分别做四个四分之一圆，其中以A为圆心的四分之一圆和以D为圆心的四分之一圆交于点E，四个四分之一圆共同围城的区域内有阴影。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4048.jpg" }, { "index": 4049, "ocr_en": "In right triangle ABC, angle BCA is a right angle, rotate triangle ABC with point B as the center of the circle to get triangle BC1A1, and BC1 and BA are on a straight line, the motion of point C is arc CC1, the motion of point A is arc AA1, and the region formed by arc CC1, arc AA1, line segment A1C1 and AC is shaded.", "ocr_zh": "直角三角形ABC中，角BCA为直角，将三角形ABC以点B为圆心旋转得到三角形BC1A1，且BC1和BA在一条直线上，点C的运动轨迹为弧CC1，点A的运动轨迹为弧AA1，由圆弧CC1、圆弧AA1、线段A1C1和AC构成的区域有阴影。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4049.jpg" }, { "index": 4050, "ocr_en": "There is a sector with O as the center of the circle, OI as the radius, two straight sides are OI and OA, arc IA is an inferior arc, point E is the midpoint of arc IA, connect IA, on the arc to make three strings HB, GC, FD are parallel to IA, the string FD, the string CG and the arc FG, the arc DC constitutes the region of the shadows, the string HB, the string IA and the arc HI, the arc AB constitutes the region of the shadows.", "ocr_zh": "有一个扇形以O为圆心，以OI为半径，两条直线边分别为OI和OA，弧IA为劣弧，点E为弧IA的中点，连接IA，在弧上做三条弦HB、GC、FD均平行于IA，弦FD、弦CG和弧FG、弧DC构成的区域有阴影，弦HB、弦IA和弧HI、弧AB构成的区域有阴影。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4050.jpg" }, { "index": 4051, "ocr_en": "The sector AOI. E is the midpoint of arc AI, and C and G are the midpoints of arcs AE and EI respectively. B, D, F, H are the midpoints of arcs AC, CE, EG, GI respectively. Connect AI, BH, CG, DF, and draw auxiliary lines OE, OF, OG, OH. OE intersects DF, CG, DH, and AI at M, J, K, and N respectively. OF intersects AI at Q, and OG intersects BH at P.", "ocr_zh": "扇形AOI，E为圆弧AI的中点，C和G分别为圆弧AE和圆弧EI的中点，B、D、F、H分别为圆弧AC、圆弧CE、圆弧EG、圆弧GI的中点，连接AI、BH、CG、DF，做辅助线OE、OF、OG、OH，OE分别与DF、CG、DH和AI相交于M、J、K和N，OF与AI相交于Q，OG与BH相交于P", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4051.jpg" }, { "index": 4052, "ocr_en": "In triangle ABC, construct auxiliary triangles. Draw circles with centers at A, B, and C respectively. Outside triangle ABC, draw an auxiliary line parallel to AB and tangent to circle A and circle B. Outside triangle ABC, draw an auxiliary line parallel to BC and tangent to circle B and circle C. Outside triangle ABC, draw an auxiliary line parallel to AC and tangent to circle A and circle C.", "ocr_zh": "三角形ABC，作辅助三角形，分别以A,B,C为圆心画圆，在三角形ABC外做平行于AB且与圆A圆B相切的辅助线，在三角形ABC外做平行于BC且与圆B圆C相切的辅助线，在三角形ABC外做平行于AC且与圆A圆C相切的辅助线", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4052.jpg" }, { "index": 4053, "ocr_en": "In a regular pentagon ABCDE, considering the arcs: arc BE with center A, arc AC with center B, arc BD with center C, arc CE with center D, and arc AD with center E. Let N be the intersection of arc BD and arc CE, M the intersection of arc AD and arc CE, R the intersection of arc AD and arc BE, Q the intersection of arc AC and arc BE, and P the intersection of AC and BD. Draw auxiliary lines CN and DN.", "ocr_zh": "正五边形ABCDE，以A为圆心的圆弧BE，以B为圆心的圆弧AC，以C为圆心的圆弧BD，以D为圆心的圆弧CE，以E为圆心的圆弧AD，N为圆弧BD和圆弧CE的交点，M为圆弧AD和圆弧CE的交点，R为圆弧AD和圆弧BE的交点，Q为圆弧AC和BE交点，P为AC和BD交点，做辅助线CN、DN", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4053.jpg" }, { "index": 4054, "ocr_en": "The circle O is inscribed in square PTXA and triangle IJH. The intersection point of triangle IJH and PT is R, the intersection point with TX is Z, the intersection points with XA are Y and C, and the intersection points with AQ are B and Q.", "ocr_zh": "圆O内切于正方形PTXA和三角形IJH，三角形IJH与PT的交点为R，与TX交点为Z，与XA交点为Y和C，与AQ交点为B和Q", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4054.jpg" }, { "index": 4055, "ocr_en": "In ∠CAB, point A1 is on AB. Draw an arc with center A and radius AA1, intersecting AC at C1. Construct C1A2 perpendicular to AC, intersecting AB at A2. Draw an arc with center A and radius AA2, intersecting AC at C2. Construct C2A3 perpendicular to AC, intersecting AB at A3. Draw an arc with center A and radius AA3, intersecting AC at C3. Construct C3A4 perpendicular to AC, intersecting AB at A4. Draw an arc with center A and radius AA4, intersecting AC at C4. Construct C3A5 perpendicular to AC, intersecting AB at A5.", "ocr_zh": "∠CAB，A1在AB上，以AA1为半径A为圆心做圆弧与AC交于C1，作C1A2垂直于AC交AB于A2，以AA2为半径A为圆心做圆弧与AC交于C2，作C2A3垂直于AC交AB于A3，以AA3为半径A为圆心做圆弧与AC交于C3，作C3A4垂直于AC交AB于A4，以AA4为半径A为圆心做圆弧与AC交于C4，作C4A5垂直于AC交AB于A5", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4055.jpg" }, { "index": 4056, "ocr_en": "In a right triangle ABC, point D lies on AB. With A as the center and AD as the radius, draw an arc intersecting BC at E and the extension of AC at F.", "ocr_zh": "三角形ABC，D在AB上，AD为半径、A为圆心画圆弧交BC于E，交AC延长线于F", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4056.jpg" }, { "index": 4057, "ocr_en": "Equilateral triangle ABC. O is the centroid of triangle ABC. Draw arcs passing through points A, O, C; arcs passing through points B, O, C; and arcs passing through points A, O, B.", "ocr_zh": "正三角形ABC，O为三角形ABC的中心，过A、O、C三点做圆弧，过B、O、C三点做圆弧，过A、O、B三点做圆弧", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4057.jpg" }, { "index": 4058, "ocr_en": "AB is the diameter of semicircle O. Draw OD perpendicular to AB, intersecting semicircle O at point D. With points A and B as centers and AB as the radius, draw arcs intersecting the extensions of AD and BD at F and E respectively. Then, with point D as the center and DE as the radius, draw an arc. Connect E and F. ", "ocr_zh": "AB是半圆O的直径，作OD垂直于AB交半圆O于点D，分别以点A、B为圆心，AB为半径画弧交AD、BD延长线于F、E，再以点D为圆心，DE为半径画弧，连接E、F", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4058.jpg" }, { "index": 4059, "ocr_en": "Equilateral triangle ABC. Draw a circle with center A and radius AB, draw a circle with center B and radius BC, and draw a circle with center C and radius AC.", "ocr_zh": "正三角形ABC，以A为圆心AB为半径画圆，以B为圆心BC为半径画圆，以C为圆心AC为半径画圆", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4059.jpg" }, { "index": 4060, "ocr_en": "AB and CD are two mutually perpendicular chords on circle O, and their foot of the perpendicular is E.", "ocr_zh": "AB与CD是圆O上两条互相垂直的弦，垂足为E", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4060.jpg" }, { "index": 4061, "ocr_en": "The line segment CD is a diameter of the circle O, and the line segment AB is a chord of the circle O, and AB is perpendicular to CD.", "ocr_zh": "线段CD是圆O的一条直径，线段AB是圆O上垂直于直径CD的一条弦", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4061.jpg" }, { "index": 4062, "ocr_en": "AB and CD are two mutually perpendicular chords on circle O, with the foot of the perpendicular being E. Draw the auxiliary line A₁B₁ such that A₁B₁ is a chord of circle O parallel to AB, intersecting CD at H. Draw the auxiliary line C₁D₁ such that C₁D₁ is a chord of circle O parallel to CD, intersecting AB at F and A₁B₁ at G.", "ocr_zh": "AB与CD是圆O上两条互相垂直的弦，垂足为E，做辅助线A1B1，使A1B1为圆O上平行于AB的弦，与CD交于H，做辅助线C1D1，使C1D1为圆O上平行于CD的弦，与AB交于F，与A1B1交于G", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4062.jpg" }, { "index": 4063, "ocr_en": "In triangle ABC, D is the midpoint of BC, and E is a point on AB. Connect ED. With D as the center and DB as the radius, draw the auxiliary circle D. F is a point on circle D. Draw the auxiliary lines EF, FD, AF, and AD.", "ocr_zh": "三角形ABC，D为BC中点，E为AB上一点，连接ED，以D为圆心DB为半径做辅助圆D，F为圆D上的一点，做辅助线EF、FD、AF和AD", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4063.jpg" }, { "index": 4064, "ocr_en": "In triangle ABC, D is the midpoint of BC. E is a point on AB, and F is a point outside triangle ABC. Connect ED, and draw auxiliary lines EF and FD.", "ocr_zh": "三角形ABC，D为BC中点，E为AB上一点，F为三角形ABC外的一点，连接ED，做辅助线EF和FD", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4064.jpg" }, { "index": 4065, "ocr_en": "EF is the diameter of circle O. Point B is on OE, point C is on OF, and point A is on circle O. Connect AB and AC.", "ocr_zh": "EF是圆O的直径，B是OE上的一点，C是OF上的一点，A是圆O上的一点，连接AB、AC", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4065.jpg" }, { "index": 4066, "ocr_en": "AB is the diameter of circle O, CD is a chord of circle O intersecting AB at G. Draw AE perpendicular to CD through A with the foot of the perpendicular at E, draw BF perpendicular to CD through B with the foot of the perpendicular at F, draw the auxiliary line ON perpendicular to CD through O with the foot of the perpendicular at N, and draw the auxiliary line OC.", "ocr_zh": "AB是圆O的直径，CD是圆O的弦与AB交于G，过A做AE垂直CD,垂足为E，过B做BF垂直CD，垂足为F，过O做辅助线ON垂直CD，垂足为N，做辅助线OC。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4066.jpg" }, { "index": 4067, "ocr_en": "AB is the diameter of circle O, CD is a chord of circle O intersecting AB at G. Through point A, draw AE perpendicular to CD with the foot of the perpendicular at E; through point B, draw BF perpendicular to CD with the foot of the perpendicular at F.", "ocr_zh": "AB是圆O的直径，CD是圆O的弦与AB交于G，过A做AE垂直CD,垂足为E，过B做BF垂直CD，垂足为F。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4067.jpg" }, { "index": 4068, "ocr_en": "Circles O₁ and O₂ intersect at points P and Q. Connect O₁O₂. Through point P, draw a secant line parallel to O₁O₂, intersecting circle O₁ at point A and circle O₂ at point B. Through point P, draw another secant line, intersecting circle O₁ at point C and circle O₂ at point D.", "ocr_zh": "圆O1和圆O2交于P、Q，连接O1O2，过P做平行于O1O2的割线交圆O1于A，交圆O2于B，过P做割线交圆O1于C，交圆O2于D。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4068.jpg" }, { "index": 4069, "ocr_en": "Circles O₁ and O₂ intersect at points P and Q. Connect O₁O₂. Draw a secant through P parallel to O₁O₂, intersecting circle O₁ at A and circle O₂ at B. Draw another secant through P, intersecting circle O₁ at C and circle O₂ at D. Draw the auxiliary line O₁G perpendicular to CD with the foot of the perpendicular at G, draw the auxiliary line O₂H perpendicular to CD with the foot of the perpendicular at H, draw the auxiliary line O₁E perpendicular to AB with the foot of the perpendicular at E, draw the auxiliary line O₂F perpendicular to AB with the foot of the perpendicular at F, and draw the auxiliary line O₁M perpendicular to O₂H with the foot of the perpendicular at M.", "ocr_zh": "圆O1和圆O2交于P、Q，连接O1O2，过P做平行于O1O2的割线交圆O1于A，交圆O2于B，过P做割线交圆O1于C，交圆O2于D，过O1做辅助线O1G垂直CD，垂足为G，过O1做辅助线O2H垂直CD，垂足为H，过O1做辅助线O1E垂直AB，垂足为E，过O1做辅助线O2F垂直AB，垂足为F，过O1做辅助线O1M垂直O2H，垂足为M。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4069.jpg" }, { "index": 4070, "ocr_en": "EF is the diameter of circle O. Point B is on OE, point C is on OF, and point A is on circle O. Connect AB and AC, draw the auxiliary diameter AD of circle O, and draw the auxiliary lines BD and CD.", "ocr_zh": "EF是圆O的直径，B是OE上的一点，C是OF上的一点，A是圆O上的一点，连接AB、AC，做圆O的辅助线直径AD，做辅助线BD、CD", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4070.jpg" }, { "index": 4071, "ocr_en": "OA and OB are two radii of circle O. Extend BO to make the diameter of circle O. Draw AC perpendicular to BO through A with the foot of the perpendicular at C. Connect AB. Let P be a point on AB, and connect OA, CP, and OP. Draw the auxiliary line OH perpendicular to AB through O with the foot of the perpendicular at H.", "ocr_zh": "OA、OB是圆O的两条半径，延长BO做圆O的直径，过A做AC垂直BO，垂足为C，连接AB，P为AB上的一点，连接OA、CP、OP,过O做辅助线OH垂直AB，垂足为H。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4071.jpg" }, { "index": 4072, "ocr_en": "OA and OB are two radii of circle O. Extend BO to make the diameter of circle O. Draw AC perpendicular to BO through A with the foot of the perpendicular at C. Connect AB. Let P be a point on AB, and connect OA, CP, and OP.", "ocr_zh": "OA、OB是圆O的两条半径，延长BO做圆O的直径，过A做AC垂直BO，垂足为C，连接AB，P为AB上的一点，连接OA、CP、OP。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4072.jpg" }, { "index": 4073, "ocr_en": "Triangle ABC is inscribed in circle O. M is the midpoint of BC. H is a point outside triangle ABC but inside circle O. Connect AH and HM. The intersection of OA and HM is N. Draw auxiliary line ED as the diameter of circle O, and draw auxiliary lines OC, AE, and DH.", "ocr_zh": "三角形ABC内接于圆O，M为BC的中点，H为三角形ABC外、圆O内的一点，连接AH、HM，连接OA于HM交于N，做辅助线ED为圆O直径，做辅助线OC、AE、DH", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4073.jpg" }, { "index": 4074, "ocr_en": "AB is a chord of circle O. Connect OB. Let E be a point on circle O, and connect BE. Take the minor arc AB and use AB as the axis of symmetry to construct an auxiliary arc AB, which intersects BE at D. Draw auxiliary lines EF, OF, AE, AD, and draw an auxiliary line AH perpendicular to BE with the foot of the perpendicular at H.", "ocr_zh": "AB是圆O的弦，连接OB，E为圆O上的点，连接BE，将劣弧AB以AB为对称轴做辅助圆弧AB交BE于D，做辅助线EF、OF、AE、AD，做辅助线AH垂直BE，垂足为H。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4074.jpg" }, { "index": 4075, "ocr_en": "Triangle ABC is inscribed in circle O. M is the midpoint of BC. H is a point outside triangle ABC but inside circle O. Connect AH and HM, and the connection of OA intersects HM at N.", "ocr_zh": "三角形ABC内接于圆O，M为BC的中点，H为三角形ABC外、圆O内的一点，连接AH、HM，连接OA于HM交于N", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4075.jpg" }, { "index": 4076, "ocr_en": "AB is a chord of circle O. Connect OA and OB. Let E be a point on circle O, and connect BE. Take the minor arc AB as the original arc, and construct the auxiliary arc AB with AB as the axis of symmetry, which intersects BE at point D.", "ocr_zh": "AB是圆O的弦，连接OA、OB，E为圆O上的点，连接BE，将劣弧AB以AB为对称轴做辅助圆弧AB交BE于D。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4076.jpg" }, { "index": 4077, "ocr_en": "AB, CD and EF are chords of the circle O, AB is parallel to CD, and CD is parallel to EF.", "ocr_zh": "AB、CD、EF为圆O上的弦，AB平行于CD，CD平行于EF", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4077.jpg" }, { "index": 4078, "ocr_en": "AB, CD, and EF are chords of circle O. AB intersects CD at G, and AB intersects EF at H.", "ocr_zh": "AB、CD、EF为圆O上的弦，AB与CD相交于G，AB与EF相交于H", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4078.jpg" }, { "index": 4079, "ocr_en": "Points A and C are two points on the circle O, and point B is inside the circle O. Connect AB and BC.", "ocr_zh": "A、C是圆O上的两点，B在圆O内，连接AB、BC", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4079.jpg" }, { "index": 4080, "ocr_en": "Draw a semicircle with AB as the diameter. Let C and D be two points on the semicircle. Connect CD and AD. Let E be the midpoint of AD, and connect CE and BE.", "ocr_zh": "以AB为直径画半圆，C、D为半圆上的两点，连接CD、AD，E为AD的中点，连接CE、BE", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4080.jpg" }, { "index": 4081, "ocr_en": "BC is a chord of circle O, E is the midpoint of BC, A is a point inside circle O. Connect AB, AC, OA, and F is the midpoint of OA.", "ocr_zh": "BC为圆O的一条弦，E为BC中点，A为圆O内的一点，连接AB、AC、OA，F为OA的中点", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4081.jpg" }, { "index": 4082, "ocr_en": "Equilateral triangle ABC, with O as the center of triangle ABC. Draw a circle with center O such that points A, B, and C are all inside the circle. The part enclosed by line AB, line AC, and circle O is S₁; the part enclosed by line AC, line BC, and circle O is S₂; the part enclosed by line AB, line BC, and circle O is S₃.", "ocr_zh": "正三角形ABC，三角形ABC的中心为O，以O为圆心做圆使A、B、C三点均在圆内，直线AB、直线AC和圆O所围成部分为S1，直线AC、直线BC和圆O所围成部分为S2，直线AB、直线BC和圆O所围成部分为S3。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4082.jpg" }, { "index": 4083, "ocr_en": "Point O is a point on circle O₁. A circle (denoted as circle O) is drawn with O as the center, intersecting circle O₁ at points A and B. Point C is on circle O. Connect BC, which intersects circle O₁ at point E; connect AC, which intersects circle O₁ at point D. Connect AB, DE, OC, AD, and BE.", "ocr_zh": "O为圆O1上一点，以O为圆心做圆O与圆O1相交于A、B，C为圆O上一点，连接BC与圆O1相交于E，连接AC与圆O1相交于D，连接AB、DE、OC、AD、BE。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4083.jpg" }, { "index": 4084, "ocr_en": "Points A, B, C, and D are four points on the same circle. Connect AB, AC, AD, BC, BD, and CD. Let I be a point on AD, and extend AD to I₁.", "ocr_zh": "A、B、C、D为同一个圆上的四个点，连接AB、AC、AD、BC、BD、CD，I为AD上的一点，延长AD到I1。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4084.jpg" }, { "index": 4085, "ocr_en": "Points A, B, C, and D are four points on the same circle. Connect AB, AC, AD, BC, BD, and CD. Let I be a point on AD. Extend AD to I₁ and draw auxiliary lines CI, BI, and CI₁. Extend AC to E and draw the auxiliary line CE.", "ocr_zh": "A、B、C、D为同一个圆上的四个点，连接AB、AC、AD、BC、BD、CD，I为AD上的一点，延长AD到I1，做辅助线CI、BI、CI1，延长AC到E，做辅助线CE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4085.jpg" }, { "index": 4086, "ocr_en": "Points A, K, C, H, B, G are points on circle O. Connect AK, KC, CH, HB, BG, GA. Connect AB, BC, CA to form triangle ABC. Let I be a point inside triangle ABC, and connect BI. With I as the center, draw the incircle of triangle ABC, which intersects AB, BC, CA at F, D, E respectively. Connect DF, FE, ED.", "ocr_zh": "A、K、C、H、B、G为圆O上的点，连接AK、KC、CH、HB、BG、GA，连接AB、BC、CA做三角形ABC，I为三角形ABC内的一点，连接BI，以I为圆心做三角形ABC的内切圆分别交AB、BC、CA于F、D、E，连接DF、FE、ED。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4086.jpg" }, { "index": 4087, "ocr_en": "Quadrilateral ABCD is inscribed in a circle. Connect AC, and let M be the midpoint of AC. Extend AB and DC to intersect at E, and let F be the midpoint of CE. Connect BM, BF, and MD to form quadrilateral BFDM. Construct the circumcircle of quadrilateral BFDM, and draw auxiliary lines BD and MF.", "ocr_zh": "四边形ABCD内接于圆，连接AC，M为AC中点，延长AB和DC相交于E，F为CE中点，连接BM、BF、MD构成四边形BFDM，做四边形BFDM的外接圆，做辅助线BD、MF。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4087.jpg" }, { "index": 4088, "ocr_en": "Quadrilateral ABCD is inscribed in a circle. Connect AC, and let M be the midpoint of AC. Extend AB and DC to intersect at E, and let F be the midpoint of CE. Connect BM, BF, and MD to form quadrilateral BFDM. Construct the circumcircle of quadrilateral BFDM, ", "ocr_zh": "四边形ABCD内接于圆，连接AC，M为AC中点，延长AB和DC相交于E，F为CE中点，连接BM、BF、MD构成四边形BFDM，做四边形BFDM的外接圆。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4088.jpg" }, { "index": 4089, "ocr_en": "Circle O is the circumcircle of quadrilateral ABCD. Connect AC and BD, which intersect at F. Let H be the midpoint of AD, and connect OH. Draw the auxiliary line DO and extend it to intersect circle O at E, then draw the auxiliary line AE.", "ocr_zh": "圆O为四边形ABCD的外接圆，连接AC和BD交于F，H为AD中点，连接OH，做辅助线DO并延长交圆O于E，做辅助线AE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4089.jpg" }, { "index": 4090, "ocr_en": "Circle O is the circumcircle of quadrilateral ABCD. Connect AC and BD, which intersect at point F. Let H be the midpoint of AD, and connect OH.", "ocr_zh": "圆O为四边形ABCD的外接圆，连接AC和BD交于F，H为AD中点，连接OH。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4090.jpg" }, { "index": 4091, "ocr_en": "In quadrilateral ABCD, connect AC and BD.", "ocr_zh": "四边形ABCD，连接AC、BD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4091.jpg" }, { "index": 4092, "ocr_en": "In quadrilateral ABCD, connect AC and BD to form triangle ABD. Construct an auxiliary circumcircle of triangle ABD. Extend AC and draw an auxiliary line CE to intersect the auxiliary circumcircle at E. Draw auxiliary lines BE and DE.", "ocr_zh": "四边形ABCD，连接AC、BD并构成三角形ABD，做三角形ABD的辅助外接圆，延长AC做辅助线CE交辅助外接圆于E，做辅助线BE、DE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4092.jpg" }, { "index": 4093, "ocr_en": "In quadrilateral ABCD, connect AC and BD to form triangle BCD. Construct an auxiliary circumcircle of triangle BCD. Extend CA and draw an auxiliary line AF to intersect the auxiliary circumcircle at F. Draw auxiliary lines BF and DF.", "ocr_zh": "四边形ABCD，连接AC、BD并构成三角形BCD，做三角形BCD的辅助外接圆，延长CA做辅助线AF交辅助外接圆于F，做辅助线BF、DF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4093.jpg" }, { "index": 4094, "ocr_en": "In quadrilateral ABCD, connect AC and BD to form triangle BCD. Construct an auxiliary circumcircle O of triangle BCD, and draw auxiliary lines OA, OB, OC, and OD.", "ocr_zh": "四边形ABCD，连接AC、BD并构成三角形BCD，做三角形BCD的辅助外接圆O，做辅助线OA、OB、OC、OD。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4094.jpg" }, { "index": 4095, "ocr_en": "In quadrilateral ABCD, connect AC and BD.", "ocr_zh": "四边形ABCD，连接AC、BD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4095.jpg" }, { "index": 4096, "ocr_en": "Points A, D, B, P, F, C are points on a circle. Connect AB and CD, which intersect at Q. Connect BF and DP, which intersect at E. Connect CF. Connect AP, which intersects CD at G. Connect QE. Draw auxiliary lines AD, BD, BP. Extend FB to draw an auxiliary line BN to point N.", "ocr_zh": "A、D、B、P、F、C为圆上的点，连接AB和CD交于Q，连接BF和DP交于E，连接CF，连接AP交CD于G，连接QE，做辅助线AD、BD、BP，延长FB做辅助线BN于N", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4096.jpg" }, { "index": 4097, "ocr_en": "AB is a chord of circle O. Connect OA and OB. Let M and N be points on circle O, and P be a point outside circle O. Connect AP, which intersects circle O at C. Connect BP. Connect MP, which intersects AB at E. Connect NP, which intersects AB at F. Draw auxiliary lines OC, OM, ON, AM, AN, and MN. Extend PA and NM as auxiliary lines, which intersect at Q.", "ocr_zh": "AB为圆O的弦，连接OA和OB，M、N为圆O上的点，P为圆O外一点，连接AP交圆O于C，连接BP，连接MP交AB于E，连接NP交AB于F，做辅助线OC、OM、ON、AM、AN、MN，分别做PA和NM的延长辅助线交于Q。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4097.jpg" }, { "index": 4098, "ocr_en": "Points A, D, B, P, F, C are points on a circle. Connect AB and CD, which intersect at Q. Connect BF and DP, which intersect at E. Connect CF. Connect AP. Connect QE. Draw auxiliary lines AD, BD, BP. Extend FB to draw an auxiliary line BN to point N.", "ocr_zh": "A、D、B、P、F、C为圆上的点，连接AB和CD交于Q，连接BF和DP交于E，连接CF，连接AP，连接QE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4098.jpg" }, { "index": 4099, "ocr_en": "AB is a chord of circle O. Connect OA and OB. Let M and N be points on circle O, and P be a point outside circle O. Connect AP and BP. Connect MP, which intersects AB at E, and connect NP, which intersects AB at F.", "ocr_zh": "AB为圆O的弦，连接OA和OB，M、N为圆O上的点，P为圆O外一点，连接AP、BP，连接MP交AB于E，连接NP交AB于F。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4099.jpg" }, { "index": 4100, "ocr_en": "Points A, P, C, Q, B, K, M, L are points on circle O. Connect AP, PQ, QB, AB, AC, BC, PK, CM, LK, LQ. Let PQ intersect BC at D, and connect AD. Let AC intersect PQ at E, and connect BE. Draw auxiliary lines CP and CQ.", "ocr_zh": "A、P、C、Q、B、K、M、L为圆O上的点，连接AP、PQ、QB、AB、AC、BC、PK、CM、LK、LQ，PQ与BC交于D，连接AD，AC与PQ交于E，连接BE，做辅助线CP和CQ。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4100.jpg" }, { "index": 4101, "ocr_en": "Points A, E, C, B, D are points on a circle. Connect AC, BC, AB, DE, CD, BE.", "ocr_zh": "A、E、C、B、D为圆上的点，连接AC、BC、AB、DE、CD、BE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4101.jpg" }, { "index": 4102, "ocr_en": "In quadrilateral ABCD, connect AC and BD.", "ocr_zh": "四边形ABCD，连接AC、BD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4102.jpg" }, { "index": 4103, "ocr_en": "Circles O₁ and O₂ intersect at A and B. Let C be a point on circle O₁. Connect AC, which intersects circle O₂ at E. Let D be a point on circle O₂. Connect AD, which intersects circle O₁ at F. Connect AB.", "ocr_zh": "圆O1和圆O2交于A和B，C为圆O1上的一点，连接AC交圆O2于E，D为圆O2上的一点，连接AD交圆O1于F，连接AB。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4103.jpg" }, { "index": 4104, "ocr_en": "Regular pentagon ABCDE. Connect AC, AD, BE, CE, BD, and construct the circumcircle of pentagon ABCDE.", "ocr_zh": "正五边形ABCDE，连接AC、AD、BE、CE、BD，做五边形ABCDE的外接圆。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4104.jpg" }, { "index": 4105, "ocr_en": "Points A, B, C, and D are points on circle O. Connect AB, BC, CD, OA, OD, AD, OB, and OC. Extend AB and CD to intersect at point P.", "ocr_zh": "A、B、C、D为圆O上的点，连接AB、BC、CD、OA、OD、AD、OB、OC，延长AB和CD交于P。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4105.jpg" }, { "index": 4106, "ocr_en": "Circle O is the circumcircle of quadrilateral ABCD. Connect AD, let E be a point on AD, and connect BE.", "ocr_zh": "圆O为四边形ABCD的外接圆，连接AD，E为AD上的点，连接BE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4106.jpg" }, { "index": 4107, "ocr_en": "AB is the diameter of circle O, and CD is a chord of circle O. Connect OC and BC. Let E be a point on the minor arc BOC. Connect AE, which intersects OC at M, and connect ED, which intersects BC at N.", "ocr_zh": "AB是圆O的直径，CD是圆O的弦，连接OC、BC，E为劣弧BOC上的点，连接AE交OC于M，连接ED交BC于N。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4107.jpg" }, { "index": 4108, "ocr_en": "Construct a semicircle with AB as the diameter, let C be a point on the arc AB, connect BC, and take BC as the axis of symmetry to make a symmetric auxiliary arc of arc BC, which intersects AB at D.", "ocr_zh": "以AB为直径做半圆，C为圆弧AB上的点，连接BC，圆弧BC以BC为对称轴做对称辅助圆弧交AB于D。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4108.jpg" }, { "index": 4109, "ocr_en": "In triangle ABC, let D be a point on BC. Connect AD to form triangle ABD. Construct the circumcircle O of triangle ABC, and connect OA. Let the circumcircle of triangle ABD intersect OA at point E, then connect BE and OC.", "ocr_zh": "三角形ABC，D为BC上的一点，连接AD构成三角形ABD，做三角形ABC的外接圆O，连接OA，做三角形ABD的外接圆交OA于E，连接BE、OC", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4109.jpg" }, { "index": 4110, "ocr_en": "Circle O is the circumcircle of the equilateral triangle ABC. Let M be a point on the minor arc AC. Connect AM, BM, and CM. Let N be a point on BM and K be a point on CM. Connect AN and NK. Let P be a point on the minor arc CM, and connect KP.", "ocr_zh": "圆O为正三角形ABC的外接圆，M为劣弧AC上的一点，连接AM、BM、CM，N为BM上的一点，K为CM上一点，连接AN、NK，P为劣弧CM上的一点，连接KP。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4110.jpg" }, { "index": 4111, "ocr_en": "Point O is a point on circle D. A circle is drawn with center O, intersecting circle D at points A and B. Connect AB and BD. Point C is a point on circle O, and connect BC.", "ocr_zh": "O是圆D上的一点，以O为圆心画圆与圆D相交于A、B，连接AB、BD，C为圆O上的一点，连接BC。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4111.jpg" }, { "index": 4112, "ocr_en": "Point O is a point on circle D. A circle is drawn with center O, intersecting circle D at points A and B. Connect AB and BD. Point C is a point on circle O, and connect BC. Draw auxiliary lines OA, OB, OC, and OD.", "ocr_zh": "O是圆D上的一点，以O为圆心画圆与圆D相交于A、B，连接AB、BD，C为圆O上的一点，连接BC，做辅助线OA、OB、OC、OD。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4112.jpg" }, { "index": 4113, "ocr_en": "AB is a chord of circle O, and Q is a point on the minor arc AB. Connect AQ and BQ. P is a point outside circle O, connect AP and BP. C is a point inside circle O (point C is not shown in the figure and is assumed), connect AC and BC.", "ocr_zh": "AB是圆O的弦，Q是劣弧AB上的一点，连接AQ、BQ，P为圆O外的一点，连接AP、BP，C为圆O内的一点（图中没有C点，自己假设的C点），连接AC、BC。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4113.jpg" }, { "index": 4114, "ocr_en": "Isosceles trapezoid ABCD, with AB parallel to CD. Let O be the midpoint of AD. Draw circle O with AD as the diameter (and O as the center), which intersects AB at point E. Let F be the midpoint of BC, and connect OF, DF, and EF.", "ocr_zh": "等腰梯形ABCD，AB平行于CD，O为AD的中点，以O为圆心AD为直径做圆O交AB于E，F为BC中点，连接OF、DF、EF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4114.jpg" }, { "index": 4115, "ocr_en": "In isosceles trapezoid ABCD, AB is parallel to CD. Let O be the midpoint of AD. With O as the center and AD as the diameter, draw circle O, which intersects AB at point E. Let F be the midpoint of BC, and connect OF, DF, and EF. Draw auxiliary lines OE and DE.", "ocr_zh": "等腰梯形ABCD，AB平行于CD，O为AD的中点，以O为圆心AD为直径做圆O交AB于E，F为BC中点，连接OF、DF、EF，做辅助线OE、DE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4115.jpg" }, { "index": 4116, "ocr_en": "Circles A, B, and C are pairwise externally tangent and all tangent to the line l. The point of tangency of circle A with l is D, that of circle B with l is F, and that of circle C with l is E. Draw auxiliary lines AD, AC, AB, BC, BF, and CE.", "ocr_zh": "圆A、圆B、圆C两两外切，且都与直线l相切，圆A与l的切点为D，圆B与l的切点为F，圆C与l的切点为E，做辅助线AD、AC、AB、BC、BF、CE", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4116.jpg" }, { "index": 4117, "ocr_en": "Circles A, B and C are pairwise externally tangent and all tangent to the line l. The point of tangency of circle A with l is D, that of circle B with l is F, and that of circle C with l is E.", "ocr_zh": "圆A、圆B、圆C两两外切，且都与直线l相切，圆A与l的切点为D，圆B与l的切点为F，圆C与l的切点为E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4117.jpg" }, { "index": 4118, "ocr_en": "In triangle ABC, point D is outside triangle ABC. Connect BD to intersect AC at E, and connect CD. Draw an auxiliary line EG through E perpendicular to BC, with the foot of the perpendicular being G. Let O₃ be the midpoint of BC, and draw the auxiliary line EO₃.", "ocr_zh": "三角形ABC，D为三角形ABC外的一点，连接BD交AC于E，连接CD，过E做辅助线EG垂直于BC，垂足为G，O3为BC的中点，做辅助线EO3。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4118.jpg" }, { "index": 4119, "ocr_en": "In quadrilateral ABCD, connect AC and BD, and AC intersects BD at E.", "ocr_zh": "四边形ABCD，连接AC与BD，AC与BD相交于E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4119.jpg" }, { "index": 4120, "ocr_en": "In quadrilateral ABCD, connect AC and BD, which intersect at E. Let O₁ be a point on AB and O₂ be a point on CD. Draw the auxiliary line O₁O₂. Draw an auxiliary line O₁H through O₁ perpendicular to CD, with the foot of the perpendicular being H.", "ocr_zh": "四边形ABCD，连接AC与BD，AC与BD相交于E，O1为AB上一点，O2为CD上一点，做辅助线O1O2，过O1做辅助线O1H垂直于CD，垂足为H。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4120.jpg" }, { "index": 4121, "ocr_en": "Point B is a point on AD, and point C is a point on BD. O₁ and O₂ are the midpoints of AB and CD respectively. Draw a semicircle with O₁ as the center and AB as the diameter, and draw another semicircle with O₂ as the center and CD as the diameter. Draw a tangent from point A to the semicircle O₂, with the tangent point being E, and draw a tangent from point D to the semicircle O₁, with the tangent point being F. Point O₃ is a point on AB, draw a semicircle with O₃ as the center such that it is tangent to AE, with the tangent point being H. Point O₄ is a point on CD, draw a semicircle with O₄ as the center such that it is tangent to DF. Draw auxiliary lines O₃H and O₂E.", "ocr_zh": "B是AD上的一点，C是BD上的一点，O1和O2分别是AB和CD的中点，以O1为圆心AB为直径做半圆，以O2为圆心CD为直径做半圆，过A做半圆O2的切线，切点为E，过D做半圆O1的切线，切点为F，O3为AB上一点，以O3为圆心做半圆使其与AE相切，切点为H，O4为CD上一点，以O4为圆心做半圆使其与DF相切，做辅助线O3H、O2E。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4121.jpg" }, { "index": 4122, "ocr_en": "Point B is a point on AD, and point C is a point on BD. O₁ and O₂ are the midpoints of AB and CD respectively. Draw a semicircle with O₁ as the center and AB as the diameter, and draw another semicircle with O₂ as the center and CD as the diameter. Draw a tangent from point A to the semicircle O₂, with the tangent point being E, and draw a tangent from point D to the semicircle O₁, with the tangent point being F. Point O₃ is a point on AB, draw a semicircle with O₃ as the center such that it is tangent to AE, with the tangent point being H. Point O₄ is a point on CD, draw a semicircle with O₄ as the center such that it is tangent to DF.", "ocr_zh": "B是AD上的一点，C是BD上的一点，O1和O2分别是AB和CD的中点，以O1为圆心AB为直径做半圆，以O2为圆心CD为直径做半圆，过A做半圆O2的切线，切点为E，过D做半圆O1的切线，切点为F，O3为AB上一点，以O3为圆心做半圆使其与AE相切，切点为H，O4为CD上一点，以O4为圆心做半圆使其与DF相切。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4122.jpg" }, { "index": 4123, "ocr_en": "Let O be the midpoint of AB. Draw a semicircle with O as the center and AB as the diameter. Let C be a point on the semicircle O. Connect AC and BC to form triangle ABC. Construct circle O₁ such that it is tangent to both AC and the semicircle O. Construct circle O₂ such that it is tangent to both BC and the semicircle O. Construct the incircle O₃ of triangle ABC.", "ocr_zh": "O为AB的中点，以O为圆心AB为直径做半圆，C为半圆O上的一点，连接AC、BC构成三角形ABC，做圆O1使其与AC和半圆O同时相切，做圆O2使其与BC和半圆O同时相切，做三角形ABC的内切圆O3。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4123.jpg" }, { "index": 4124, "ocr_en": "Five circles of the same size are distributed along the diagonal direction in a square, and each circle is tangent to the other circles and the sides of the square.", "ocr_zh": "五个相同大小的圆，在一个正方形中按对角线方向分布，且每个圆与其他圆和正方形的边均相切。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4124.jpg" }, { "index": 4125, "ocr_en": "Circles O₁ and O₂ are separate (or: disjoint). Points A and B lie on circle O₁, while points C and D lie on circle O₂. Connect O₁O₂, AB, CD, BC, AD, O₁A, O₁B, O₂C, and O₂D.", "ocr_zh": "圆O1与圆O2相离，A、B在圆O1上，C、D在圆O2上，连接O1O2、AB、CD、BC、AD、O1A、O1B、O2C、O2D。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4125.jpg" }, { "index": 4126, "ocr_en": "Points A, D, and E are on the same circle, while points B and C are inside the circle. Connect AB, BC, AC, BD, DE, and CE, where AB = 2 and BD = 2.", "ocr_zh": "A、D、E三点在同一圆上，B、C在该圆内，连接AB、BC、AC、BD、DE、CE，AB=2，BD=2。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4126.jpg" }, { "index": 4127, "ocr_en": "Circles O₁, O₂, and O₃ are pairwise externally tangent. Circle O₄ is internally tangent to circles O₁, O₂, and O₃. Connect O₃O₄ and extend it to intersect circle O₃.", "ocr_zh": "圆O1、圆O2和圆O3两两外切，圆O3与圆O1、圆O2和圆O3均内切，连接O3O4并延长与圆O3相交。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4127.jpg" }, { "index": 4128, "ocr_en": "Points A, B, and D are on circle O. Connect AB, BD, and AD. Point C is outside circle O. Connect AC. Draw the auxiliary line AO and extend it to intersect circle O at E. Draw the auxiliary line BE.", "ocr_zh": "A、B、D为圆O上的点，连接AB、BD、AD，C为圆O外一点，连接AC，做辅助线AO并延长交圆O于E，做辅助线BE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4128.jpg" }, { "index": 4129, "ocr_en": "Point P is a point outside circle O. Connect OP. Draw two tangent lines from P to circle O, with the tangent points being E and F. Extend PE and PF to form the rays PE and PF respectively.", "ocr_zh": "P为圆O外一点，连接OP，过P做圆O的两条切线，切点为E、F，延长PE、PF分别构成射线PE、射线PF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4129.jpg" }, { "index": 4130, "ocr_en": "Point P is a point outside the circle, and point B is on the circle. Connect BP and let it intersect the circle at point A. Draw a tangent from point P to the circle, with the tangent point being C. Extend PC to form the ray PC.", "ocr_zh": "P为圆外一点，B为圆上一点，连接BP交圆于A，过P做圆的切线，切点为C，延长PC构成射线PC。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4130.jpg" }, { "index": 4131, "ocr_en": "Circle I is the incircle of the right isosceles triangle ABC, where AB is perpendicular to AC. The points of tangency of circle I with AB, BC, and AC are E, M, and F respectively. Connect CI and extend it to intersect AB at P. Point D is on circle I, connect DM. Draw auxiliary lines IE, IF, ID, and IM.", "ocr_zh": "圆I为直角等腰三角形ABC的内切圆，AB垂直于AC，与AB、BC、AC的切点分别为E、M、F，连接CI并延长交AB于P，D为圆I上一点，连接DM，做辅助线IE、IF、ID、IM。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4131.jpg" }, { "index": 4132, "ocr_en": "Circle I is the incircle of the right isosceles triangle ABC, where AB is perpendicular to AC. The points of tangency of circle I with AB, BC, and AC are E, M, and F respectively. Connect CI and extend it to intersect AB at P. Point D is on circle I, connect DM. ", "ocr_zh": "圆I为直角等腰三角形ABC的内切圆，AB垂直于AC，与AB、BC、AC的切点分别为E、M、F，连接CI并延长交AB于P，D为圆I上一点，连接DM。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4132.jpg" }, { "index": 4133, "ocr_en": "Circle O is the incircle of triangle ABC, and the points of tangency of circle O with AB, BC, and AC are F, D, and E respectively.", "ocr_zh": "圆O为三角形ABC的内切圆，圆O与AB、BC、AC的切点分别为F、D、E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4133.jpg" }, { "index": 4134, "ocr_en": "Circle O is the incircle of quadrilateral ABCD.", "ocr_zh": "圆O为四边形ABCD的内切圆。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4134.jpg" }, { "index": 4135, "ocr_en": "Circle O is the incircle of triangle ABC.", "ocr_zh": "圆O为三角形ABC的内切圆", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4135.jpg" }, { "index": 4136, "ocr_en": "Construct the circumcircle of triangle ABC. Let Q be on the extension of CA, and connect AQ and BQ. Let P be on the extension of CB, and connect BP and AP. Let R be a point on AP, and connect BR. Let X be a point on BQ, and let Y be a point on AP.", "ocr_zh": "做三角形ABC的外接圆，Q在CA延长线上，连接AQ、BQ，P在CB延长线上，连接BP、AP，R为AP上的点，连接BR，X为BQ上的点，Y为AP上的点。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4136.jpg" }, { "index": 4137, "ocr_en": "Construct the circumcircle of triangle ABC. Let Q be on the extension of CA, and connect AQ and BQ. Let P be on the extension of CB, and connect BP and AP. Let R be a point on AP, and connect BR. Let X be a point on BQ, and let Y be a point on AP. Connect XY. Let D be the midpoint of AB, and draw auxiliary lines DX and DY.", "ocr_zh": "做三角形ABC的外接圆，Q在CA延长线上，连接AQ、BQ，P在CB延长线上，连接BP、AP，R为AP上的点，连接BR，X为BQ上的点，Y为AP上的点，连接XY，D为AB中点，做辅助线DX、DY。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4137.jpg" }, { "index": 4138, "ocr_en": "Construct the incircle of triangle ABC, with the points of tangency on AB, BC, and AC being F, D, and E respectively. Connect EF, let G be a point on EF, and connect BG, DG, and CG. Draw an auxiliary line BM through B perpendicular to the extended line of GF, with the foot of the perpendicular being M. Draw an auxiliary line CN through C perpendicular to the extended line of GE, with the foot of the perpendicular being N.", "ocr_zh": "做三角形ABC的内切圆，与AB、BC、AC的切点分别为F、D、E，连接EF，G为EF上的点，连接BG、DG、CG，过B做辅助线BM垂直GF辅助延长线，垂足为M，过C做辅助线CN垂直GE辅助延长线，垂足为N。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4138.jpg" }, { "index": 4139, "ocr_en": "Construct the incircle of triangle ABC, with the points of tangency on AB, BC, and AC being F, D, and E respectively. Connect EF, let G be a point on EF, and connect BG, DG, and CG.", "ocr_zh": "做三角形ABC的内切圆，与AB、BC、AC的切点分别为F、D、E，连接EF，G为EF上的点，连接BG、DG、CG。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4139.jpg" }, { "index": 4140, "ocr_en": "In triangle ABC, D is the midpoint of BC. Connect AD. The incircle O of triangle ABC intersects AD at points X and Y. Connect OX and OY.", "ocr_zh": "三角形ABC，D为BC中点，连接AD，做三角形ABC的内切圆O与AD相交于X、Y，连接OX、OY。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4140.jpg" }, { "index": 4141, "ocr_en": "In triangle ABC, D is the midpoint of BC. Connect AD. Let the incircle O of triangle ABC intersect AD at points X and Y. Connect OX and OY. Draw the auxiliary line OF perpendicular to BC from O with the foot at F, draw the auxiliary line OE perpendicular to AB from O with the foot at E, and draw the auxiliary line OH perpendicular to AD from O with the foot at H.", "ocr_zh": "三角形ABC，D为BC中点，连接AD，做三角形ABC的内切圆O与AD相交于X、Y，连接OX、OY，过O做辅助线OF垂直于BC，垂足为F，过O做辅助线OE垂直于AB，垂足为E，过O做辅助线OH垂直于AD，垂足为H。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4141.jpg" }, { "index": 4142, "ocr_en": "In quadrilateral EHGF, point A lies on EH, point B on EF, point C on FG, and point D on GH. Given AH = a, BF = b, CG = c, and DG = d, connect AB, BC, CD, and AD to form quadrilateral ABCD. Inscribe circle O in quadrilateral ABCD, and connect OA, OB, OC, and OD.", "ocr_zh": "四边形EHGF，A在EH上，B在EF上，C在FG上，D在GH上，AH=a，BF=b，CG=c，DG=d，连接AB、BC、CD、AD构成四边形ABCD，做四边形ABCD的内切圆O，连接OA、OB、OC、OD。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4142.jpg" }, { "index": 4143, "ocr_en": "In quadrilateral EHGF, point A is on EH, point B on EF, point C on FG, and point D on GH. Given AH = a, BF = b, CG = c, and DG = d, connect AB, BC, CD, and AD to form quadrilateral ABCD. Inscribe circle O in quadrilateral ABCD, connect OA, OB, OC, and OD, and draw the auxiliary line OE.", "ocr_zh": "四边形EHGF，A在EH上，B在EF上，C在FG上，D在GH上，AH=a，BF=b，CG=c，DG=d，连接AB、BC、CD、AD构成四边形ABCD，做四边形ABCD的内切圆O，连接OA、OB、OC、OD，做辅助线OE。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4143.jpg" }, { "index": 4144, "ocr_en": "There are two circles inscribed, with the tangent point being P. The chord AB of the larger circle is tangent to the smaller circle at the tangent point Q. Connect AP and it intersects the smaller circle at C. Connect BP and it intersects the smaller circle at D. Connect CD and BP, and they intersect at H.", "ocr_zh": "有两圆内切，切点为P，大圆弦AB与小圆相切，切点为Q，连接AP与小圆相交于C，连接BP与小圆相交于D，连接CD和PQ并相交于H。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4144.jpg" }, { "index": 4145, "ocr_en": "The line segment AmAm+1, with O being a point outside the line segment AmAm+1. Connect OAm and OAm+1. Draw a perpendicular line from O to AmAm+1 with the foot of the perpendicular being Hm. Draw a ray from Am and draw a perpendicular line from O to the ray with the foot of the perpendicular being Hm-1. Draw a ray from Am+1 and draw a perpendicular line from O to the ray with the foot of the perpendicular being Hm+1.", "ocr_zh": "线段AmAm+1，O为线段AmAm+1外一点，连接OAm、OAm+1,过O做AmAm+1的垂线垂足为Hm，过Am做射线并过O做射线的垂线垂足为Hm-1，过Am+1做射线并过O做射线的垂线垂足为Hm+1。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4145.jpg" }, { "index": 4146, "ocr_en": "Construct a semicircle O with AB as the diameter. Let P be a point on AB. Draw an arc with center A and radius AP, intersecting the semicircle O at point C. Draw an arc with center B and radius BP, intersecting the semicircle O at point D. Connect CD, and draw a perpendicular line from P to AB, intersecting CD at point M.", "ocr_zh": "以AB为直径做半圆O，P为AB上一点，以A为圆心AP为半径画圆弧与半圆O交于C，以B为圆心BP为半径画圆弧与半圆O交于D，连接CD，过P做AB的垂线交CD于M。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4146.jpg" }, { "index": 4147, "ocr_en": "Construct a semicircle O with AB as the diameter. Let P be a point on AB. Draw an arc with center A and radius AP, intersecting the semicircle O at point C. Draw an arc with center B and radius BP, intersecting the semicircle O at point D. Connect CD, and draw a perpendicular line from P to AB, intersecting CD at point M. Draw auxiliary lines AC, BC, AD, and BD. Draw an auxiliary perpendicular line from C to AB with the foot at E, and draw an auxiliary perpendicular line from D to AB with the foot at F.", "ocr_zh": "以AB为直径做半圆O，P为AB上一点，以A为圆心AP为半径画圆弧与半圆O交于C，以B为圆心BP为半径画圆弧与半圆O交于D，连接CD，过P做AB的垂线交CD于M，做辅助线AC、BC、AD、BD，过C做AB的辅助垂线垂足为E，过D做AB的辅助垂线垂足为F。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4147.jpg" }, { "index": 4148, "ocr_en": "Two circles O₁ and O₂ are externally tangent at P. Let A be a point on circle O₂ and B be a point on circle O₁. Connect BP and extend it to intersect circle O₁ at D. Connect O₁O₂ and AB. Draw a perpendicular line from P to O₁O₂, intersecting AB at C. Draw auxiliary lines AP, O₂C, O₁C, O₁B, and AD, with O₂ lying on AD.", "ocr_zh": "圆O1和圆O2外切于P，A为圆O2上一点，B为圆O1上一点，连接BP并延长交圆O1于D，连接O1O2、AB，过P做O1O2的垂线交AB于C，做辅助线AP、O2C、O1C、O1B、AD，且O2在AD上。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4148.jpg" }, { "index": 4149, "ocr_en": "Two circles O₁ and O₂ are externally tangent at point P. Let A be a point on circle O₂ and B be a point on circle O₁. Connect BP and extend it to intersect circle O₁ at point D. Connect O₁O₂ and AB. Draw a perpendicular line from P to O₁O₂, intersecting AB at point C.", "ocr_zh": "圆O1和圆O2外切于P，A为圆O2上一点，B为圆O1上一点，连接BP并延长交圆O1于D，连接O1O2、AB，过P做O1O2的垂线交AB于C。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4149.jpg" }, { "index": 4150, "ocr_en": "In quadrilateral ABCD, connect BD to form triangle ABD and triangle CBD. Inscribe incircle O₁ in triangle ABD, where circle O₁ intersects AB, BD, and AD at E, Q, and H respectively. Inscribe incircle O₂ in triangle CBD, where circle O₂ intersects BC, CD, and BD at F, G, and P respectively, and Q coincides with P.", "ocr_zh": "四边形ABCD，连接BD构成三角形ABD和三角形CBD，做三角形ABD的内切圆O1，圆O1与AB、BD、AD分别相交于E、Q、H，做三角形CBD的内切圆O2，圆O2与BC、CD、BD分别相交于F、G、P，其中Q与P重合。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4150.jpg" }, { "index": 4151, "ocr_en": "In quadrilateral ABCD, connect BD to form triangle ABD and triangle CBD. Inscribe the incircle O₁ of triangle ABD and the incircle O₂ of triangle CBD.", "ocr_zh": "四边形ABCD，连接BD构成三角形ABD和三角形CBD，做三角形ABD的内切圆O1，做三角形CBD的内切圆O2。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4151.jpg" }, { "index": 4152, "ocr_en": "Two circles are externally tangent at A. Point B is on the smaller circle, and point C is on the larger circle. Connect BC, which intersects the smaller circle at D and the larger circle at E. Connect AB, AD, AE, AC, and extend BA to intersect the larger circle at F.", "ocr_zh": "两圆外切于A，B在小圆上，C在大圆上，连接BC交小圆于D、大院于E，连接AB、AD、AE、AC，延长BA交大圆于F。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4152.jpg" }, { "index": 4153, "ocr_en": "In triangle ABC, D is the midpoint of BC. Connect AD to form triangles ABD and ACD. Inscribe incircles in triangles ABD and ACD, and draw the common tangent of the two circles, which intersects AD at point K.", "ocr_zh": "三角形ABC，D为BC的中点，连接AD构成三角形ABD和三角形ACD，做三角形ABD和三角形ACD的内切圆，做两圆的公共切线与AD相交于K。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4153.jpg" }, { "index": 4154, "ocr_en": "BC is the diameter of circle O, and A is a point outside circle O. Connect AB and AC. F is a point on the extension of AC, connect CF. Draw a perpendicular line from F to AB with the foot at E, and draw a tangent from A to circle O with the tangent point at D.", "ocr_zh": "BC为圆O的直径，A为圆O外一点，连接AB、AC，F为AC延长线上一点，连接CF，过F做AB垂线垂足为E，过A做圆O的切线切点为D。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4154.jpg" }, { "index": 4155, "ocr_en": "In the right triangle ABC, where AB is perpendicular to BC, an incircle of triangle ABC is constructed, intersecting AB, BC, and AC at D, E, and F respectively. Connect AD, which intersects the incircle at P. Then connect PF, PE, DF, DE, and PC.", "ocr_zh": "直角三角形ABC，AB垂直BC，做三角形ABC的内切圆与AB、BC、AC分别交于F、D、E，连接AD与内切圆交于P，连接PF、PE、DF、DE、PC。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4155.jpg" }, { "index": 4156, "ocr_en": "In quadrilateral ABCD, point O is on AB. Draw a semicircle with center O such that it is tangent to AD, CD, and BC.", "ocr_zh": "四边形ABCD，O在AB上，以O为圆心做半圆使其与AD、CD、BC相切。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4156.jpg" }, { "index": 4157, "ocr_en": "Construct the circumcircle of triangle ABC. Draw a line l through point C such that l is tangent to the circle. Let D be the midpoint of BC, and connect AD. Draw a perpendicular line from B to AC with the foot at E, a perpendicular line from E to l with the foot at G, and a perpendicular line from D to l with the foot at F.", "ocr_zh": "做三角形ABC的外接圆，过C做直线l使其与圆相切，D为BC中点，连接AD，过B做AC垂线垂足为E，过E做l的垂线垂足为G，过D做l的垂线垂足为F。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4157.jpg" }, { "index": 4158, "ocr_en": "In triangle ABC, draw a perpendicular from B to AC with the foot at B₁, forming triangle ABB₁. Inscribe the incircle of triangle ABB₁, then draw a tangent from C to the circle intersecting BB₁ at D and AB at C₁. Connect AD.", "ocr_zh": "三角形ABC，过B做AC的垂线垂足为B1，构成三角形ABB1，做三角形ABB1的内切圆，过C做圆的切线与BB1交于D、与AB交于C1，连接AD。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4158.jpg" }, { "index": 4159, "ocr_en": "Two circles intersect at A and M. Let B be a point on the larger circle. Connect AB, which intersects the smaller circle at C₁. Connect BM and extend it, intersecting the smaller circle at C. Connect AC, which intersects the larger circle at B₁. Connect BB₁, which intersects the smaller circle at P, and extend BB₁ to intersect the smaller circle again at Q. Connect CC₁, which intersects BQ at M, and extend CC₁ to intersect the larger circle at N.", "ocr_zh": "两圆相交于A和M，B为大圆上一点，连接AB与小圆交于C1，连接BM并延长与小圆交于C，连接AC与大圆交于B1，连接BB1与小圆交于P，延长BB1与小圆交于Q，连接CC1与BQ交于M，延长CC1与大圆交于N。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4159.jpg" }, { "index": 4160, "ocr_en": "Two circles intersect at points A and M. Let B be a point on the larger circle. Connect AB, which intersects the smaller circle at C₁. Connect BM and extend it, intersecting the smaller circle at C. Connect AC, which intersects the larger circle at B₁. Connect BB₁, which intersects the smaller circle at P, and extend BB₁ to intersect the smaller circle again at Q. Connect CC₁, which intersects BQ at M, and extend CC₁ to intersect the larger circle at N. Draw auxiliary lines AM and BM.", "ocr_zh": "两圆相交于A和M，B为大圆上一点，连接AB与小圆交于C1，连接BM并延长与小圆交于C，连接AC与大圆交于B1，连接BB1与小圆交于P，延长BB1与小圆交于Q，连接CC1与BQ交于M，延长CC1与大圆交于N，做辅助线AM、BM。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4160.jpg" }, { "index": 4161, "ocr_en": "In triangle ABC, let P be a point on BC. Draw a line through P parallel to AB, intersecting AC at R. Let Q be a point on AB. Connect QR and QP. Let P₁ be a point outside triangle ABC. Connect P₁B, P₁Q, and P₁A. Draw auxiliary lines P₁R, P₁P, and P₁C.", "ocr_zh": "三角形ABC，P为BC上一点，过P做AB平行线交AC于R，Q为AB一点，连接QR、QP，P1为三角形ABC外一点，连接P1B、P1Q、P1A，做辅助线P1R、P1P、P1C。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4161.jpg" }, { "index": 4162, "ocr_en": "In triangle ABC, let P be a point on BC. Draw a line through P parallel to AB, intersecting AC at R. Let Q be a point on AB. Connect QR and QP. Let P₁ be a point outside triangle ABC. Connect P₁B, P₁Q, and P₁A.", "ocr_zh": "三角形ABC，P为BC上一点，过P做AB平行线交AC于R，Q为AB一点，连接QR、QP，P1为三角形ABC外一点，连接P1B、P1Q、P1A。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4162.jpg" }, { "index": 4163, "ocr_en": "Triangle ABC is inside triangle DFE. Extend CA to intersect DF at A₁, extend BA to intersect DE at A₂, extend AB to intersect EF at B₁, extend CB to intersect DF at B₂, extend BC to intersect DE at C₁, and extend AC to intersect EF at C₂.", "ocr_zh": "三角形ABC在三角形DFE内，延长CA与DF交于A1，延长BA与DE交于A2，延长AB与EF交于B1，延长CB与DF交于B2，延长BC与DE交于C1，延长AC与EF交于C2。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4163.jpg" }, { "index": 4164, "ocr_en": "In triangle ABC which is inside triangle DFE, extend CA to intersect DF at A₁, extend BA to intersect DE at A₂, extend AB to intersect EF at B₁, extend CB to intersect DF at B₂, extend BC to intersect DE at C₁, and extend AC to intersect EF at C₂. Draw auxiliary perpendicular lines from A to DF and DE with feet at G and H respectively, and draw auxiliary lines AA₁, BB₁, and CC₁.", "ocr_zh": "三角形ABC在三角形DFE内，延长CA与DF交于A1，延长BA与DE交于A2，延长AB与EF交于B1，延长CB与DF交于B2，延长BC与DE交于C1，延长AC与EF交于C2，过A做DF和DE的辅助垂线，垂足分别为G和H，做辅助线A1A2、B1B2、C1C2。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4164.jpg" }, { "index": 4165, "ocr_en": "Construct the circumcircle O of triangle ABC. Let FG be a chord of circle O that intersects AB and AC at D and E respectively. Draw the tangent HI to circle O at point A, and connect HD and IE. Extend HD as an auxiliary line to intersect BC at K, and draw the auxiliary line KE. Points K, E, and I are collinear, forming triangle KIH. Construct the auxiliary circumcircle of triangle KIH, with F and G also lying on this circumcircle.", "ocr_zh": "做三角形ABC的外接圆O，FG为圆O的弦且与AB和AC分别交于D和E，过A做圆O的切线HI，连接HD、IE，做HD的辅助延长线与BC交于K，做辅助线KE，K、E、I三点共线，构成三角形KIH，做三角形KIH的辅助外接圆，且F、G也在外接圆上。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4165.jpg" }, { "index": 4166, "ocr_en": "Construct the circumcircle O of triangle ABC. Let FG be a chord of circle O that intersects AB and AC at D and E respectively. Draw the tangent HI to circle O at point A, and connect HD and IE.", "ocr_zh": "做三角形ABC的外接圆O，FG为圆O的弦且与AB和AC分别交于D和E，过A做圆O的切线HI，连接HD、IE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4166.jpg" }, { "index": 4167, "ocr_en": "In quadrilateral ABCD, M is the midpoint of AB. Connect CM and DM. N is the midpoint of CM. Connect DN, and connect AC to intersect DN at E.", "ocr_zh": "四边形ABCD，M为AB中点，连接CM、DM，N为CM中点，连接DN，连接AC与DN交于E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4167.jpg" }, { "index": 4168, "ocr_en": "In triangle ABC, let P be a point inside the triangle. Connect PA, PB, and PC.", "ocr_zh": "三角形ABC，P为三角形ABC内一点，连接PA、PB、PC。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4168.jpg" }, { "index": 4169, "ocr_en": "In triangle ABC, let P be a point inside triangle ABC. Connect PA, PB, and PC. Let E be a point outside triangle ABC, and draw the auxiliary lines BE and PE. Let D be the midpoint of BE, and then draw the auxiliary lines AD, PD, and CD.", "ocr_zh": "三角形ABC，P为三角形ABC内一点，连接PA、PB、PC，E为三角形ABC外一点，做辅助线BE、PE，D为BE中点，做辅助线AD、PD、CD。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4169.jpg" }, { "index": 4170, "ocr_en": "Construct the circumcircle of triangle ABC. Draw a perpendicular from A to BC with the foot at D, a perpendicular from B to AC with the foot at E, and a perpendicular from C to AB with the foot at F. Let AD, BE, and CF intersect at H, forming triangle EFH. Connect EF and let it intersect AD at G. Let K be a point on the circle, connect AK and let it intersect BC at M. Connect GM and HK, draw the auxiliary lines BK and CK, and construct the circumcircle of triangle EFH as an auxiliary circle.", "ocr_zh": "做三角形ABC的外接圆，过A做BC的垂线垂足为D，过B做AC的垂线垂足为E，过C做AB的垂线垂足为F，AD、BE、CF交于H，构成三角形EFH，连接EF与AD交于G，K为圆上一点，连接AK与BC交于M，连接GM、HK，做辅助线BK、CK，做三角形EFH的辅助外接圆。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4170.jpg" }, { "index": 4171, "ocr_en": "Construct the circumcircle of triangle ABC. Draw a perpendicular from A to BC with the foot at D, a perpendicular from B to AC with the foot at E, and a perpendicular from C to AB with the foot at F. Let AD, BE, and CF intersect at H, forming triangle GHF. Connect EF and let it intersect AD at G. Let K be a point on the circle, connect AK and let it intersect BC at M. Connect GM and HK.", "ocr_zh": "做三角形ABC的外接圆，过A做BC的垂线垂足为D，过B做AC的垂线垂足为E，过C做AB的垂线垂足为F，AD、BE、CF交于H，构成三角形GHF，连接EF与AD交于G，K为圆上一点，连接AK与BC交于M，连接GM、HK。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4171.jpg" }, { "index": 4172, "ocr_en": "In triangle ABC, let M be the midpoint of BC. Connect AM to form triangles ABM and ACM. Construct the circumcircle of triangle ABM, which intersects AC at point B₁. Construct the circumcircle of triangle ACM, which intersects AB at point C₁. Connect B₁C₁. Let O be a point inside triangle ABC, and connect OC₁, OB₁. Draw the auxiliary lines OB, OC, and OM.", "ocr_zh": "三角形ABC，M为BC中点，连接AM构成三角形ABM和三角形ACM，做三角形ABM的外接圆与AC交于B1，做三角形ACM的外接圆与AB交于C1，连接B1C1，O为三角形ABC内一点，连接OC1、OB1，做辅助线OB、OC、OM。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4172.jpg" }, { "index": 4173, "ocr_en": "In triangle ABC, let M be the midpoint of BC. Connect AM to form triangles ABM and ACM. Construct the circumcircle of triangle ABM, which intersects AC at point B₁. Construct the circumcircle of triangle ACM, which intersects AB at point C₁. Connect B₁C₁. Let O be a point inside triangle ABC, and connect OC₁, OB₁.", "ocr_zh": "三角形ABC，M为BC中点，连接AM构成三角形ABM和三角形ACM，做三角形ABM的外接圆与AC交于B1，做三角形ACM的外接圆与AB交于C1，连接B1C1，O为三角形ABC内一点，连接OC1、OB1。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4173.jpg" }, { "index": 4174, "ocr_en": "AB is the diameter of circle O, and D is a point outside circle O. Draw two tangent lines from D to circle O, which intersect the circle at C and E respectively. Connect AC, let P be a point on AC, and connect DP.", "ocr_zh": "AB为圆O直径，D为圆O外一点，过D做圆O的两条切线分别交于C、E，连接AC，P为AC上一点，连接DP。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4174.jpg" }, { "index": 4175, "ocr_en": "AB is the diameter of circle O, and D is a point outside circle O. Draw two tangent lines from D to circle O, which intersect the circle at C and E respectively. Connect AC, let P be a point on AC, and connect DP. Draw the auxiliary lines OD, BE, HE, and OE.", "ocr_zh": "AB为圆O直径，D为圆O外一点，过D做圆O的两条切线分别交于C、E，连接AC，P为AC上一点，连接DP，做辅助线OD、BE、HE、OE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4175.jpg" }, { "index": 4176, "ocr_en": "In triangle ABC, let G be a point inside triangle ABC. Connect AG and extend it to intersect BC at D, connect BG and extend it to intersect AC at E, and connect CG and extend it to intersect AB at F. Take G's symmetric point G' with respect to BC as the axis of symmetry, and connect BG' and CG'. Extend AD to draw the auxiliary line DK to point K. Let S be a point on BC, and draw the auxiliary lines GS, BK, and CK.", "ocr_zh": "三角形ABC，G为三角形ABC内一点，连接AG并延长交BC于D，连接BG并延长交AC于E，连接CG并延长交AB于F，G以BC为对称轴做对称点G'，连接BG'、CG'，延长AD做辅助线DK到K，S为BC上的点，做辅助线GS、BK、CK。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4176.jpg" }, { "index": 4177, "ocr_en": "In triangle ABC, let G be a point inside triangle ABC. Connect AG and extend it to intersect BC at D, connect BG and extend it to intersect AC at E, and connect CG and extend it to intersect AB at F. Take the symmetric point G' of G with respect to BC as the axis of symmetry, and connect BG' and CG'.", "ocr_zh": "三角形ABC，G为三角形ABC内一点，连接AG并延长交BC于D，连接BG并延长交AC于E，连接CG并延长交AB于F，G以BC为对称轴做对称点G'，连接BG'、CG'。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4177.jpg" }, { "index": 4178, "ocr_en": "In triangle ABC, let H be a point inside triangle ABC. Connect AH and extend it to intersect BC at D. Connect BH and extend it to intersect AC at E. Let P be the midpoint of AB. Connect EP and HP. Extend PH to point Q, where point Q is outside triangle ABC. Connect CQ.", "ocr_zh": "三角形ABC，H为三角形ABC内一点，连接AH并延长交BC于D，连接BH并延长交AC于E，P为AB中点，连接EP、HP，延长PH至点Q，点Q在三角形ABC外，连接CQ。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4178.jpg" }, { "index": 4179, "ocr_en": "In quadrilateral ABCD, let K be a point on AD and M be a point on BC. Connect DM, AM, BK, and CK.", "ocr_zh": "四边形ABCD，K为AD上的点，M为BC上的点，连接DM、AM、BK、CK。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4179.jpg" }, { "index": 4180, "ocr_en": "In triangle ABC, construct the circumcircle of triangle ABC. Draw a circle with AC as the diameter, which intersects AB at K and BC at L. Connect CK and extend it to intersect the circumcircle of triangle ABC at F. Connect AL and extend it to intersect the circumcircle of triangle ABC at D. Pass through B and the intersection point of AD and CF, then extend the line to intersect AC at N and the circumcircle of triangle ABC at E. Connect KN and LN.", "ocr_zh": "三角形ABC，做三角形ABC的外接圆，以AC为直径画圆与AB交于K、与BC交于L，连接CK并延长与三角形ABC的外接圆交于F，连接AL并延长与三角形ABC的外接圆交于D，过B与AD和CF的交点并延长与AC交于N、与三角形ABC的外接圆交于E，连接KN、LN。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4180.jpg" }, { "index": 4181, "ocr_en": "In quadrilateral ABCD, connect AC and BD intersecting at E. Construct the circumcircle of quadrilateral ABCD. Draw a tangent from B to the circle, which intersects the extension of DA at F. Extend BA to G, connect FG and DG. Draw a perpendicular from C to the extension of GF with the foot at H.", "ocr_zh": "四边形ABCD，连接AC和BD交于E，做四边形ABCD的外接圆，过B做圆的切线与DA延长线交于F，延长BA于G，连接FG、DG，过C做GF延长线的垂线垂足为H。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4181.jpg" }, { "index": 4182, "ocr_en": "In triangle ABC, construct the circumcircle of triangle ABC. Let I be a point inside triangle ABC, and connect BI and CI. Let O be a point on the circumcircle of triangle ABC. Draw a circle through points B, I, and O, and draw another circle through points C, I, and O.", "ocr_zh": "三角形ABC，做三角形ABC的外接圆，I为三角形ABC内一点，连接BI、CI，O为三角形ABC外接圆上一点，以B、I、O三点画圆，以C、I、O三点画圆。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4182.jpg" }, { "index": 4183, "ocr_en": "In triangle ABC, construct the incircle of triangle ABC, which is tangent to AB at E and tangent to AC at F. Connect EF. Let D be the midpoint of BC, and let M and N be points on EF. Connect DM and DN, and let CM and BN intersect at I.", "ocr_zh": "三角形ABC，做三角形ABC的内切圆，与AB的切点为E，与AC的切点为F，连接EF，D为BC的中点，M、N在EF上，连接DM、DN，连接CM和BN相交于I。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4183.jpg" }, { "index": 4184, "ocr_en": "Points A, B, C, and D are on a circle. Connect AD and BC. Connect AB and CD, which intersect at O. Draw a chord of the circle through O, intersecting the circle at M and N, AD at F, and BC at E.", "ocr_zh": "A、B、C、D为圆上的点，连接AD、BC，连接AB、CD交于O，过O做圆的弦与圆交于M和N、与AD交于F、与BC交于E。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4184.jpg" }, { "index": 4185, "ocr_en": "Points A, B, C, and D are on a circle. Connect BA and DC, and extend them to intersect at point P. Draw a ray from P that is tangent to the circle, with the point of tangency being T.", "ocr_zh": "A、B、C、D为圆上的点，连接BA、DC并延长交于P，过P做圆的一条射线使其与圆相切，切点为T。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4185.jpg" }, { "index": 4186, "ocr_en": "Point B is inside circle O. Connect OB and extend it as an auxiliary line to intersect circle O at A. Point C is on circle O. Connect BC and extend it to P. Connect AP. Points A and E coincide. Extend CB as an auxiliary line to intersect circle O at D. Extend BO as an auxiliary line to intersect circle O at F.", "ocr_zh": "B为圆O内一点，连接OB并做辅助延长线交圆O于A，C为圆O上的点，连接BC并延长到P，连接AP，A与E两点重合，做CB的辅助延长线交圆O于D，做BO的辅助延长线交圆O于F。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4186.jpg" }, { "index": 4187, "ocr_en": "Points A, B, C, and D are on the circle O. Connect AB and CD, which intersect at point P.", "ocr_zh": "A、B、C、D为圆O上的点，连接AB和CD交于P。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4187.jpg" }, { "index": 4188, "ocr_en": "Point B is a point inside circle O. Connect OB. Points A and C are on circle O. Connect BC and extend it to point P. Connect AP.", "ocr_zh": "B为圆O内一点，连接OB，A、C为圆O上的点，连接BC并延长到P，连接AP。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4188.jpg" }, { "index": 4189, "ocr_en": "Points B and C are on the circle O. Connect OC, connect CB and extend it to A. Draw a tangent from A to the circle O with the tangent point D. Connect OA. Let E be a point on OA, and connect BE and DE.", "ocr_zh": "B、C为圆O上的点，连接OC，连接CB并延长到A，过A做圆O的一条切线切点为D，连接OA，E为OA上的点，连接BE、DE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4189.jpg" }, { "index": 4190, "ocr_en": "Points B and C are on circle O. Connect OC, connect CB and extend it to point A. Draw a tangent from A to circle O with the tangent point D. Connect OA. Let E be a point on OA, and connect BE and DE. Draw the auxiliary line OD.", "ocr_zh": "B、C为圆O上的点，连接OC，连接CB并延长到A，过A做圆O的一条切线切点为D，连接OA，E为OA上的点，连接BE、DE，做辅助线OD。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4190.jpg" }, { "index": 4191, "ocr_en": "AB is the diameter of circle O, and C, D are points on circle O. Connect CD, extend BA to intersect CD at P. E is a point inside circle O, connect DE to intersect AB at F.", "ocr_zh": "AB为圆O直径，C、D为圆O上的点，连接CD，延长BA与CD交于P，E为圆O内的点，连接DE与AB交于F。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4191.jpg" }, { "index": 4192, "ocr_en": "AB is the diameter of circle O, and C, D are points on circle O. Connect CD, extend BA to intersect CD at P. E is a point inside circle O, connect DE to intersect AB at F. Draw the auxiliary line OC.", "ocr_zh": "AB为圆O直径，C、D为圆O上的点，连接CD，延长BA与CD交于P，E为圆O内的点，连接DE与AB交于F，做辅助线OC。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4192.jpg" }, { "index": 4193, "ocr_en": "Points B and C are on circle A. Connect AB, BC, and AC. Points G and H are on circle A. Connect BH. Connect BG and let it intersect AC at D. Connect GH. Extend AC to intersect GH at X. Draw the auxiliary lines AG, AH, and DH.", "ocr_zh": "B、C为圆A上的点，连接AB、BC、AC，G、H为圆A上的点，连接BH，连接BG与AC交于D，连接GH，延长AC与GH交于X，做辅助线AG、AH、DH。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4193.jpg" }, { "index": 4194, "ocr_en": "Points B and C are on circle A. Connect AB, BC, and AC. Points G and H are on circle A. Connect BH. Connect BG and let it intersect AC at D. Connect GH. Extend AC to intersect GH at X.", "ocr_zh": "B、C为圆A上的点，连接AB、BC、AC，G、H为圆A上的点，连接BH，连接BG与AC交于D，连接GH，延长AC与GH交于X。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4194.jpg" }, { "index": 4195, "ocr_en": "The line m is a tangent to the circle O with the point of tangency at A, and the line n is a tangent to the circle O with the point of tangency at C. Connect AC, let P be a point on AC. Draw a perpendicular from P to m with the foot at Q, and draw a perpendicular from P to n with the foot at R. Draw the auxiliary line AB, which is the diameter of the circle O, through A, draw the auxiliary line DE, which is the diameter of the circle O, through P, and draw the auxiliary line BC.", "ocr_zh": "直线m为圆O的切线切点为A，直线n为圆O的切线切点为C，连接AC，P为AC上的点，过P做m的垂线垂足为Q，过P做n的垂线垂足为R，过A做圆O的辅助线直径AB，过P做圆O的辅助线直径DE，做辅助线BC。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4195.jpg" }, { "index": 4196, "ocr_en": "The line m is a tangent to the circle O with the point of tangency at A, and the line n is a tangent to the circle O with the point of tangency at C. Connect AC, let P be a point on AC. Draw a perpendicular from P to m with the foot at Q, and draw a perpendicular from P to n with the foot at R.", "ocr_zh": "直线m为圆O的切线切点为A，直线n为圆O的切线切点为C，连接AC，P为AC上的点，过P做m的垂线垂足为Q，过P做n的垂线垂足为R。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4196.jpg" }, { "index": 4197, "ocr_en": "Construct the circumcircle of quadrilateral ABCD, with P being a point on AB and Q being a point on CD.", "ocr_zh": "做四边形ABCD的外接圆，P为AB上的点，Q为CD上的点。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4197.jpg" }, { "index": 4198, "ocr_en": "The circle T1 and the circle T2 intersect at two points AB, C is a point on the AB arc on the circle T1, connect CA and extend the intersection T2 at the point F, connect the CB and extend the intersection T2 at the point D, connect the AD intersection circle T1 at the point E, and O is the internal point of T2 and connect BO", "ocr_zh": "圆T1与圆T2相交于AB两点，C为圆T1上AB大弧上一点，连接CA并延长交T2于点F，连接CB并延长交T2于点D，连接AD交圆T1于点E，O为T2内部一点，连接BO", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4198.jpg" }, { "index": 4199, "ocr_en": "The circle T1 and the circle T2 intersect at two points AB, C is a point on the AB large arc on the circle T1, connects CA and extends the intersection T2 at the point F, connects the CB and extends the intersection T2 at the point D, connects the AD intersection circle T1 at the point E, O is the inner point of T2, connects BO, passes through the point of O as OH perpendicular to CF, perpendicular foot is H, OG perpendicular to AD, perpendicular foot is G", "ocr_zh": "圆T1与圆T2相交于AB两点，C为圆T1上AB大弧上一点，连接CA并延长交T2于点F，连接CB并延长交T2于点D，连接AD交圆T1于点E，O为T2内部一点，连接BO，过O点作虚线OH垂直于CF，垂足为H，过O点作虚线OG垂直于AD，垂足为G", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4199.jpg" }, { "index": 4200, "ocr_en": "O as the center of the circle, the line segment CD as the radius of the circle, the line segment AB is another diameter, P is a point on the arc CB, connect PD to cross AB at point F, connect AP to cross CD at point E, with a dotted line to connect DB", "ocr_zh": "以O为圆心，线段CD为半径作圆，线段AB为另一条直径，P为弧CB上一点，连接PD交AB于点F，连接AP交CD于点E，做辅助线DB", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4200.jpg" }, { "index": 4201, "ocr_en": "O as the center of the circle, the line segment CD as the radius of the circle, the line segment AB is another diameter, P is a point on the arc CB, connect PD to cross AB at point F, connect AP to cross CD at point E", "ocr_zh": "以O为圆心，线段CD为半径作圆，线段AB为另一条直径，P为弧CB上一点，连接PD交AB于点F，连接AP交CD于点E", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4201.jpg" }, { "index": 4202, "ocr_en": "There is a circle, A is a point outside the circle, over the point A to make a ray AB and ray AT tangent to the circle, the ray AT is tangent to the circle and the point T, C is a point on the line segment AT, BC is tangent to the circle, over the point A to make a straight line inside the ∠A intersecting with the circle at the point P and Q.", "ocr_zh": "有一个圆，A为圆外一点，过A点做射线AB和射线AT于圆相切，射线AT与圆相切与T点，C为线段AT上一点，BC与圆相切，过A点做∠A内的一条直线与圆相交于P点和Q点", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4202.jpg" }, { "index": 4203, "ocr_en": "A, B, P, D are points on circle O, C, D are points on circle P. Circle O intersects with circle P at point D. Connect AB, the extension of AB is tangent to circle P at point C. Connect PB, PA, AD, AD intersects with circle P at point E. Take a point F on circle O, and make an auxiliary line PF perpendicular to the line AP, with the vertical foot at point P. Make auxiliary lines FA, PE, PD, PC.", "ocr_zh": "A、B、P、D均为圆O上的点，C、D为圆P上的点，圆O与圆P相交于点D。连接AB、AB的延长线与圆P相切于点C，连接PB、PA、AD，AD与圆P相交于点E。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4203.jpg" }, { "index": 4204, "ocr_en": "In a circle, Q, P1, R, Q1, P, R1 are points on the circle. connect PP1, QQ1, RR1. the intersection of PP1 and QQ1 is A. the intersection of PP1 and RR1 is B. the intersection of RR1 and QQ1 is C. ", "ocr_zh": "在一个圆中，Q、P1、R、Q1、P、R1均为圆上的点，连接PP1、QQ1、RR1，PP1与QQ1的交点为A，PP1与RR1的交点为B，RR1与QQ1的交点为C。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4204.jpg" }, { "index": 4205, "ocr_en": "Point A, O1, B, O2, C, O3 common line, O1 is the midpoint of AB, O2 is the midpoint of BC, AB as the diameter of the circle O1, BC as the diameter of the circle O2, CO3 as the radius of the circle O3, through the point A as a straight line AG and the circle O3 tangent to the point G, the straight line AG and the intersection of the circle O2 at the point E, F.", "ocr_zh": "点A、O1、B、O2、C、O3共线，O1为AB中点，O2为BC中点，以AB为直径作圆O1，以BC为直径作圆O2，以CO3为半径作圆O3，过点A作直线AG与圆O3相切于点G，直线AG与圆O2相交于点E、F。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4205.jpg" }, { "index": 4206, "ocr_en": "A, M, N, B are points on circle O with center O. Connect BA, BN, BM, NA, NM, AM, extend line segments BA, NM and the two extension lines intersect at point P outside the circle.", "ocr_zh": "A、M、N、B都是圆O上的点，圆心为O，连接BA、BN、BM、NA、NM、AM，延长线段BA、NM，两条延长线相交于圆外的点P", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4206.jpg" }, { "index": 4207, "ocr_en": "Points A and C are points on a circle, line segment AP is tangent to the circle at the point A, and point M is the midpoint of line segment AP. Connect MC, line MC intersects the circle at point B, connect PB, the extension of PB intersects the circle at point D, connect PC, line PC intersects the circle at point E, connect DE.", "ocr_zh": "点A、C均为一个圆上的点，线段AP与圆相切于点A，点M是线段AP的中点。连接MC、线段MC与圆相交于点B，连接PB、PB延长线与圆相交于点D，连接PC、线段PC与圆相交于点E，连接DE", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4207.jpg" }, { "index": 4208, "ocr_en": "There are two circles in the plane whose intersection points are C and N. Make the common tangent AB to the two circles, the midpoint of AB is M. Connect AC and BC.", "ocr_zh": "平面内有两个圆的交点为C、N，作两个圆的公切线AB，AB的中点为M。连接AC、BC", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4208.jpg" }, { "index": 4209, "ocr_en": "There is a square triangle ABC and a circle in the plane. Line AB intersects the circle at points H and J. Line AC intersects the circle at points G and F. Line BC intersects the circle at points D and E.", "ocr_zh": "平面内有正三角形ABC和一个圆，线段AB与圆相交于点H、J，线段AC与圆相交于点G、F，线段BC与圆相交于点D、E", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4209.jpg" }, { "index": 4210, "ocr_en": "There is a semicircle in the plane with line segment AB as its diameter, C and D are points on the arc of the semicircle, connecting AC and BD, and line segment AC intersects BD at point E.", "ocr_zh": "平面内有一个以线段AB为直径的半圆，C、D均为半圆圆弧上的点，连接AC、BD，线段AC与BD相交于点E", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4210.jpg" }, { "index": 4211, "ocr_en": "Points A and B are points on the circle O, connect AB, P is a point on the line segment AB, connect OP. The line segment OH is perpendicular to the line segment AB and H is the pendant.", "ocr_zh": "点A、B均为圆O上的点，连接AB，P为线段AB上的一点，连接OP。线段OH垂直于线段AB，H为垂足。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4211.jpg" }, { "index": 4212, "ocr_en": "There is a convex quadrilateral ABCD in the plane, extend line segment AB, line segment CD, and the two extension lines intersect at point E. Make a circle with dashed lines through points A, D, and E. Make an auxiliary line AC, extend AC to intersect with the dashed circle determined by the three points of ADE at point F, and make an auxiliary line EF, and make a circle with dashed lines through points E, B, and C.", "ocr_zh": "平面中有凸四边形ABCD，延长线段AB、线段CD，两条延长线相交于点E。过点A、D、E三点以虚线作圆，作辅助线AC、延长AC与ADE三点确定的虚线圆相交于点F，作辅助线EF，过点E、B、C以虚线作圆。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4212.jpg" }, { "index": 4213, "ocr_en": "In the plane, B is a point on the circle O, the center of the circle is O. P and Q are two points outside the circle O. Connecting BP, BQ, OP, PQ and OQ, the line BP intersects the circle O at the point A, and the line BQ intersects the circle O at the point C.", "ocr_zh": "在平面内，B为圆O上的一点，圆心为O，P、Q为圆O外的两个点，连接BP、BQ、OP、OQ、PQ，线段BP与圆O相交于点A，线段BQ与圆O相交于点C。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4213.jpg" }, { "index": 4214, "ocr_en": "There is a circle in the plane, and points A, D, C, and B are points on the circle. Connect AD, DC, CB, BA, DB, and AC, line segment DB intersects AC at point Q. Extend the line segment BA and CD to intersect point P. Make the auxiliary line QP, and extend the auxiliary line to a point E in triangle QBC to make the auxiliary line EB.", "ocr_zh": "在平面内有一个圆，点A、D、C、B均为圆上的点，连接AD、DC、CB、BA、DB、AC，线段DB与AC相交于点Q，延长线段BA、CD相交于点P。作辅助线QP、延长辅助线至三角形QBC内一点E，作辅助线EB。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4214.jpg" }, { "index": 4215, "ocr_en": "There is a circle in the plane and points A, D, C, and B are points on the circle.Connect AD, DC, CB, BA, DB, and AC.The line segment DB intersects AC at point Q.Extend the line segments BA and CD to intersect at point P.", "ocr_zh": "在平面内有一个圆，点A、D、C、B均为圆上的点，连接AD、DC、CB、BA、DB、AC，线段DB与AC相交于点Q，延长线段BA、CD相交于点P。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4215.jpg" }, { "index": 4216, "ocr_en": "Points P, M are points on circle O, there is a straight line l outside circle O. Points B, A, C are points on straight line l. A is the midpoint of line BC, connecting the center of the circle O with point A. Line OA is perpendicular to straight line l, with A as the pendant. Connect PB, MC, line PB and circle O intersect at point Q, line MC and circle O intersect at N, PQ and MN parallel. Connect PM, the extension of line PM intersects line l at point R. Connect QN, the extension of line QN intersects line l at point S.", "ocr_zh": "点P、M均为圆O上的点，圆O外有一条直线l，点B、A、C均为直线l上的点，A为线段BC中点，连接圆心O与点A，线段OA垂直于直线l，A为垂足。连接PB、MC，线段PB与圆O相交于点Q，线段MC与圆O相交于N。连接PM，线段PM的延长线与直线l相交于点R，连接QN，线段QN的延长线与直线l相交于点S。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4216.jpg" }, { "index": 4217, "ocr_en": "CD is a horizontal line segment, the line segment CD as the diameter of the CD midpoint O as the center of the circle to make a circle, straight line AB is perpendicular to the CD of a chord, point A in the line segment CD below, point E is a point on the arc BD, connect AE intersected with the CD at the point F, connected to the DE, the extension of the CB and DE intersected at the point G, connected to the GF.", "ocr_zh": "CD为水平线段，以线段CD为直径，以CD的中点O为圆心做圆，直线AB为垂直于CD的一条弦，A点在线段CD下方，点E为弧BD上一点，连接AE交CD于点F，连接DE，延长CB和DE交于点G，连接GF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4217.jpg" }, { "index": 4218, "ocr_en": "There is a circle, the center of the circle is O, P is a point outside the circle, over the point P to make the line PS and line PY tangent to the circle with the point S and the point T, connecting ST, to make the auxiliary line connecting OP and OS, over the point P to make the circumcircle cut line PAB intersecting the circle at the point A and B, and intersecting ST at the point C.", "ocr_zh": "有一个圆，圆心为O，P为圆外一点，过P点做线段PS和线段PY与圆相切与点S和点T，连接ST，做辅助线连接OP与OS，过P点做圆的割线PAB交圆于A点和B点，交ST于C点。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4218.jpg" }, { "index": 4219, "ocr_en": "There is a circle, the center of the circle is O, P is a point outside the circle, over the point P make the line PS and line PY tangent to the circle with the point S and the point T, connect ST, over the point P make the circumcision line of the circle PAB intersecting the circle at the point A and the point B, and intersecting ST at the point C.", "ocr_zh": "有一个圆，圆心为O，P为圆外一点，过P点做线段PS和线段PY与圆相切与点S和点T，连接ST，过P点做圆的割线PAB交圆于A点和B点，交ST于C点。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4219.jpg" }, { "index": 4220, "ocr_en": "Triangle ABC is a non-isosceles triangle, do the external circle of triangle ABC, M is the midpoint of BC, line CD, line BE for the side AB and AC on the height, the pendant for D and E, connect ME, MD, DE, L and K are the midpoint of the line ML and ME, respectively, over the point A to do the parallel line of BC intersection of the extension of KL at the point T, connect TA, TM, do the auxiliary circle of the triangle ADE, The center of the circle is O.", "ocr_zh": "三角形ABC为非等腰三角形，做三角形ABC的外接圆，M为BC中点，线段CD、线段BE为边AB和AC上的高，垂足为D和E，连接ME、MD、DE，L和K分别为线段ML和ME的中点，过A点做BC的平行线交KL的延长线于点T，连接TA、TM，做三角形ADE的辅助圆,圆心为O。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4220.jpg" }, { "index": 4221, "ocr_en": "Triangle ABC is a non-isosceles triangle, do the external circle of triangle ABC, M is the midpoint of BC, line CD, line BE for the side AB and AC on the height, the pendant for D and E, connect ME, MD, DE, L and K are the midpoint of the line ML and ME, respectively, over the point A to do the parallel line of BC intersection of the extension of KL at the point T, connect TA, TM.", "ocr_zh": "三角形ABC为非等腰三角形，做三角形ABC的外接圆，M为BC中点，线段CD、线段BE为边AB和AC上的高，垂足为D和E，连接ME、MD、DE，L和K分别为线段ML和ME的中点，过A点做BC的平行线交KL的延长线于点T，连接TA、TM。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4221.jpg" }, { "index": 4222, "ocr_en": "Triangle ABC is isosceles triangle, AB length is equal to AC, D is a point on AB, extend CD to point E so that the length of AC is equal to CE, do the outer circle of the triangle ABC, over the point ADE to make a circle, the two circles intersect at point A and point F, extend the AF and CB intersect at point G to connect DG.", "ocr_zh": "三角形ABC为等腰三角形，AB长度等于AC，D为AB上一点，延长CD至E点使AC长度等于CE，做三角形ABC的外接圆，过点ADE做一个圆，两个圆相交于A点和F点，延长AF和CB相交于点G连接DG。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4222.jpg" }, { "index": 4223, "ocr_en": "In triangle ABC, P and Q are points on line segments AB and AC respectively, PQ is parallel to BC, point D is a point in triangle APQ, connect BD and intersect PQ at point E, connect CD and intersect PQ at point F, connect AD and extend it on line segment BC and intersect it at point L, AL and intersect PQ at K.", "ocr_zh": "三角形ABC中，P、Q分别为线段AB、AC上的点，PQ平行于BC，点D为三角形APQ中一点，连接BD交PQ于点E，连接CD交PQ于点F，连接AD并延长于线段BC上，交于点L，AL交PQ于K。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4223.jpg" }, { "index": 4224, "ocr_en": "Triangle ABC is an isosceles triangle, AB length is equal to AC, D is a point on AB, extend CD to point E so that the length of AC is equal to CE, make the outer circle of triangle ABC, cross point ABD and then make a circle, the two circles intersect at point A and point F, extend AF and CB to intersect at point G to connect DG, point I is the inner part of the triangle ACD, connect the auxiliary lines IA, IC and ID.", "ocr_zh": "三角形ABC为等腰三角形，AB长度等于AC，D为AB上一点，延长CD至E点使AC长度等于CE，做三角形ABC的外接圆，过点ABD再做一个圆，两个圆相交于A点和F点，延长AF和CB相交于点G连接DG，点I为三角形ACD的内心，连接辅助线IA、IC和ID。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4224.jpg" }, { "index": 4225, "ocr_en": "O，H are two points inside the triangle ABC, line CD for the side AB on the high, the pendant foot is D, connect OD, over the D point to do the line DE perpendicular to the OD, E point for DE in the line CB extension of the intersection. The external circle of triangle BCH intersects straight line AB at point B, F, connect EF, FO, OH. make the external circle of triangle DHF, where the inferior arc DF segment is a solid line, the rest of the segment is dashed, the circle intersects DE at point P. make the external circle of triangle DBC with a dashed line, where the inferior arc DB is a solid line, and the rest of the segment is a dashed. Make the external circle of triangle BHC, where the inferior arc BH is the dashed line and the remaining segments are solid lines. m is the midpoint of FH, N is the midpoint of BC, connect ON, MN, DN, and point K is the midpoint of segment DP. Make the auxiliary line CO and extend it to intersect the outer circle of triangle ABC at point Q. Connect the auxiliary lines AQ and BQ.", "ocr_zh": "三角形ABC内有两点O、H，线段CD为边AB上的高，垂足为D，连接OD，过D点做线段DE垂直于OD，E点为DE于线段CB延长线的交点。三角形BCH的外接圆交直线AB于点B、F，连接EF、FO、OH。做三角形DHF的外接圆，其中劣弧DF段为实线，其余段为虚线，该圆交DE于点P。用虚线做三角形DBC的外接圆，劣弧DB为实线，其余段为虚线。做三角形BHC的外接圆，其中劣弧BH为虚线，其余段为实线。M为FH的中点，N为BC中点，连接ON、MN、DN，K点为线段DP中点。做辅助线CO并延长，交三角形ABC的外接圆于点Q，连接辅助线AQ、BQ。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4225.jpg" }, { "index": 4226, "ocr_en": "O，H are two points inside the triangle ABC, line CD for the side AB on the high, the pendant foot is D, connect OD, over the D point to do the line DE perpendicular to the OD, E point for DE in the line CB extension of the intersection. The outer circle of triangle BCH intersects the line AB at point B, F, connect EF, FO, OH.", "ocr_zh": "三角形ABC有两点O、H，线段CD为边AB上的高，垂足为D，连接OD，过D点做线段DE垂直于OD，E点为DE于线段CB延长线的交点。三角形BCH的外接圆交直线AB于点B、F，连接EF、FO、OH。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4226.jpg" }, { "index": 4227, "ocr_en": "In triangle ABC, AD is the bisector of angle BAC, point D is on BC, AM is the median on the side of BC, and the outer circle of triangle AMD intersects AB at E and AC at F.", "ocr_zh": "三角形ABC中，AD为角BAC的平分线，点D在BC上，AM为BC边上的中线，三角形AMD的外接圆交AB于E，交AC于F。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4227.jpg" }, { "index": 4228, "ocr_en": "Triangle ABC, AD is the side BC on the high, the pendant foot is D, AD as the diameter of the circle, intersection of AB at point M, intersection of AC at point N. Connection MN intersection of AD at point G, extend AD to intersect the outer circle of the triangle ABC in point E.", "ocr_zh": "三角形ABC中，AD为边BC上的高，垂足为D，以AD为直径做圆，交AB于点M，交AC于点N。连接MN交AD于点G，延长AD交三角形ABC的外接圆于点E。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4228.jpg" }, { "index": 4229, "ocr_en": "Convex quadrilateral ABCD is interior to a circle. Extend BA and CD to intersect at point P. Extend DA and BC to intersect at point Q.", "ocr_zh": "凸四边形ABCD内接于圆，延长BA和CD相交于点P，延长DA和BC相交于点Q。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4229.jpg" }, { "index": 4230, "ocr_en": "Line AB is the chord of circle O, M is the midpoint of the inferior arc AB, C is a point outside the circle, from point C to make a cut line of the circle to intersect the circle with two points PQ, connect MP to intersect AB at R to connect RQ, Z is the chord PQ above the point O below the point S is a point on the inferior arc BP, T is a point on the inferior arc PQ, to connect MT to intersect AB at F, to connect MS to intersect AB at E. From F to make the perpendicular line of AB to intersect with the OT at point Y, X is a point below AB in the circle, X is a point below AB. From point E, make the vertical segment EX of AB, X is a point below AB in the circle.", "ocr_zh": "线段AB为圆O的弦，M为劣弧AB的中点，C为圆外一点，由C点做一条圆的割线交圆与PQ两点，连接MP交AB于R连接RQ，Z为弦PQ上方，点O下方的一点S为劣弧BP上一点，T为劣弧PQ上一点，连接MT交AB于F，连接MS交AB于E。由F做AB的垂线与OT交于点Y，由E点做AB的垂线段EX，X为圆中AB下方一点。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4230.jpg" }, { "index": 4231, "ocr_en": "Circle O1 and circle O2 outside away, AB and CD were circle O1, O2 two diameters, and AB is parallel to CD, connect AD, BC, DB, AC, AC intersection of circle O1 and point E, intersection of circle O2 and point F, connect DF and BE.", "ocr_zh": "圆O1与圆O2外离，AB和CD分别为圆O1、O2两条直径，且AB平行于CD，连接AD、BC、DB、AC，AC交圆O1与点E，交圆O2与点F，连接DF和BE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4231.jpg" }, { "index": 4232, "ocr_en": "In a convex quadrilateral ABCD, connect AC and BD, there is a point E in the triangle formed by the intersection of the two lines and BC, connect BE and EC and make the auxiliary line ED.", "ocr_zh": "凸四边形ABCD中，连接AC与BD，在两条线的交点和BC构成的三角形中有一点E，连接BE和EC，做辅助线ED。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4232.jpg" }, { "index": 4233, "ocr_en": "G and H are two points in the convex hexagon ABCDEF, with point G above point H. X and Y are two points outside the convex hexagon ABCDEF, with point X above line BA and point Y below line DE. Connect GB, GA, GH, HD, HE and make the auxiliary lines BD, AE, XB, XA, YD, YE.", "ocr_zh": "G、H是凸六边形ABCDEF中的两个点，点G位于点H的上方，X、Y是在凸六边形ABCDEF外的两个点，X点在BA线的上方，Y点在DE线的下方。连接GB、GA、GH、HD、HE，作辅助线BD、AE、XB、XA、YD、YE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4233.jpg" }, { "index": 4234, "ocr_en": "G and H are two points in the convex hexagon ABCDEF, and point G lies above point H. Connect GB, GA, GH, HD, HE.", "ocr_zh": "G、H是凸六边形ABCDEF中的两个点，点G位于点H的上方，连接GB、GA、GH、HD、HE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4234.jpg" }, { "index": 4235, "ocr_en": "A, B, and C are points on a circle, connect AB, BC, and CA. Make a tangent to the circle with B as the tangent point, extend the line segment CA, the extension of CA intersects the tangent line with B as the tangent point at point P. Make another tangent to the circle through point P and tangent to the circle at point D, connect BD and CD.", "ocr_zh": "A、B、C都是圆上的点，连接AB、BC、CA。以B为切点作圆的切线，延长线段CA，CA的延长线与以B为切点的切线相交于点P，过点P作圆的另一条切线与圆相切于点D，连接BD、CD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4235.jpg" }, { "index": 4236, "ocr_en": "A, B, C are points on the circle, connect AB, BC, CA. Make a tangent to the circle with B as the tangent point, extend the line CA, the extension of CA intersects with the tangent to B at point P, make another tangent to the circle through point P and tangent to the circle at point D, connect BD, CD. Make the auxiliary line AD.", "ocr_zh": "A、B、C都是圆上的点，连接AB、BC、CA。以B为切点作圆的切线，延长线段CA，CA的延长线与以B为切点的切线相交于点P，过点P作圆的另一条切线与圆相切于点D，连接BD、CD。作辅助线AD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4236.jpg" }, { "index": 4237, "ocr_en": "A, B, C, A1, B1, C1 are points on a circle in the plane, A1 is a point on the arc BC, B1 is a point on the arc AC, C1 is a point on the arc AB, connect AB, BC, CA, AA1, BB1, CC1, make auxiliary line A1B, A1C.", "ocr_zh": "A、B、C、A1、B1、C1均为平面内一个圆上的点，A1是位于圆弧BC上的一点，B1是位于圆弧AC上的一点，C1是位于圆弧AB上的一点，连接AB、BC、CA、AA1、BB1、CC1，作辅助线A1B、A1C.", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4237.jpg" }, { "index": 4238, "ocr_en": "There is a triangle ABC in the plane, make an external circle through points A, B, and C. D is a point on the circular arc BC, make the auxiliary lines DB, DA, and DC.", "ocr_zh": "平面内有三角形ABC，过点A、B、C作外接圆，D是位于圆弧BC上的一点，作辅助线DB、DA、DC.", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4238.jpg" }, { "index": 4239, "ocr_en": "There is a triangle ABC in the plane, O, I are two points in the triangle ABC, connect OA, OI, AI, through the point A, B, C for the outer circle, D is located in the arc BC on the point, with a dotted line for the extension of the line segment AI and the outer circle of the triangle ABC intersected with the other point D, make auxiliary lines DO, DC, DB.", "ocr_zh": "平面内有三角形ABC，O、I是在三角形ABC内的两个点，连接OA、OI、AI，过点A、B、C作外接圆，D是位于圆弧BC上的一点，用虚线作线段AI的延长线与三角形ABC的外接圆相交于另一点D，作辅助线DO、DC、DB.", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4239.jpg" }, { "index": 4240, "ocr_en": "There is a triangle ABC in the plane, O and I are two points in triangle ABC, connect OA, OI, AI, and make the external circle through points A, B, and C.", "ocr_zh": "平面内有三角形ABC，O、I是在三角形ABC内两个点，连接OA、OI、AI，过点A、B、C作外接圆。 ", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4240.jpg" }, { "index": 4241, "ocr_en": "plane with the same point O as the center of the two circles, the radius of the larger called the outer circle, smaller called the inner circle. A1, A2, A3, A4 are the inner circle of the point, connect A2A3, A3A4, A4A1, A1A2, for the extension of A2A3 and the outer circle intersected at point B3, for the extension of A3A4 and the outer circle of the extension intersects the point B4, as the extension of A4A1 and the outer circle intersects at point B1, for the extension of A1A2 and the outer circle intersects at point B2, connecting B2B3, B3B4, B4B1, B1B2. make auxiliary lines OB2, OA1, OB1.", "ocr_zh": "平面内有以同一点O为圆心的两个圆，半径较大的称之为外圆、较小的称之为内圆。A1、A2、A3、A4都是内圆上的点，连接A2A3、A3A4、A4A1、A1A2，作A2A3的延长线与外圆相交于点B3，作A3A4的延长线与外圆相交于点B4，作A4A1的延长线与外圆相交于点B1，作A1A2的延长线与外圆相交于点B2，连接B2B3、B3B4、B4B1、B1B2.作辅助线OB2、OA1、OB1.", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4241.jpg" }, { "index": 4242, "ocr_en": "plane with the same point O as the center of the two circles, the larger radius is called the outer circle, the smaller is called the inner circle. A1, A2, A3, A4 are points on the inner circle, connecting A2A3, A3A4, A4A1, A1A2, for the extension of A2A3 and the outer circle intersects the point B3, for the extension of A3A4 and the outer circle intersects the point B4, as the extension of A4A1 and the outer circle intersects at point B1, as the extension of A1A2 and the outer circle intersects at point B2, connecting B2B3, B3B4, B4B1, B1B2.", "ocr_zh": "平面内有以同一点O为圆心的两个圆，半径较大的称之为外圆、较小的称之为内圆。A1、A2、A3、A4都是内圆上的点，连接A2A3、A3A4、A4A1、A1A2，作A2A3的延长线与外圆相交于点B3，作A3A4的延长线与外圆相交于点B4，作A4A1的延长线与外圆相交于点B1，作A1A2的延长线与外圆相交于点B2，连接B2B3、B3B4、B4B1、B1B2.", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4242.jpg" }, { "index": 4243, "ocr_en": "point D is a point in the plane triangle ABC, connect DA, DC, DB, through the point D to the dotted line for the line segment BC parallel line DE, through the point B to the dotted line for the line segment DC parallel line BE, the two dotted line segments intersection for E. Through the point E to the dotted line for the line segment AD parallel line EF, through the point A to the dotted line for the auxiliary line DE parallel line AF, the intersection of two dotted line segments for the F, as the The intersection of the two dashed line segments is F. Make the auxiliary line AE.", "ocr_zh": "点D是平面内三角形ABC内的一点，连接DA、DC、DB，过点D以虚线作线段BC的平行线DE，过点B以虚线作线段DC的平行线BE，两条虚线线段交点为E。过点E以虚线作线段AD的平行线EF，过点A以虚线作辅助线DE的平行线AF，两条虚线线段交点为F，作辅助线AE.", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4243.jpg" }, { "index": 4244, "ocr_en": "There are two circles in the plane, the left is a large circle, the right is a small circle, point A, D is the intersection of the two circles, B is a point on the large circle, I is in the two circles and the triangle ABC common area. Connect BD, the extension of line BD and the small circle intersects at point C, connect BA, AC, BI, CI, the extension of line BI and the small circle intersects at point P, the extension of line CI and the large circle intersects at point Q. Make the auxiliary line PQ, connect ID, the extension of line ID and the auxiliary line PQ intersects at point M, make the auxiliary lines AM, AQ, AP, over the point A, Q, P with the dotted line to make a circle.", "ocr_zh": "平面内有两个圆，左边为一个大圆、右边为一个小圆，点A、D是两个圆的交点，B是大圆上的一点，I是处于两圆和三角形ABC公共区域内的一点。连接BD，线段BD的延长线与小圆相交于点C，连接BA、AC、BI、CI，线段BI的延长线与小圆相交于点P，线段CI的延长线与大圆相交于点Q。作辅助线PQ，连接ID，线段ID的延长线与辅助线PQ相交于点M，作辅助线AM、AQ、AP，过点A、Q、P以虚线作圆。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4244.jpg" }, { "index": 4245, "ocr_en": "There are two circles in the plane, a large circle on the left and a small circle on the right, points A and D are the intersection of the two circles, B is a point on the large circle, and I is a point that is in the common area of the two circles and triangle ABC. Connect BD, the extension of line BD intersects the small circle at point C. Connect BA, AC, BI, CI, the extension of line BI intersects the small circle at point P, and the extension of line CI intersects the large circle at point Q.", "ocr_zh": "平面内有两个圆，左边为一个大圆、右边为一个小圆，点A、D是两个圆的交点，B是大圆上的一点，I是处于两圆和三角形ABC公共区域内的一点。连接BD，线段BD的延长线与小圆相交于点C，连接BA、AC、BI、CI，线段BI的延长线与小圆相交于点P，线段CI的延长线与大圆相交于点Q。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4245.jpg" }, { "index": 4246, "ocr_en": "Point O is a point outside triangle ABC in the plane lying below line segment BC, connect OB, OC, OA. Point M is a point inside triangle ABC lying on line segment OA, connect MB, MC.", "ocr_zh": "点O是平面内三角形ABC外位于线段BC下方的一个点，连接OB、OC、OA。点M是三角形ABC内位于线段OA上的一点，连接MB、MC。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4246.jpg" }, { "index": 4247, "ocr_en": "The inner part of triangle ABC is I. Make the outer circle of triangle ABC with center O. Connect AI and extend it to intersect BC at D, and intersect the outer circle of ABC at E. Make the auxiliary line EO and extend it to intersect the circle O at J. Make the auxiliary lines EB, EC and JC. Make the auxiliary line IF perpendicular to AC at point I, with the foot of the pendant at F.", "ocr_zh": "三角形ABC的内心为I，连接IA，IB，IC，做三角形ABC的外接圆，圆心为O。连接AI并延长交BC于D，交ABC的外接圆于E。做辅助线EO并延长交圆O于J，做辅助线EB、EC和JC。过I点做辅助线IF垂直于AC，垂足为F。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4247.jpg" }, { "index": 4248, "ocr_en": "The inner part of triangle ABC is I. Make the outer circle of triangle ABC with center O. Connect AI and extend it to intersect BC at D, and intersect the outer circle of ABC at E.", "ocr_zh": "三角形ABC的内心为I，连接IA，IB，IC，做三角形ABC的外接圆，圆心为O。连接AI并延长交BC于D，交ABC的外接圆于E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4248.jpg" }, { "index": 4249, "ocr_en": "The three angle bisectors of triangle ABC are AK, CM, and BL. points K, L, and M are on line segments BC, AC, and AB. take points Q and P on MC and LB, and connect AQ, QK, AP, and PK so that AP=PK, and AQ=QK. make the auxiliary line QP and extend it toward the ends, intersecting AB at point D, and intersecting AC at point E. Make the auxiliary lines DK and KE.", "ocr_zh": "三角形ABC的三条角平分线分别为AK、CM、BL，点K、L、M分别在线段BC、AC和AB上，在MC、LB上取点Q、P，连接AQ、QK、AP、PK，使AP=PK，AQ=QK。做辅助线QP并向两端延长，分别交AB于点D，交AC于点E。做辅助线DK和KE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4249.jpg" }, { "index": 4250, "ocr_en": "The three angle bisectors of triangle ABC are AK, CM, and BL, and points K, L, and M are on line segments BC, AC, and AB, respectively; take points Q and P on MC and LB, and connect AQ, QK, AP, and PK such that AP = PK and AQ = QK.", "ocr_zh": "三角形ABC的三条角平分线分别为AK、CM、BL，点K、L、M分别在线段BC、AC和AB上，在MC、LB上取点Q、P，连接AQ、QK、AP、PK，使AP=PK，AQ=QK。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4250.jpg" }, { "index": 4251, "ocr_en": "The inner part of triangle ABC is I. Make the outer circle of triangle ABC with center O. Connect AI and extend it to intersect BC at D, and intersect the outer circle of ABC at E. Make the auxiliary line OI and extend it to both ends to intersect the circle O at the points MN.", "ocr_zh": "三角形ABC的内心为I，连接IA，IB，IC，做三角形ABC的外接圆，圆心为O。连接AI并延长交BC于D，交ABC的外接圆于E。做辅助线OI并向两端延长，交圆O于MN两点。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4251.jpg" }, { "index": 4252, "ocr_en": "Take points F, E, D on the sides AB, AC, BC of triangle ABC and connect EF, FD, DE.O1, O2, O3 are three points inside triangles AEF, FBD, EDC respectively. connect O1, O2, O3. make the auxiliary lines O1F, FO2, O2D, DO3, O3E, EO1. rotate O3O2 clockwise around the point O3 by the angle DO3E to get Auxiliary line O3P, connect PE and PO1.", "ocr_zh": "三角形ABC的边AB、AC、BC上分别取点F、E、D，连接EF、FD、DE，O1、O2、O3分别为三角形AEF、FBD、EDC内的一点。连接O1、O2、O3。做辅助线O1F、FO2、O2D、DO3、O3E、EO1。将O3O2绕O3点顺时针旋转角DO3E的角度得到辅助线O3P，连接PE和PO1。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4252.jpg" }, { "index": 4253, "ocr_en": "Take points F, E, and D on sides AB, AC, and BC of triangle ABC, connect EF, FD, and DE, and O1, O2, O3 are three points inside triangles AEF, FBD, EDC respectively. connect O1, O2, and O3.", "ocr_zh": "三角形ABC的边AB、AC、BC上分别取点F、E、D，连接EF、FD、DE，O1、O2、O3分别为三角形AEF、FBD、EDC内的一点。连接O1、O2、O3。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4253.jpg" }, { "index": 4254, "ocr_en": "In the non-isosceles triangle ABC, use the dotted line to make the height AH on the side of BC, the vertical foot is H, make the auxiliary line AG and extend it to cross BC at M, the point G is a point on AM, I is a point in the triangle directly below the point G, connect GI and extend it to cross BC at the point F, where the IF segment is the dotted line, and the point D is a point in FM.", "ocr_zh": "在非等腰三角形ABC中，用虚线做BC边上的高AH，垂足为H，做辅助线AG并延长交BC于M，点G为AM上一点，I为三角形内点G正下方一点，连接GI并延长，交BC于点F，其中IF段为虚线，点D为FM中一点。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4254.jpg" }, { "index": 4255, "ocr_en": "Triangle ABC has a point D on the BC side, point O is a point inside triangle ADC, point O1 is a point outside the triangle to the left of AB, and point O2 is a point in triangle ADC near the AC side. Connect AD, 0102, make OM parallel to AD over point O and point M is on BC.", "ocr_zh": "三角形ABC的BC边上有一点D，点O为三角形ADC内一点，点O1为三角形外一点，在AB左侧，点O2为三角形ADC中靠近AC边的一点。连接AD、0102，过O点做OM平行于AD，且点M在BC上。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4255.jpg" }, { "index": 4256, "ocr_en": "Triangle ABC has a point D on the BC side, point O is a point inside triangle ADC, point O1 is a point outside the triangle to the left of AB, and point O2 is a point in triangle ADC near the AC side. Connect AD, 0102, make OM parallel to AD over point O and point M is on BC. Make the auxiliary lines AO2, AO, C0, CO2, MO2. connect the auxiliary line OO2 and extend it to intersect the AC side at point H.", "ocr_zh": "三角形ABC的BC边上有一点D，点O为三角形ADC内一点，点O1为三角形外一点，在AB左侧，点O2为三角形ADC中靠近AC边的一点。连接AD、0102，过O点做OM平行于AD，且点M在BC上。做辅助线AO2、AO、C0、CO2、MO2。连接辅助线OO2并延长交AC边于点H。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4256.jpg" }, { "index": 4257, "ocr_en": "Do the external circle of triangle ABC, the center of the circle is O, AM, AT for the triangle's median and angle bisector, point M and point T are in BC, through point B and point C do respectively circle O tangent intersects at point P, connect PA intersection of the circle O at point E, intersection of BC at point D, connect ME. connect auxiliary lines OC, OA, OE, OP. do auxiliary line CF perpendicular to AB, the foot of the pituitary is F, connect auxiliary line FM. FM.", "ocr_zh": "做三角形ABC的外接圆，圆心为O，AM、AT分别为三角形的中线和角平分线，点M和点T都在BC上，过点B和点C做分别圆O的切线相交于点P，连接PA交圆O于点E，交BC于点D，连接ME。连接辅助线OC、OA、OE、OP。做辅助线CF垂直AB，垂足为F，连接辅助线FM。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4257.jpg" }, { "index": 4258, "ocr_en": "Do the external circle of the triangle ABC, the center of the circle O, AM, AT, respectively, for the triangle of the median and the angle bisector, point M and point T are in the BC, over the point B and point C to do respectively, the tangent to the circle O intersects at the point P, connecting PA intersection of the circle O at the point E, intersecting the BC at the point D, connect ME.", "ocr_zh": "做三角形ABC的外接圆，圆心为O，AM、AT分别为三角形的中线和角平分线，点M和点T都在BC上，过点B和点C做分别圆O的切线相交于点P，连接PA交圆O于点E，交BC于点D，连接ME。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4258.jpg" }, { "index": 4259, "ocr_en": "Do the triangle ABC high line AD, BE, the vertical foot is point D and point E, two high line intersect at point H, do the triangle ABC external circle, the center of the circle is O, through the point O to do the line segment OF parallel to BC point F on AB, point M is the midpoint of AH, connect MC, FM, CF.", "ocr_zh": "做三角形ABC的高线AD、BE，垂足为点D和点E，两条高线交于点H，做三角形ABC的外接圆，圆心为O，过点O做线段OF平行于BC点F在AB上，点M为AH的中点，连接MC、FM、CF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4259.jpg" }, { "index": 4260, "ocr_en": "Make the high line AD, BE of triangle ABC, the pendant is point D and point E, the two high lines intersect at point H, make the outer circle of triangle ABC, the center of the circle is O, cross point O to make the line OF parallel to BC point F on AB, point M is the midpoint of AH, connect MC, FM, CF, make the auxiliary line BO and extend to intersect the circle O at point G, connect the auxiliary lines GC, GA, CH, OB. cross point O to make the auxiliary line ON perpendicular to BC, the pendant is N, over the point F to do the auxiliary line FT perpendicular to BC, the pendant is T, connect the auxiliary line TM, TH.", "ocr_zh": "做三角形ABC的高线AD、BE，垂足为点D和点E，两条高线交于点H，做三角形ABC的外接圆，圆心为O，过点O做线段OF平行于BC点F在AB上，点M为AH的中点，连接MC、FM、CF，做辅助线BO并延长交圆O于点G，连接辅助线GC、GA、CH、OB。过点O做ON垂直于BC，垂足为N，过点F做辅助线FT垂直于BC，垂足为T，连接辅助线TM、TH。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4260.jpg" }, { "index": 4261, "ocr_en": "There is a triangle ABC in the plane, points I and P are two points inside the triangle ABC, and point I lies to the upper left of point P. Make the outer circle of triangle ABC and connect IC, IA, IB, PC, PA, PB.", "ocr_zh": "平面内有三角形ABC，点I、P为三角形ABC内的两个点，I点位于P点左上方。作三角形ABC的外接圆，连接IC、IA、IB、PC、PA、PB.", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4261.jpg" }, { "index": 4262, "ocr_en": "There is a circle in the plane called the large circle. Points A and D are both points on the large circle, and point O is a point inside the large circle. Draw an inscribed circle of the great circle with point O as the center and point D as the tangent point. Draw two tangent lines AB and AC to the inscribed circle through point A. Segment AB is tangent to the inscribed circle at point P and intersects the great circle at point B. Segment AC is tangent to the inscribed circle at point Q and intersects the great circle at point C. Connect PQ and BC, and point I is the midpoint of segment PQ.", "ocr_zh": "平面内有一个圆称之为大圆，点A、D都是大圆上的一点，点O是大圆内的一点。以点O为圆心、D为切点作大圆的一个内切圆，过点A作该内切圆的两条切线AB、AC，线段AB与内切圆相切于点P、与大圆相交于点B，线段AC与内切圆相切于点Q、与大圆相交于点C。连接PQ、BC，点I为线段PQ中点", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4262.jpg" }, { "index": 4263, "ocr_en": "In a plane, there is a triangle ABC. Points I and P are two points inside triangle ABC, with point I located to the upper left of point P. Construct the circumcircle of triangle ABC. Connect IC, IA, IB, PC, PA, and PB. Extend AI with a dashed line until it intersects the circumcircle of triangle ABC at point M. Draw auxiliary lines MC, MB, and MP. With point M as the center, draw an arc BIC with a dashed line, where point B is the starting point, passing through point I, and ending at point C.", "ocr_zh": "平面内有三角形ABC，点I、P为三角形ABC内的两个点，I点位于P点左上方。作三角形ABC的外接圆，连接IC、IA、IB、PC、PA、PB.用虚线延长AI与三角形ABC的外接圆相交于点M，作辅助线MC、MB、MP,以点M为圆心以虚线作圆弧BIC，点B为起始、经过点I、终止于点C", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4263.jpg" }, { "index": 4264, "ocr_en": "Points R, Q, and P are all points on circle O, with O as the center. Connect OR, OQ, and OP. Line segments OP, OQ, and arc QP form sector OPQ. Points A and B are two points located in the region of sector OPQ. Connect AB.", "ocr_zh": "点R、Q、P都是圆O上的点，点O为圆心，连接OR、OQ、OP。线段OP、OQ和圆弧QP构成了扇形OPQ，点A、B是位于扇形OPQ区域的两个点，连接AB", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4264.jpg" }, { "index": 4265, "ocr_en": "In a plane, there is a square ABCD. Point A1 lies on segment AD, and point B1 lies on segment BC. Draw auxiliary line A1B1 such that segment A1B1 is parallel to segment AB. Point E is the midpoint of segment DC, and point F is the midpoint of segment A1B1. Draw auxiliary lines A1E, DF, EF, EB1, FC, A1B, and AB1. Segment A1E intersects with DF at point O1, and segment EB1 intersects with CF at point O2.", "ocr_zh": "在平面内有正方形ABCD，A1是线段AD上的一点，B1是线段BC上的一点，作辅助线A1B1、线段A1B1与线段AB平行。点E是线段DC中点，点F是线段A1B1中点，作辅助线A1E、DF、EF、EB1、FC、A1B、AB1，线段A1E与DF相交于点O1，线段EB1与CF相交于点O2。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4265.jpg" }, { "index": 4266, "ocr_en": "There are two intersecting circles O and O1 in a plane. Circle O has a larger radius and its center is O. Circle O1 has a smaller radius and its center is O1. Points A and B are two points on circle O. Connect OO1 and AB. Line segment AB is perpendicular to OO1, and the foot of the perpendicular is point O. Connect AO1.", "ocr_zh": "平面内有两个相交的圆O和O1，圆O半径较大、圆心为O，圆O1半径较小、圆心为O1。点A、B为圆O上的两个点，连接OO1、AB，线段AB与OO1相垂直，垂足为点O。连接AO1。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4266.jpg" }, { "index": 4267, "ocr_en": "Point E is a point inside the square ABCD in the plane. Connect EA and EB, and the line segments EA and EB are equal. Point F is a point outside the square ABCD. Connect FD and FC, and the line segments FD and FC are equal in length. Connect EF.", "ocr_zh": "点E是平面内正方形ABCD内的一点，连接EA、EB，线段EA和EB相等。点F是正方形ABCD外的一点， 连接FD、FC，线段FD和FC长度相等，连接EF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4267.jpg" }, { "index": 4268, "ocr_en": "In an isosceles triangle ABE in the plane, segment AB is equal to segment BF. segment BC is the altitude of isosceles triangle ABF, with foot C; segment DC is the altitude of triangle ABC, with foot D; segment CE is the altitude of triangle BCF, with foot E; draw an arc DE with point C as the center and segment DC as the radius, and draw an arc AF with point C as the center and segment AC as the radius. The regions enclosed by arc AB and segment AB, arc BF and segment BF, and arc DE and segments BD and BE are shaded regions.", "ocr_zh": "在平面内的等腰三角形ABE中，线段AB与线段BF相等，线段BC是等腰三角形ABF的高，垂足为C；线段DC是三角形ABC的高，垂足为D；线段CE是三角形BCF的高，垂足为E；以点C为圆心、线段DC为半径作圆弧DE，以点C为圆心、线段AC为半径作圆弧AF。圆弧AB和线段AB围成的区域、圆弧BF和线段BF围成的区域、圆弧DE和线段BD跟线段BE围成的区域为阴影区域。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4268.jpg" }, { "index": 4269, "ocr_en": "In an isosceles right triangle ABC, line segment AC is equal to line segment BC. Extend line segment AC, draw an arc DF with point A as the center and line segment AD as the radius, the intersection point of the arc and line segment BC is E, and the intersection point of the arc and the extension of AC is F. The area enclosed by arc DE, line segment DB, and line segment BE, as well as the area enclosed by arc EF, line segment EC, and line segment CF, are shaded areas.", "ocr_zh": "在等腰直角三角形ABC中，线段AC与线段BC相等。延长线段AC，以点A为圆心、线段AD为半径作圆弧DF，圆弧与线段BC交点为E，与AC延长线交点为F。圆弧DE、线段DB、线段BE围成的区域以及圆弧EF、线段EC、线段CF围成的区域为阴影区域。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4269.jpg" }, { "index": 4270, "ocr_en": "In the isosceles triangle ABD in the plane, segment AD is equal to segment BD. Segment OD is the altitude of the isosceles triangle ABD, with the foot of the perpendicular at O. Draw an arc AB with O as the center and OA as the radius. Extend AD and BD, and draw an arc AE with B as the center and AB as the radius. The arc AE intersects the extension of segment BD at point E. Draw an arc BF with center A and radius AB. The arc BF intersects the extended line segment AD at point F. Draw an arc EF with center D and radius DE. The region enclosed by arcs AE, EF, FB, and AB is the shaded region.", "ocr_zh": "在平面内的等腰三角形ABD中，线段AD与线段BD相等，线段OD是等腰三角形ABD的高，垂足为O，以O为圆心、OA为半径作圆弧AB。延长AD、BD，以B为圆心、AB为半径作圆弧AE，圆弧AE与线段BD延长线交于点E。以A为圆心、AB为半径作圆弧BF，圆弧BF与线段AD延长线交于点F。以D为圆心，DE为半径作圆弧EF。圆弧AE、圆弧EF、圆弧FB与圆弧AB所围成的区域为阴影区域。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4270.jpg" }, { "index": 4271, "ocr_en": "In an equilateral triangle ABC, D is a point on segment BC. Circles are drawn with centers at A, B, and C, each with a radius equal to the length of segment BD, and each extending to the point where it intersects the sides of the equilateral triangle. The circle with center C intersects the circle with center B at point N. Draw a dotted line segment NM perpendicular to segment BC, with the foot of the perpendicular at M, and draw an auxiliary line NB. The three arcs and the three sides of the triangle divide triangle ABC into seven regions. The common region formed by each arc and the two sides of the triangle it intersects is called z. The three regions formed by connecting the intersection points between any two arcs, the two points on the intersection of the two arcs and the triangle's sides that are closer to the intersection of the two arcs, and the triangle's vertex closest to the intersection of the two arcs are all called x, while the remaining three regions are all called y.", "ocr_zh": "在正三角形ABC中，D为线段BC上的一点。分别以A、B、C为圆心作圆弧，半径都与线段BD大小相等，都到和正三角形的边的交点终止。以C为圆心的圆弧与以B为圆心的圆弧相交于点N。过N作虚线线段NM垂直于线段BC，垂足为M，作辅助线NB。三条圆弧和三角形的三条边将三角形ABC分割为七个区域，其中每条圆弧和其与三角形相交的两边围成的三个扇形区域的公共区域称为z，每两条圆弧间的交点、这两条圆弧和三角形两边的交点里靠近两条圆弧交点的两个点、靠近两条圆弧交点的三角形顶点这四个点所连接成的三个区域都称为x，剩下的三个区域都称为y.", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4271.jpg" }, { "index": 4272, "ocr_en": "In triangle ABC in the plane, there is an inscribed circle. Points A2 and B1 lie on segment AB, points C2 and B3 lie on segment BC, and points A3 and C1 lie on segment AC. Segments B1C1, A2C2, and A3B3 are three tangents to the incircle. Segment B1C1 is parallel to segment BC, segment A2C2 is parallel to segment AC, and segment A3B3 is parallel to segment AB. Each of triangles AB1C1, CA3B3, and BA2C2 has an incircle.", "ocr_zh": "在平面内的三角形ABC中有一个内切圆，A2、B1是线段AB上的点，C2、B3是线段BC上的点，A3、C1是线段AC上的点，线段B1C1、A2C2、A3B3是内切圆的三条切线，线段B1C1平行于线段BC，线段A2C2平行于线段AC，线段A3B3平行于线段AB。在三角形AB1C1、三角形CA3B3、三角形BA2C2中各有一个内切圆。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4272.jpg" }, { "index": 4273, "ocr_en": "In a plane, there is an equilateral triangle ABC. E is the midpoint of segment AC. Circles A, B, and C are drawn with the vertices A, B, and C of the equilateral triangle as their centers and the side length of the equilateral triangle as their radii. Circle A intersects circle C at point D, which is the other intersection point besides point B. DC is connected, and F is the midpoint of segment DC. The region enclosed by the arcs AB, points A and B, and the other intersection point of circles A and B (excluding point C) forms the shaded region. The region enclosed by the arcs BC, points B and C, and the other intersection point of circles B and C (excluding point A) forms the shaded region. The region enclosed by the arcs AC, points A and C, and point D forms the shaded region.", "ocr_zh": "在平面内有正三角形ABC，E为线段AC上的中点，分别以正三角形的顶点A、B、C为圆点并以正三角形的边长为半径画圆A、圆B、圆C，圆A与圆C除了点B外的另一交点为点D，连接DC，F为线段DC中点；圆弧AB、点A和点B到圆A与圆B除了点C外的另一交点形成两条的圆弧围成的区域为阴影区域，圆弧BC、点B和点C到圆B与圆C除了点A外的另一交点形成的两条圆弧围成的区域为阴影区域，圆弧AC、点A和点C到点D形成的两条圆弧围成的区域为阴影区域。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4273.jpg" }, { "index": 4274, "ocr_en": "Points E, F, C, D, A, and B are all points on circle O, with center O. Connect EF, CD, and AB. The line segments EF, CD, and AB are pairwise parallel. Draw a dashed line segment OH1 through point O perpendicular to line segment EF, with the foot of the perpendicular being point H1. Extend OH1 with a dashed line until it intersects circle O, intersecting line segment CD at point H2 and line segment AB at point H3. Draw auxiliary lines OE, OC, and OA.", "ocr_zh": "点E、F、C、D、A、B都是圆O上的点，圆心为O。连接EF、CD、AB，线段EF、CD、AB两两平行。过点O以虚线作线段OH1垂直于线段EF，垂足为点H1。以虚线延长OH1至与圆O相交，交线段CD于点H2，交线段AB于点H3。作辅助线OE、OC、OA。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4274.jpg" }, { "index": 4275, "ocr_en": "Points E, F, C, D, A, and B are all points on circle O, with center O. Connect AB, CD, and EF. The line segment CD intersects AB at point G, and the line segment EF intersects AB at point H. Draw a line segment OM perpendicular to the line segment CD through point O, with the foot of the perpendicular being point M. Draw a line segment ON perpendicular to the line segment EF through point O, with the foot of the perpendicular being point N. Draw a line segment OP perpendicular to the line segment AB through point O, with the foot of the perpendicular being point P. Draw auxiliary lines OG and OH.", "ocr_zh": "点E、F、C、D、A、B都是圆O上的点，圆心为O。连接AB、CD、EF，线段CD与AB相交于点G，线段EF与AB相交于点H。过点O以虚线作线段OM垂直于线段CD，垂足为点M，以虚线作线段ON垂直于线段EF，垂足为点N，以虚线作线段OP垂直于线段AB，垂足为点P。作辅助线OG、OH。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4275.jpg" }, { "index": 4276, "ocr_en": "Point C is the midpoint of the arc of the semicircle AB. Point D lies on the arc CB. Point E is a point within the space enclosed by the semicircle AB and the diameter AB. Points F and G are two points outside the space enclosed by the semicircle AB and the diameter AB. Connect CD, CE, ED, EA, and EB. Segment CE is perpendicular to segment CD, with the foot of the perpendicular at C. Draw auxiliary lines AF and CG, with segment AF perpendicular to segment CG, and the foot of the perpendicular at F. Draw auxiliary lines AG, FE, and FB.", "ocr_zh": "点C是半圆AB的圆弧中点，D是位于圆弧CB上的点，E是半圆AB与直径AB围成的空间内的一点，点F、G是半圆AB与直径AB围成的空间外的两点。连接CD、CE、ED、EA、EB，线段CE与线段CD垂直，垂足为C。作辅助线AF、CG，线段AF与线段CG垂直，垂足为F。作辅助线AG、FE、FB。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4276.jpg" }, { "index": 4277, "ocr_en": "Points A and C are points on circle O, with center O. Point B is a point inside circle O. Connect AB and BC, and extend segment AB with a dashed line to intersect circle O at point D. Draw auxiliary lines CD, AC, OA, and OD. Draw a dashed line OE through point O perpendicular to segment AC, with foot E. Draw a dashed line OF through point O perpendicular to segment AB, with foot F.", "ocr_zh": "点A、C是圆O上的点，圆心为O，点B是圆O内的一点。连接AB、BC，以虚线延长线段AB交圆O于点D，作辅助线CD、AC、OA、OD。过点O作虚线OE与线段AC垂直，垂足为E。过点O作虚线OF与线段AB垂直，垂足为F。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4277.jpg" }, { "index": 4278, "ocr_en": "In a plane, there are two circles with the same center. The circle with the smaller radius is called the smaller circle, and the circle with the larger radius is called the larger circle. Draw three tangents AB, AC, and BC to the smaller circle, each with its endpoints on the larger circle. The three intersection points of the three tangents are A, B, and C, respectively. The plane inside the larger circle and outside the triangle ABC is divided into six regions by the three tangents and the arc of the larger circle. Among the three larger regions, the region with BC as its vertex is called T1, the region with AC as its vertex is called T2, and the region with AB as its vertex is called T3. Among the three smaller regions, the region with point B as its vertex is called S2, the region with A as its vertex is called S1, and the region with C as its vertex is called S3.", "ocr_zh": "在平面内有圆心相同的两个圆，半径较小的称之为小圆，半径较大的称之为大圆。作三条以大圆上的两个点为线段始末的小圆切线AB、AC、BC，三条切线两两间的三个交点分别是A、B、C。大圆内、三角形ABC外的平面被三条切线和大圆圆弧分割成了六个区域。面积较大的三个区域中，以BC为顶边的区域称为T1，以AC为顶边的区域称为T2，以AB为顶边的区域称为T3。面积较小的三个区域中，以点B为顶点的区域称为S2，以A为顶点的区域称为S1，以C为顶点的区域称为S3。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4278.jpg" }, { "index": 4279, "ocr_en": "Points A1, A, C, C1, and B are points on circle O, with center O. Connect A1B, BC1, A1C1, AC, AB, and BC. Segment A1C1 is parallel to segment AC, segment A1B is equal to BC1, and segment AB is equal to BC. K is the midpoint of segment AB, M is the midpoint of segment AC, and N is the midpoint of segment A1C1. Draw arcs through points B, K, and C. Draw an auxiliary line OK. Draw a dashed line segment NT perpendicular to segment AB through point N, with the foot of the perpendicular at T.", "ocr_zh": "点A1、A、C、C1、B是圆O上的点，圆心为O，连接A1B、BC1、A1C1、AC、AB、BC，线段A1C1与线段AC平行，线段A1B与BC1相等，线段AB与BC相等，K是线段AB的中点,M是线段AC的中点，N是线段A1C1的中点，过点B、K、C作圆弧。作辅助线OK，过点N作虚线线段NT垂直于线段AB，垂足为T。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4279.jpg" }, { "index": 4280, "ocr_en": "Points E, B, C, D, and F are all points on the dotted circle A, with center A. Draw auxiliary lines AE, AF, BE, and DE, and connect AB, AC, AD, BC, CD, and BD. Line segment AD is perpendicular to AB, with the foot of the perpendicular at A.", "ocr_zh": "点E、B、C、D、F都是虚线圆A上的点，圆心为A。作辅助线AE、AF、BE、DE，连接AB、AC、AD、BC、CD、BD，线段AD垂直于AB，垂足为A。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4280.jpg" }, { "index": 4281, "ocr_en": "Make the outer circle of triangle ABC, DE is a chord of the square circle on BC, where the point D is on the inferior arc AB, the point E is on the inferior arc AC, connect DC to cross AB at the point H, connect BE to cross DC at the point F.", "ocr_zh": "做三角形ABC的外接圆，DE为BC上方圆的一条弦，其中点D在劣弧AB上，点E在劣弧AC上，连接DC交AB于点H，连接BE交DC于点F。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4281.jpg" }, { "index": 4282, "ocr_en": "Outside the circle O there is a point P, respectively, lead two cut line were intersected in the circle at the point A, B and point C, D, where the chord AB and CD are respectively in the center of the circle on both sides, connecting the AO, BO, CO, DO and AD, extend the AO to do the auxiliary line OE to intersect the circle at the point E, connecting the auxiliary line DE.", "ocr_zh": "圆O外有一点P，分别引两条割线分别交圆于点A、B和点C、D，其中弦AB和CD分别在圆心的两侧，连接AO、BO、CO、DO和AD，延长AO做辅助线OE交圆于点E，连接辅助线DE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4282.jpg" }, { "index": 4283, "ocr_en": "Draw a semicircle with line segment AB as its diameter. Let C be a point on the semicircle closest to A. Connect CB. Draw an auxiliary line CH perpendicular to AB through point C, with the foot of the perpendicular being H. Connect the auxiliary line AC. Using CH as the axis, draw the symmetrical auxiliary line segment CD of line segment AC. Let E be a point on the minor arc BC. Draw the auxiliary lines CE and EB. Draw dotted arcs through points C, D, and B.", "ocr_zh": "以线段AB为直径做半圆，C为半圆上靠近A的一点，连接CB，过C点做辅助线CH垂直AB，垂足为H。连接辅助线AC，以CH为轴做线段AC的对称辅助线段CD，点E为劣弧BC上一点，作辅助线CE和EB。过点C、D、B做虚线圆弧。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4283.jpg" }, { "index": 4284, "ocr_en": "Line AD is a chord on the circle O, C is a point on the superior arc AD, B is a point on the inferior arc AD, connect the line segments AC, AB, DC, DB, connect the auxiliary line BC to AD at point F, point E is a point on the line segment FD, connect the line segment BE.", "ocr_zh": "线段AD为圆O上的一条弦，C为优弧AD上一点，B为劣弧AD上一点，连接线段AC、AB、DC、DB，连接辅助线BC交AD于点F，点E为线段FD上一点，连接线段BE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4284.jpg" }, { "index": 4285, "ocr_en": "Make the outer circle of a regular pentagon ABCDE and connect AD, AC, BE, BD, CE.", "ocr_zh": "做正五边形ABCDE的外接圆，连接AD、AC、BE、BD、CE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4285.jpg" }, { "index": 4286, "ocr_en": "Circle O with vertical segment AB as the diameter, O as the center of the circle, CD is a chord perpendicular to AB below the center of the circle O, connect CB, point E is a point on the inferior arc CB, connect DE to intersect CB at point N, connect AE, connect OC to intersect AE at point M. Connect the auxiliary lines DB, AC.", "ocr_zh": "圆O以竖直的线段AB为直径、以O为圆心，CD为圆心O下方垂直于AB的一条弦，连接CB，点E为劣弧CB上一点，连接DE交CB于点N，连接AE，连接OC交AE于点M。连接辅助线DB、AC。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4286.jpg" }, { "index": 4287, "ocr_en": "Make the outer circle of the orthogonal triangle ABC, the center of the circle is O. Point M is the point near A on the inferior arc AC, connect AM, BM, CM, point N is the midpoint of the line segment BM, connect AN. Connect the auxiliary line BO and extend it to intersect the circle O at point P, connect auxiliary lines PA, PC, OA, ON. point K is the point near M on the line segment MC, connect the line segments NK and KP.", "ocr_zh": "做正三角形ABC的外接圆，圆心为O，点M为劣弧AC上靠近A的一点，连接AM、BM、CM，点N为线段BM的中点，连接AN。连接辅助线BO并延长交圆O于点P，连接辅助线PA、PC、OA、ON。点K为线段MC上靠近M的一点，连接线段NK和KP。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4287.jpg" }, { "index": 4288, "ocr_en": "Circle O1 and circle O2 intersect at point A and point B, connect the line segment AB. draw the circumcision line AF of circle O1 on the right side of AB and extend it to intersect O2 at point D, and the circumcision line AE of circle O2 on the left side of AB and extend it to intersect O1 at point C. Connect the auxiliary lines CB, DB, BF and BE.", "ocr_zh": "圆O1和圆O2相交于点A和点B，连接线段AB。过点A分别引圆O1在AB右侧的的割线AF并延长交O2于点D，圆O2在AB左侧的的割线AE并延长交O1于点C。连接辅助线CB、DB、BF、BE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4288.jpg" }, { "index": 4289, "ocr_en": "Make a semicircle with segment AB as its diameter and C as a point on the semicircle near point A. Connect CA and BC. make a circle tangent to triangle ABC with center O3. make circle O1 tangent to segment AC and inferior arc AC and circle O2 tangent to segment BC and inferior arc BC respectively.Connect O1O2,the centers of the two circles.", "ocr_zh": "以线段AB为直径做半圆，C为半圆上靠近点A的一点，连接CA和BC。做三角形ABC的内切圆，圆心为O3。分别做相切于线段AC和劣弧AC的圆O1和相切于线段BC和劣弧BC的圆O2，连接两个圆的圆心O1O2.", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4289.jpg" }, { "index": 4290, "ocr_en": "Circle O1 and circle O2 tangent to the diameter of circle O1 and circle O2 and for the diameter of the two circles to do the outer circle, the center of the circle for O3. do circle O4 in circle O1 and circle O2, respectively, tangent to the circle O3, and circle O3 tangent to the circle, the center of the circle for the O4. connecting the auxiliary lines O1O2, O2O4, O4O1, connecting auxiliary line O3O4 and extend the auxiliary line O3 to the circle on the O3.", "ocr_zh": "圆O1和圆O2相切，以圆O1和圆O2的直径和为直径做两圆的外接圆，圆心为O3。做圆O4分别于圆O1和圆O2外切，和圆O3内切，圆心为O4.连接辅助线O1O2、O2O4、O4O1，连接辅助线O3O4并延长至圆O3上。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4290.jpg" }, { "index": 4291, "ocr_en": "In the Cartesian coordinate system xOy, O is the origin of the coordinate system, O1 and O2 are two points on the x-axis, with O1 located on the negative half-axis and O2 on the positive half-axis. Two circles are drawn with O1 and O2 as centers, respectively, such that they are tangent to the y-axis. Circle O1 has a smaller radius, while circle O2 has a larger radius. Line AB is another common tangent line to circles O1 and O2, passing through the first, second, and third quadrants. It is tangent to circle O1 at point B, tangent to circle O2 at point A, intersects the X-axis at point M, and intersects the Y-axis at point C. Connect BO and AO2. The extension of line segment BO intersects circle O2 at point D. Connect DO2. Draw an auxiliary line BO1. Draw a dashed line segment O1N perpendicular to AD, with the foot of the perpendicular at N.", "ocr_zh": "在直角坐标系xOy中，O为坐标系原点，O1、O2是X轴上的两个点，O1位于负半轴，O2位于正半轴。分别以O1、O2为圆心作和Y轴相切的两个圆，圆O1半径较小，圆O2半径较大。直线AB是圆O1和O2的另一条公切线，穿过第一、第二、第三象限，与圆O1相切于点B，与圆O2相切于点A，与X轴相交于点M，与Y轴相交于点C。连接BO、AO2，线段BO的延长线与圆O2相交于点D，连接DO2，作辅助线BO1，以虚线作线段O1N垂直于AD，垂足为N。", "class": "Analytic Geometry", "difficulty": 2, "image_path": "fig_4291.jpg" }, { "index": 4292, "ocr_en": "There are two circles O1 and O2 in the plane. A and B are two points on circle O1, with center O1. Connect AB, O1B, O1A, and O1O2. The line segment O1B is perpendicular to O1A, with the foot of the perpendicular at O1. C and D are two points on circle O2. Connect O2C, O2D, and CD. The segment O2C is parallel to O1A, and the segment O2D is parallel to O1B. Draw the auxiliary line BD, which intersects O1O2 at point O. Point M is the midpoint of segment AB. Draw the auxiliary lines O1M and OM.", "ocr_zh": "平面内有圆O1、O2，A、B是圆O1上的两个点，圆心为O1，连接AB、O1B、O1A、O1O2，线段O1B垂直于O1A，垂足为O1。C、D是圆O2上的两个点，连接O2C、O2D、CD，线段O2C与O1A平行，线段O2D与O1B平行。作辅助线BD，线段BD与O1O2相交于点O。点M是线段AB中点，作辅助线O1M、OM。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4292.jpg" }, { "index": 4293, "ocr_en": "In triangle ABC in the plane, D is a point on segment BC, and AD is drawn. The incircle of triangle ABD is constructed, with segment AB tangent to the circle at point B1, segment AD tangent at point A1, and segment BD tangent at point D1. Construct the incircle of triangle ADC. The point where segment AC is tangent to the circle is C2, the point where segment AD is tangent to the circle is A2, and the point where segment DC is tangent to the circle is D2. Construct another common tangent line GH between the two circles. The point where segment GH is tangent to the incircle of triangle ABD is G, the point where segment GH is tangent to the incircle of triangle ADC is H, and the point where segment GH intersects segment AD is K.", "ocr_zh": "在平面内的三角形ABC中，D是线段BC上的一点，连接AD。作三角形ABD的内切圆，线段AB与该圆切点为B1，线段AD与该圆切点为A1，线段BD与该圆切点为D1。作三角形ADC的内切圆，线段AC与该圆切点为C2，线段AD与该圆切点为A2，线段DC与该圆切点为D2。作两圆的另一条公切线GH，线段GH与三角形ABD内接圆的切点为G，与三角形ADC内接圆的切点为H，与线段AD的交点为K。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4293.jpg" }, { "index": 4294, "ocr_en": "Point C is a point on circle O, with O as the center. AB is a diameter of circle O. Connect CA and CB, and draw an auxiliary line CO. H is a point on segment BC. Draw a line HG perpendicular to AB, with the foot of the perpendicular being G. Point D is a point on line HG outside circle O. Connect DC. Segment DC is tangent to circle O, with the point of tangency being C.", "ocr_zh": "点C是圆O上的点，点O为圆心，AB是圆O的一条直径，连接CA、CB，作辅助线CO。H是线段BC上的一点，作直线HG垂直于AB，垂足为G。点D为直线HG上在圆O外的一点，连接DC，线段DC与圆O相切，切点为C。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4294.jpg" }, { "index": 4295, "ocr_en": "There is a circle in a plane. Points A, B, and C are all points on the circle. Line l is tangent to the circle at point C. Connect AC, BC, and AB. Draw line segment AD perpendicular to BC, with the foot of the perpendicular at D. Draw line segment BE perpendicular to AC, with the foot of the perpendicular at E. Draw line segment EG perpendicular to line l, with the foot of the perpendicular at G. Draw line segment DF perpendicular to line l, with the foot of the perpendicular at F.", "ocr_zh": "在平面内有一个圆，点A、B、C都是圆上的点，直线l与圆相切，切点为C，连接AC、BC、AB。作线段AD垂直于BC，垂足为D.作线段BE垂直于AC，垂足为E.作线段EG垂直于直线l,垂足为G.作线段DF垂直于直线l,垂足为F，", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4295.jpg" }, { "index": 4296, "ocr_en": "In a right triangle ABC in a plane, the right angle vertex is B. Triangle ABC has an inscribed circle that is tangent to line segment AB at point F, tangent to line segment BC at point D, and tangent to line segment AC at point E. Connect AD, and line segment AD intersects the circle at point P. Connect PF, PE, FD, ED, and PC.", "ocr_zh": "在平面内的直角三角形ABC中，直角顶点为B。三角形ABC有一个内切圆，与线段AB相切于点F，与线段BC相切于点D，与线段AC相切于点E。连接AD，线段AD与圆相交于点P。连接PF、PE、FD、ED、PC。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4296.jpg" }, { "index": 4297, "ocr_en": "In a plane, there is a quadrilateral ABCD. O is a point on line segment AB. A semicircle with center O is tangent to the three sides of quadrilateral ABCD except AB. Connect the center O, line segment AD, and the tangent point of the semicircle with a dotted line. Extend line segments AD and BC with a dotted line so that they intersect at point E. Draw auxiliary lines DO and OC.", "ocr_zh": "平面内有四边形ABCD，O是线段AB上的一点，以O为圆心的半圆与四边形ABCD除AB外的三条边相切,以虚线连接圆心O、线段AD与半圆的切点，以虚线延长线段AD、BC相交于点E，作辅助线DO、OC。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4297.jpg" }, { "index": 4298, "ocr_en": "In a plane, there is an acute-angled triangle ABC. O is the midpoint of BC. With O as the center and line segment BC as the diameter, draw a circle O. Circle O intersects line segment AB at point G. E is a point on line segment BG. Extend line segment AC to a point F outside circle O. Draw line segment FE perpendicular to AB, with the foot of the perpendicular at E. Draw auxiliary line GC. Draw line segment AD through point A, which is tangent to circle O at point D.", "ocr_zh": "在平面内有锐角三角形ABC，O为BC中点，以O为圆心、线段BC为直径作圆O，圆O与线段AB相交于点G.E是线段BG上的一点，延长线段AC至圆O外的一点F，作线段FE垂直AB，垂足为E。作辅助线GC，过点A作线段AD与圆O相切于点D。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4298.jpg" }, { "index": 4299, "ocr_en": "In a plane, there is a triangle ABC. B1 is a point on segment AC, and C1 is a point on segment AB. Connect BB1 and CC1; segment BB1 intersects CC1 at point D. Construct the incircle of quadrilateral AC1DB1. The points of tangency between segment AC1 and the incircle are E, between AB1 and the incircle are F, between DC1 and the incircle are G, and between DB1 and the incircle are H. Draw the incircle of triangle ABD with dashed lines, and the point of tangency between segment AD and the incircle is X. Draw the incircle of triangle ADC with a dashed line; segment AD is tangent to the incircle at point Y.", "ocr_zh": "平面内有三角形ABC，B1是线段AC上的一点，C1是线段AB上的一点，连接BB1、CC1，线段BB1与CC1相交于点D。作四边形AC1DB1的内接圆，线段AC1与内接圆的切点为E，AB1与内接圆的切点F，DC1与内接圆的切点为G，DB1与内接圆的切点为H。以虚线作三角形ABD的内接圆，线段AD与其相切于点X。以虚线作三角形ADC的内接圆，线段AD与其相切于点Y。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4299.jpg" }, { "index": 4300, "ocr_en": "In a plane, there is a triangle ABC. B1 is a point on segment AC, and C1 is a point on segment AB. Connect BB1 and CC1; segment BB1 intersects CC1 at point D. Construct the incircle of quadrilateral AC1DB1. The points of tangency between segment AC1 and the incircle are E, between AB1 and the incircle are F, between DC1 and the incircle are G, and between DB1 and the incircle are H. Draw the incircle of triangle ABD with dashed lines, and the point of tangency between segment AD and the incircle is X. Draw the incircle of triangle ADC with a dashed line; segment AD is tangent to the incircle at point Y.", "ocr_zh": "平面内的三角形ABC中，P是线段AB的中点，作线段AD垂直于BC，垂足为D，作线段BE垂直于AC，垂足为E，连接PE。线段AD与BE的交点为H，用虚线连接CH并延长至与线段AB相交。Q是三角形ABC外的一点，连接PQ、QC，线段PQ与QC垂直，垂足为Q，作辅助线QE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4300.jpg" }, { "index": 4301, "ocr_en": "In trapezoid ABCD, AB is the lower base and DC is the upper base. K is a point on line segment AD, M is a point on line segment BC, connect DM, CK, MA, and KB, draw auxiliary line KM, and extend line segment CB downward to a point F outside the trapezoid.", "ocr_zh": "在梯形ABCD中，AB为下底，DC为上底。K是线段AD上的点，M是线段BC上的点，连接DM、CK、MA、KB，作辅助线KM，向下延长线段CB至梯形外的一点F。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4301.jpg" }, { "index": 4302, "ocr_en": "Points A, M, D, and N are points on a circle in the plane. Connect MN. Point O is the midpoint of segment NM. Connect segment AO and extend it to intersect the circle at point B. Connect segment DO to intersect the circle at point C. Connect BC and AD. The intersection point of segment BC and MN is E, and the intersection point of segment AD and MN is F. Point G lies on the arc AC. Draw auxiliary lines GA, GO, and GE.", "ocr_zh": "点A、M、D、N是平面内的一个圆上的点，连接MN，点O是线段NM的中点，连接线段AO并延长至与圆相交于点B，连接线段DO至与圆相交于点C，连接BC、AD，线段BC与MN的交点为E，线段AD与MN的交点为F。G是位于圆弧AC上的点，作辅助线GA、GO、GE。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4302.jpg" }, { "index": 4303, "ocr_en": "In a plane, there is a triangle ABC. I is a point inside triangle ABC. Connect IB and IC, and draw the circumcircle of triangle ABC. Draw a circle tangent to segment IC through points B and I, and draw a circle tangent to segment IB through points C and I. The points of tangency are both I. The two circles intersect at another point O other than I, and point O is located outside triangle ABC. Draw auxiliary lines OI, OB, and OC.", "ocr_zh": "平面内有三角形ABC，I是三角形ABC内的一点，连接IB、IC，作三角形ABC的外接圆。过点B、I作与线段IC相切的圆，过点C、I作与线IB相切的圆，切点都为I，两圆相交于除点I外的另一点O，点O位于三角形ABC之外。作辅助线OI、OB、OC。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4303.jpg" }, { "index": 4304, "ocr_en": "In a plane, there is a triangle ABC, and point D is the midpoint of segment BC. Draw the incircle of triangle ABC. The point of tangency between segment AB and the circle is E, and the point of tangency between segment AC and the circle is F. Connect EF. Draw the bisector of angle B, BN, which intersects segment EF at point N. Draw the bisector of angle C, CM, which intersects segment EF at point M. The intersection of segments BN and CM is point I. Connect DM and DN, and draw auxiliary lines IE and BM.", "ocr_zh": "平面内有三角形ABC，点D是线段BC的中点。作三角形ABC的内切圆，线段AB与圆的切点为E，线段AC与圆的切点为F，连接EF。作角B的平分线BN与线段EF相交于点N，作角C的平分线CM与线段EF相交于点M，线段BN与CM的交点为I，连接DM、DN，作辅助线IE、BM。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4304.jpg" }, { "index": 4305, "ocr_en": "In a plane, there is an acute-angled triangle ABC. AD, BE, and CF are the three altitudes of triangle ABC, with their feet at D, E, and F, respectively. The orthocenter is H, and Q is a point on segment AF. Connect DF, and the intersection of DF and BE is P. Connect QP and QE, and the intersection of segment QE and AD is N.", "ocr_zh": "平面内有锐角三角形ABC，AD、BE、CF是三角形 ABC 的三条高，垂足分别是D、E、F，垂心是H，Q是线段AF上的一点。连接DF，DF与BE交点为P，连接QP、QE，线段QE与AD的交点为N。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4305.jpg" }, { "index": 4306, "ocr_en": "Make a circle with vertical segment AC as its diameter and BD as a chord perpendicular to AC above the center of the circle, with pivot at E. Connect AB, BC, CD, DA. Make a tangent to the circle at point B and intersect the extension line of DA at point F. Extend AB to point G so that AG=AC, connect GD and GF, and make a perpendicular to CH at point C, with pivot at H, and connect HF. Connect the auxiliary lines HB, FE, GC.", "ocr_zh": "做以竖直线段AC为直径的圆，BD为圆心上方一条垂直AC的弦，垂足为E。连接AB、BC、CD、DA。过B点做圆的切线交DA的延长线于点F，延长AB至点G使AG=AC，连接GD和GF，过点C做CH垂直FG，垂足为H，连接HF。连接辅助线HB、FE、GC。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4306.jpg" }, { "index": 4307, "ocr_en": "Two circles intersect at point C and point N, do the two circles of the common tangent AB, point A and point B are in the two circles, AB midpoint is D, connect the auxiliary line CN and extend the intersection of AB at point M.", "ocr_zh": "两圆相交于点C和点N，做两圆的公切线AB，点A和点B分别在两圆上，AB中点为D，连接辅助线CN并延长交AB于点M。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4307.jpg" }, { "index": 4308, "ocr_en": "All three sides of an equilateral triangle ABC are extended on both sides by the same length, where AB is extended to points P1 and P, AC is extended to points Q and Q1, and BC is extended to points R1 and R. Make circles through the six points QQ1PP1RR1.", "ocr_zh": "等边三角形ABC的三条边都向两侧延长相同的长度，其中AB延长至P1点和P点，AC延长至Q点和Q1点，BC延长至R1和R点。过QQ1PP1RR1六个点做圆。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4308.jpg" }, { "index": 4309, "ocr_en": "Make the outer circle of the square ABCD, the center of the circle is O, connect AC, P is a point on the inferior arc AB, connect DP to intersect AC at point Q, connect the auxiliary line BD.", "ocr_zh": "做正方形ABCD的外接圆，圆心为O，连接AC，P为劣弧AB上一点，连接DP交AC于点Q，连接辅助线BD。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4309.jpg" }, { "index": 4310, "ocr_en": "The highest point of the circle is A, the lowest point is C, and DE is a horizontal chord below the center of the circle. Through the point A to do DE parallel line AP, connect CE and extend the intersection of AP at point P, connect DE, connect DP intersection of the circle at point B, connect CB and extend the intersection of AP at point M.", "ocr_zh": "圆的最高点为A，最低点为C，DE为圆心下方的一条水平的弦。过点A做DE的平行线AP，连接CE并延长交AP于点P，连接DE，连接DP交圆于点B，连接CB并延长交AP于点M。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4310.jpg" }, { "index": 4311, "ocr_en": "Isosceles trapezoid AMNB, AM parallel to BN, make the trapezoid's external circle, the center of the circle is O, connect AN and BM, extend BA and NM intersect at point P.", "ocr_zh": "等腰梯形AMNB中，AM平行于BN，做梯形的外接圆，圆心为O，连接AN和BM，延长BA和NM相交于点P。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4311.jpg" }, { "index": 4312, "ocr_en": "Make a semicircle with line AB as the diameter, take point D and point C on the arc of the circle, point D is close to point A, point C is close to point B, connect DB and AC to intersect at point E. Make an auxiliary line EF perpendicular to AB over point E, with the foot of the pendant as F. Connect the auxiliary lines AD and CB.", "ocr_zh": "以线段AB为直径做半圆，在圆弧上取点D和点C，点D靠近点A，点C靠近点B，连接DB和AC交于点E，过点E做辅助线EF垂直AB，垂足为F。连接辅助线AD和CB。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4312.jpg" }, { "index": 4313, "ocr_en": "Inside triangle ABC, D is a point on BC, F is a point on AB, connect AD and CF intersect at point H, connect BH and extend to intersect AC at point E. Connect DE and extend to intersect the extension line of BA at point P, connect FE and extend to intersect the extension line of BC at point Q, connect PQ, point O is a point in triangle BFH, connect OH.", "ocr_zh": "三角形ABC内，D为BC上一点，F为AB上一点，连接AD和CF交于点H，连接BH并延长交AC于点E。连接DE并延长交BA的延长线于P点，连接FE并延长交BC的延长线于Q点，连接PQ，点O为三角形BFH中一点，连接OH。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4313.jpg" }, { "index": 4314, "ocr_en": "Convex quadrilateral ABCD is inside a circle, extend BA and CD to intersect at point P, and extend BC and AD to intersect at point Q. Connect the auxiliary line PQ, and make the auxiliary line DM parallel to CQ over point D. M is on the line segment PQ.", "ocr_zh": "凸四边形ABCD内接于圆，延长BA和CD交于点P，延长BC和AD交于点Q。连接辅助线PQ，过点D做辅助线DM平行于CQ，M在线段PQ上。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4314.jpg" }, { "index": 4315, "ocr_en": "Take the line segment BC as the diameter to make a circle, the center of the circle is point D, point A is a point on the arc BC, connect AD and extend to intersect the circle at point E. Take AD as the diameter to make a circle, connect AB to intersect the circle at point M, connect AC to intersect the circle at point N, connect MN to intersect AD at point G, connect the auxiliary lines MD, BE and ND.", "ocr_zh": "以线段BC为直径做圆，圆心为点D，点A为弧BC上一点，连接AD并延长交圆于点E，以AD为直径做圆，连接AB交该圆于点M，连接AC交该圆于点N，连接MN交AD于点G，连接辅助线MD、BE、ND。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4315.jpg" }, { "index": 4316, "ocr_en": "Make the outer circle of the right triangle AMD, where AD is perpendicular to MD, and the vertical foot is D. Extend MD to the two ends to point B and point C respectively, connect AB to the circle at point E, and connect AC to the circle at point F.", "ocr_zh": "做直角三角形AMD的外接圆，其中AD垂直于MD，垂足为D。向两端延长MD分别至点B和点C，连接AB交圆于点E，连接AC交圆于点F。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4316.jpg" }, { "index": 4317, "ocr_en": "The line segment AB is the chord of the circle O, the tangent to the circle is made at points A and B and intersects at point P. Point N is the midpoint of the line segment AB, point C is a point on the superior arc AB, connect CN and extend it to intersect AP at point M, connect CP to intersect the circle at point D, connect ND and extend it to intersect PB at point Q. Connect the auxiliary lines OA, OC, OD and OP.", "ocr_zh": "线段AB为圆O的弦，过A点和B点分别做圆的切线并相交于点P。点N为线段AB的中点，点C为优弧AB上一点，连接CN并延长交AP于点M，连接CP交圆于点D，连接ND并延长交PB于点Q。连接辅助线OA、OC、OD、OP。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4317.jpg" }, { "index": 4318, "ocr_en": "Line AB is the chord of circle O, M is the midpoint of the inferior arc AB, C is a point outside the circle, from point C to make a circumcircle cut line intersecting the circle and PQ two points, connect MP to cross AB at R to connect RQ, Z is the chord PQ above the point O below the point S is a point on the inferior arc BP, T is an inferior arc on the point of PQ, to connect OS, to connect MT to cross AB at F, to connect MS to cross AB at E. From F to make the perpendicular line of AB and OT to intersect at point Y, from point E to make a segment EX of the perpendicular line of AB, X is a point on OS. The perpendicular of AB from F intersects with OT at point Y. The perpendicular of AB from point E is EX, and X is a point on OS. Connect the auxiliary lines AM, AP, MC, XZ, OM.", "ocr_zh": "线段AB为圆O的弦，M为劣弧AB的中点，C为圆外一点，由C点做一条圆的割线交圆与PQ两点，连接MP交AB于R连接RQ，Z为弦PQ上方，点O下方的一点S为劣弧BP上一点，T为劣弧PQ上一点，连接OS，连接MT交AB于F，连接MS交AB于E。由F做AB的垂线与OT交于点Y，由E点做AB的垂线段EX，X为OS上一点。连接辅助线AM、AP、MC、XZ、OM。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4318.jpg" }, { "index": 4319, "ocr_en": "Circle O1 and circle O2 outside away, AB and CD were circle O1, O2 two diameters, and AB is parallel to CD, connect AD, BC, DB, AC, AC intersection of circle O1 and point E, intersection of circle O2 and point F, connect DF and BE.", "ocr_zh": "圆O1与圆O2外离，AB和CD分别为圆O1、O2两条直径，且AB平行于CD，连接AD、BC、DB、AC，AC交圆O1与点E，交圆O2与点F，连接DF和BE。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4319.jpg" }, { "index": 4320, "ocr_en": "In parallelogram ABCD, take point Q and point P on AD and AB so that AQ is equal to AP, connect the auxiliary line QP, make the outer circle of triangle APQ, connect AC to intersect the circle at point R, connect the auxiliary lines RQ and RP.", "ocr_zh": "平行四边形ABCD中，在AD和AB上取点Q和点P使AQ等于AP，连接辅助线QP，做三角形APQ的外接圆，连接AC交圆于点R，连接辅助线RQ和RP。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4320.jpg" }, { "index": 4321, "ocr_en": "In a plane, there is an acute-angled triangle ABC. Draw AA1 perpendicular to segment BC and BB1 perpendicular to segment AC with dashed lines. The feet of the perpendiculars are points A1 on segment BC and B1 on segment AC, respectively. The intersection point of segments AA1 and BB1 is H. Draw auxiliary lines A1A2 perpendicular to segment AC and B1B2 perpendicular to segment BC. The feet of the perpendiculars are points A2 on segment AC and B2 on segment BC, respectively. The intersection point of segment BB2 and segment A1A2 is O. Draw auxiliary line AB2 and connect OH. G is a point on segment OH.", "ocr_zh": "平面内有锐角三角形ABC，以虚线作AA1垂直于线段BC、BB1垂直于线段AC，垂足分别是线段BC上的点A1、线段AC上的点B1，线段AA1与BB1交点为H。以虚线作A1A2垂直于线段AC、B1B2垂直于线段BC，垂足分别是线段AC上的A2、线段BC上的B2，线段BB2与线段A1A2的交点为O，作辅助线AB2，连接OH，G是线段OH上的一点。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4321.jpg" }, { "index": 4322, "ocr_en": "In a plane, there is an acute-angled triangle ABC. The line segments BE and CF are the two altitudes of triangle ABC, with their feet at E and F, respectively. The intersection point of line segments BE and CF is H. M and G are points on line segment BH, with point G closer to point B than point M. N is a point on line segment HF. O is a point inside the triangle BHF formed by the line segments BH, HF, and BF. Draw auxiliary lines OB, OG, OH, and OC.", "ocr_zh": "在平面内有锐角三角形ABC，线段BE、CF是三角形ABC的两条高，垂足分别为E、F，线段BE和CF交点为H。M、G是线段BH上的一点，点G相比点M更靠近点B，N是线段HF上的一点。O是线段BH、HF、BF围成的三角形BHF内的一点，作辅助线OB、OG、OH、OC。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4322.jpg" }, { "index": 4323, "ocr_en": "In a plane, there is an acute-angled triangle ABC. Point I is a point inside triangle ABC. Points A1, B1, and C1 are three points outside triangle ABC. Connect IA1, IB1, and IC1. Then, the three sides AB, BC, and AC of triangle ABC are respectively perpendicular bisectors of IC1, IA1, and IB1. The intersection point of IA1 and segment BC is D. Draw the auxiliary line IB, connect A1C1, A1B1, and B1C1 to form triangle A1B1C1, and draw the circumcircle of triangle A1B1C1.", "ocr_zh": "在平面内有锐角三角形ABC，I是三角形ABC内的一点，点A1、B1、C1是三角形ABC外的三个点，连接IA1、IB1、IC1，则三角形ABC的三条边AB、BC、AC分别垂直平分IC1、IA1、IB1，IA1与线段BC的交点为D。作辅助线IB，连接A1C1、A1B1、B1C1，形成三角形A1B1C1，作三角形A1B1C1的外接圆。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4323.jpg" }, { "index": 4324, "ocr_en": "In a plane, there is an acute-angled triangle ABC, where the length of segment AB is greater than the length of segment AC. Draw the circumcircle of triangle ABC. Line l is tangent to the circumcircle of ABC, with the point of tangency at A. D is a point on segment AB, E and F are two points on line l, point A is the midpoint of segment EF, point E is to the right of point A, and point F is to the left of point A. Draw auxiliary lines DC, FD, ED, and EC. I is a point on segment DE located within the region of triangle ABC. Connect IA, extend the dashed line segment DF with a dashed line, and extend the dashed line segment AI with a dashed line. The two extended lines intersect at a point I1 outside the circumcircle of triangle ABC. Draw auxiliary lines IB, IC, and I1B, and extend segment AB downward with a dashed line.", "ocr_zh": "平面内有锐角三角形ABC，线段AB长度大于线段AC。作三角形ABC的外接圆，直线l与ABC的外接圆相切，切点为A。D是线段AB上的一点，E、F是直线l上的两点，点A为线段EF的中点，E点位于A点右侧，F点位于A点左侧，作辅助线DC、FD、ED、EC。I是线段DE位于三角形ABC区域内的一点，连接IA，以虚线延长虚线线段DF、以虚线延长虚线线段AI，两条延长线相交于三角形ABC外接圆外的一点I1，作辅助线IB、IC、I1B，用虚线向下延长线段AB。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4324.jpg" }, { "index": 4325, "ocr_en": "In a plane, there is a triangle ABC. D is a point on line segment BC. Connect AD to form triangles ABD and ADC. O2 is a point inside triangle ADC, O is a point inside triangle ABD, and O1 is a point outside triangle ABC, located to the left of line segment AB. Draw auxiliary lines O1A, O1B, O1D, OA, and OB, and connect O1O2, O2O, and OO1.", "ocr_zh": "在平面内有三角形ABC，D是线段BC上的一点，连接AD，围成三角形ABD与ADC。O2是三角形ADC内的一点，O是三角形ABD内的一点，O1是三角形ABC外、位于线段AB左侧的一点。作辅助线O1A、O1B、O1D、OA、OB，连接O1O2、O2O、OO1。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4325.jpg" }, { "index": 4326, "ocr_en": "In a plane, there is a triangle ABC. I is a point inside triangle ABC. O2 is a point outside triangle ABC and to the right of line segment AC. O1 is a point outside triangle ABC and below line segment BC. O3 is a point outside triangle ABC and to the left of line segment AB. Connect IA, IB, IC, O1O2, O2O3, and O3O1.", "ocr_zh": "平面内有三角形ABC，I是三角形ABC内的一点，O2是位于三角形ABC外、线段AC右侧的点，O1是位于三角形ABC外、线段BC下方的点，O3是位于三角形ABC外、线段AB左侧的点，连接IA、IB、IC、O1O2、O2O3、O3O1.", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4326.jpg" }, { "index": 4327, "ocr_en": "Point O is the center of circle O in the plane. Point A is a point outside circle O. Line segments AB and AC are two tangents to circle O, with tangent points B and C, respectively. Connect OB, OC, BC, and OA. The intersection points of line segment OA with circle O are I and M, respectively. Extend line segment OA downward, and it intersects circle O at point I1. Point E is a point on the arc BI. Draw EM and extend it to intersect circle O at point F. Draw auxiliary lines OE and OF, and connect EA and FA.", "ocr_zh": "点O是平面内圆O的圆心，点A是圆O外的一点，线段AB、AC是圆O的两条切线，切点分别是B、C，连接OB、OC、BC、OA，线段OA与圆O的交点为I、与BC的交点为M,向下延长线段OA与圆O相交于点I1。点E是位于圆弧BI上的一点，连接EM并延长、与圆O相交于点F，做辅助线OE、OF，连接EA、FA。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4327.jpg" }, { "index": 4328, "ocr_en": "There are six points O1, O2, O3, O4, O5, and O6 in a plane. These six points are distributed around point M in the plane in clockwise order. Connect O1M, O2M, O3M, O4M, O5M, and O6M.", "ocr_zh": "平面内有O1、O2、O3、O4、O5、O6六个点，这六个点按顺时针顺序围绕平面内的点M分布，连接O1M、O2M、O3M、O4M、O5M、O6M。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4328.jpg" }, { "index": 4329, "ocr_en": "In a plane, there is a line segment AB. A circle is drawn with AB as its diameter. Lines l1 and l2 are tangent to this circle at points A and B, respectively. Point P is located to the left of line l1 and below point A. Connect P1A and P1B. Point P2 is located below the circle and within the area between lines l1 and l2. Connect P2A and P2B.", "ocr_zh": "平面内有线段AB，以AB为直径作圆，直线l1、l2与该圆相切，切点分别是A、B。点P是位于直线l1左侧、A点下方的一点，连接P1A、P1B。点P2是位于圆下方、直线l1和l2之间的区域内的一点，连接P2A、P2B。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4329.jpg" }, { "index": 4330, "ocr_en": "In a plane, there is a circle with diameter AiBi and center Oi. Point O is a point located outside the circle, connecting OAi, OBi, and OOi.", "ocr_zh": "平面内有以线段AiBi为直径、以点Oi为圆心的圆，点O是位于圆之外的一点，连接OAi、OBi、OOi。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4330.jpg" }, { "index": 4331, "ocr_en": "In a plane, there is an acute-angled triangle ABC. The three altitudes of triangle ABC are AD, CF, and BE, with their feet at D, F, and E, respectively, and the orthocenter at H. Draw the circumcircle of triangle ABC with center O and connect OH. Connect H to a point on the arc AB. Draw a circle through points D, E, and F. This circle intersects segment AB at another point F', segment AC at another point E', segment BC at another point D', segment AD at another point J, segment BE at another point K, and segment CF at another point L.", "ocr_zh": "平面内有锐角三角形ABC，三角形ABC的三条高是AD、CF、BE，垂足分别是D、F、E，垂心为H。作三角形ABC的外接圆，圆心为O，连接OH。连接H和位于圆弧AB上的某一点。过点D、E、F作圆，该圆与线段AB相交于另一点F’，与线段AC相交于另一点 E’， 与线段BC相交于另一点D’，与线段AD相交于另一点J，与线段BE相交于另一点K，与线段CF相交于另一点L。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4331.jpg" }, { "index": 4332, "ocr_en": "A, B, C for the circle O on the three points, connect AB and AC, over the O point respectively do the string AB of the pendant and extend, intersection AC at point K, over the O point respectively do the string AC of the pendant and extend, intersection AB at point J, point P for the arc BC on the one point, connect the PC and PB, extend the AB and CP intersected at point E, extend the BP and AC intersected at point F. Connection of the auxiliary line CJ and BK.", "ocr_zh": "A、B、C为圆O上三点，连接AB和AC，过O点分别做弦AB的中垂线并延长，交AC于点K，过O点分别做弦AC的中垂线并延长，交AB于点J，点P为弧BC上一点，连接PC和PB，延长AB和CP交于点E，延长BP和AC交于点F。连接辅助线CJ和BK。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4332.jpg" }, { "index": 4333, "ocr_en": "Extend the six sides of the ortho-hexagon ABCDFE to intersect at six points PQRP'Q'R' respectively. Where angle P'AF is angle 1, angle PAB is angle 2, angle PAB is angle 3, angle Q'BC is angle 4, angle Q'CB is angle 5, angle QCD is angle 6, angle QDC is angle 7, angle R'DE is angle 8, angle R'ED is angle 9, angle REF is angle 10, angle RFE is angle 11 and angle P'FA is angle 12.", "ocr_zh": "分别延长正六边形ABCDFE的六条边，分别相交于PQRP'Q'R'六个点。其中角P'AF为角1，角PAB为角2，PAB角为角3，Q'BC角为角4，Q'CB角为角5，QCD角为角6，QDC角为角7，R'DE角为角8，R'ED角为角9，REF角为角10，RFE角为角11，P'FA角为角12。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4333.jpg" }, { "index": 4334, "ocr_en": "Point B and point E for the triangle ACD outside the two points, where the point B in the side AC outside, point E in the side AD outside, connect BD to AC at point X, connect EC to AD at point Y, BD and CE intersect at point P, do the auxiliary line AP and extend the intersection of CD at point M.", "ocr_zh": "点B和点E为三角形ACD外两点，其中点B在边AC外侧，点E在边AD外侧，连接BD交AC于点X，连接EC交AD于点Y，BD和CE相交于点P，做辅助线AP并延长交CD于点M。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4334.jpg" }, { "index": 4335, "ocr_en": "Line segment AB is a chord of circle Γ2, below the center of the circle, make the mid-perpendicular MN of AB, and intersect the inferior arc AB at point M, with the foot of the perpendicular N. C is a point on the line segment AN, make the auxiliary line MC and extend it to intersect the circle Γ2 at point S, and make the perpendicular CD of AB at point C, with the foot of the perpendicular C and intersecting with the perpendicular SD of SC at point D. Make the outer circle Γ1 of triangle SCD, with the center of the circle as O1.", "ocr_zh": "线段AB是圆Γ2的一条弦，在圆心下方，做AB的中垂线MN，交劣弧AB于点M，垂足为N。C为线段AN上一点，做辅助线MC并延长交圆Γ2于点S，过C点做AB的垂线CD，垂足为C，和SC的垂线SD交于点D。做三角形SCD的外接圆Γ1，圆心为O1。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4335.jpg" }, { "index": 4336, "ocr_en": "Isosceles triangle ABC, AB is equal to AC, BC is the horizontal line, do the triangle ABC external circle, the center of the circle O1, point F is the lowest point of the circle O1, point F to do the circle O, the center of the circle O, tangent to the line segment AB and AC, the tangent point is D and E, connect FE and extend the circle O1 at point N, connect FD and extend the circle O1 at point M, connect MC and NB, MC and NB intersect at point I. Connect the auxiliary line DE. Connect the auxiliary line DE.", "ocr_zh": "等腰三角形ABC中，AB等于AC，BC为水平线，做三角形ABC的外接圆，圆心为O1，点F为圆O1的最低点，过点F做圆O，圆心为O，与线段AB和AC相切，切点分别为D和E，连接FE并延长交圆O1于点N，连接FD并延长交圆O1于点M，连接MC和NB，MC和NB相交于点I。连接辅助线DE。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4336.jpg" }, { "index": 4337, "ocr_en": "Make the diagonals AC and BD of the convex quadrilateral ABCD, AC and BD intersect at point O. Make the external circles of triangle OAD and triangle OBC, which intersect at point O and point M. Connect MD and make the external circles of triangle ABO and triangle CDO. Extend CB to intersect the external circle of triangle OAB at point T. Connect OT and extend to intersect the external circle of triangle OCD at point S. Connect CS. Connect the auxiliary lines MA, MB and MC.", "ocr_zh": "做凸四边形ABCD的对角线AC和BD，AC和BD交于点O，做三角形OAD和三角形OBC的外接圆，两圆相交于点O和点M，连接MD，做三角形ABO和三角形CDO的外接圆。延长CB交三角形OAB的外接圆于点T，连接OT并延长交三角形OCD的外接圆于点S，连接CS。连接辅助线MA、MB、MC。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4337.jpg" }, { "index": 4338, "ocr_en": "The perpendicular bisector of the vertical line segment AC is the auxiliary line BD, B is on the left side of the line segment AB, connect AB, BC, CD, DA, point F is a point on AD near point D, connect BF, make isosceles right triangle BFQ with BF as the long side, angle Q is 90 degrees, and point Q is in the triangle ABC. Point E is a point on the line segment AB near point A, connect ED, with ED as the long side of the isosceles obtuse angle triangle EPD, angle P is obtuse. Point O is a point on triangle BQF, and point Q is to the left of AC and below BD, connect the auxiliary lines OB, OE, OF, OD.", "ocr_zh": "竖直线段AC的垂直平分线为辅助线BD，B在线段AB左侧，连接AB、BC、CD、DA，点F为AD上靠近D点的一点，连接BF，做以BF为长边的等腰直角三角形BFQ，角Q为90度，且Q点在三角形ABC内。点E为线段AB上靠近点A的一点，连接ED，以ED为长边做等腰钝角三角形EPD，角P为钝角。点O为三角形BQF上一点，且点Q在AC左侧，BD下方，连接辅助线OB、OE、OF、OD。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4338.jpg" }, { "index": 4339, "ocr_en": "In triangle ABC, make the auxiliary line XY, where X is on AB, Y is on AC, and XY is parallel to BC. point D is a point on the line BC, connect to AC and extend it, make the auxiliary line BP perpendicular to the extension of AC at point B, and the pivot point is P. Connect the auxiliary line XD to the auxiliary line YD. connect to PX and extend it, make the extension of AM perpendicular to the extension of PX at point M, and make the line AN through A. make angle MAN equal to the angle BAC. Make the angle MAN equal to the angle BAC. Make the line YN perpendicular to the line AN at the point Y, with the foot of the line N, and connect it to MN. Extend the line YN to the point AP and connect it to the point Q. The point O is a point on the line AQ and is above the line NM. The area within triangle AMX and triangle ANY is shaded.", "ocr_zh": "三角形ABC中，做辅助线XY，其中X在AB上，Y在AC上，且XY平行于BC。点D为线段BC上一点，连接AC并延长，过B点做辅助线BP垂直于AC的延长线，垂足为P。连接辅助线XD和辅助线YD。连接PX并延长，过A点做AM垂直于PX的延长线于点M，过点A做线段AN使角MAN等于角BAC，过点Y做线段YN垂直于线段AN，垂足为N，连接MN。延长YN交AP于点Q，连接CQ。点O为AQ上一点，且在线段NM上方。三角形AMX和三角形ANY内的区域为阴影区。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4339.jpg" }, { "index": 4340, "ocr_en": "In triangle ABC there is a point O, connecting OA, OB and OC. An auxiliary line ED is made parallel to AC through point O. Point E is on AB and point D is on BC. Angle AOE is angle 1 and angle COD is angle 2.", "ocr_zh": "在三角形ABC中有一点O，连接OA、OB、OC。过点O做辅助线ED平行于AC，点E在AB上，点D在BC上。角AOE为角1，角COD为角2。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4340.jpg" }, { "index": 4341, "ocr_en": "Make the outer circle of triangle ABC, point P is a point on the inferior arc AB near A. Connect CP and BP and extend them to point V and point U so that angle PUV is equal to angle BCU, connect UV and extend it to point Q. Connect the auxiliary lines AV, AU, and AQ. Extend BC to make the auxiliary line CD at point D and connect the auxiliary lines AD and AQ.", "ocr_zh": "做三角形ABC的外接圆，点P为劣弧AB上靠近A的一点，连接CP和BP并延长于点V和点U，使角PUV等于角BCU，连接UV并延长于点Q。连接辅助线AV、AU、AQ。延长BC做辅助线CD于点D，连接辅助线AD和AQ。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4341.jpg" }, { "index": 4342, "ocr_en": "The center of the tangent circle of triangle ABC is I, and the center of the external circle of the triangle is O. Connect the auxiliary line AI and extend it to intersect the circle O at point D. Connect the auxiliary line OD and extend it to intersect the circle O at point E, and connect it to EC. The tangent point of the circle I and AB is F. Connect the auxiliary line IF.", "ocr_zh": "做三角形ABC的内切圆圆心为I，三角形的外接圆圆心为O，连接辅助线AI并延长交圆O于点D，连接辅助线OD并延长交圆O于点E，连接EC。圆I和AB的切点为F，连接辅助线IF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4342.jpg" }, { "index": 4343, "ocr_en": "Make three auxiliary line segments IK3, IK2, and IK1 in three directions outward from the point I so that the three segments are equal, connect the auxiliary lines K3K1, K3K2, and K2K1, and extend IK3, IK2, and IK1 twice to the points B, C, and A, respectively, and connect AB, AC, and BC. Make four circles centered on K1, K2, K3, and I, and with half the length of IK1 as the radius, respectively.", "ocr_zh": "由点I向外三个方向做三条辅助线段IK3、IK2、IK1，使三条线段相等，连接辅助线K3K1、K3K2、K2K1，分别延长IK3、IK2、IK1两倍于点B、C、A，连接AB、AC、BC。分别以K1、K2、K3、I为圆心，以IK1的长度的一半为半径，做四个圆。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4343.jpg" }, { "index": 4344, "ocr_en": "The quadrilateral ABCD is connected to the circle O, and the AB extension wire and the DC extension line intersect at the point E. The AD extension cable and the BC extension cable intersect F, the midpoint of AC is M, and the midpoint of BD is N, and the auxiliary line is connected: connect MN and EF. Extend the MN to the point P, take the EC midpoint Q, and connect MQ and QP.", "ocr_zh": "四边形ABCD内接于圆O，AB延长线与DC延长线交于点E，\nAD延长线与BC延长线交于F，AC中点为M，BD中点为N，做辅助线：连接MN，EF。延长MN交EF于点P,取EC中点Q，连接MQ和QP。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4344.jpg" }, { "index": 4345, "ocr_en": "The center of the quadrilateral ABCD inscribed circle is O, connecting AC and BD, taking the midpoint of AC as M and the midpoint of BD as N, and making auxiliary lines: let the AB extension line and the DC extension line intersect at the point E, take the point X on AB, take the point Y on the CD, connect XY, and make XY pass the point O, connect AO, DO, MN.", "ocr_zh": "四边形ABCD内切圆圆心为O，连接AC，BD，取AC中点为M，BD中点为N，做辅助线：设AB延长线和DC延长线交于点E，在AB上取点X，在CD上取点Y，连接XY，并且使得XY过点O，连接AO，DO，MN。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4345.jpg" }, { "index": 4346, "ocr_en": "The point I is a point in the triangle ABC, and the parallel lines A1B2, B1C2 and C1A2 of AB, BC and CA are respectively intersected by I, AC and BC at points A1 and B2, AB and CA are intersected at points B1 and C2, and BC and AB are intersected at points C1 and A2.", "ocr_zh": "点I为三角形ABC内一点，过I分别作AB、BC、CA的平行线A1B2、B1C2、C1A2分别交AC、BC于点A1、B2，交AB、CA于点B1、C2，交BC、AB于点C1、A2。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4346.jpg" }, { "index": 4347, "ocr_en": "In the triangle ABC, any point P is taken, and the straight lines AP, BP, and CP intersect with the edges BC, CA, and AB at the points D, E, F, respectively", "ocr_zh": "三角形ABC内任取一点P，直线AP、BP、CP分别与边BC、CA、AB相交于点D、E、F", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4347.jpg" }, { "index": 4348, "ocr_en": "The straight line and the three sides of the triangle ABC AB, BC, and CA intersect X, Y, Z, respectively. Auxiliary line: through A, B, C respectively as a straight line XYZ perpendicular line, set the vertical feet are Q, P, S", "ocr_zh": "一直线与三角形ABC的三边AB、BC、CA所在的直线分别相交于X、Y、Z。辅助线：过A、B、C分别作直线XYZ的垂线，设垂足分别为Q、P、S", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4348.jpg" }, { "index": 4349, "ocr_en": "M is a point in the triangle ABC, which connects and extends CM, AM, BM, and passes AB, BC, and AC at the points F, D, E, respectively", "ocr_zh": "M为三角形ABC内一点，连接并延长CM,AM,BM分别交AB,BC,AC于点F,D,E", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4349.jpg" }, { "index": 4350, "ocr_en": "The first figure: D, E, and F are any three points in the triangle ABC, connecting AD, BE, CF; The second figure: D, E, and F are any three points outside the triangle ABC, connecting AD, BE, CF; The third diagram: D', E, F are any three points outside the triangle ABC, D is one point inside the triangle ABC, connecting AD, AD', BE, CF.", "ocr_zh": "第一张图：D,E,F为三角形ABC内的任意三点，连接AD,BE,CF；第二张图：D,E,F为三角形ABC外任意三点，连接AD,BE,CF;第三张图：D',E,F为三角形ABC外任意三点，D为三角形ABC内一点，连接AD,AD',BE,CF.", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4350.jpg" }, { "index": 4351, "ocr_en": "D', E, F are any three points outside the triangle ABC, D is one point inside the triangle ABC, connecting AD, AD', BE, CF.", "ocr_zh": "D',E,F为三角形ABC外任意三点，D为三角形ABC内一点，连接AD,AD',BE,CF.", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4351.jpg" }, { "index": 4352, "ocr_en": "Take any point D on the side BC of the triangle ABC, and connect AD to the length BD=u, CD=v, AD=t, AB=c, AC=b.", "ocr_zh": "三角形ABC的边BC上任取一点D,连接AD设长度BD=u,CD= v,AD = t，AB=c,AC=b。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4352.jpg" }, { "index": 4353, "ocr_en": "K is the upper point of the triangle ABC side BC, D1 and D2 are two different points on the AK extension line, the AK extension line is represented as AD, the BD1 extension and the AC extension line intersect the point N1, the CD1 extension and the AB extension line intersect the point M1, and E1 is the intersection point of the ray AK and the auxiliary line M1N1. The BD2 extension intersects with the AC extension line at the point N2, the CD2 extension intersects with the AB extension line at the point M2, and E2 is the intersection point between the ray AK and the auxiliary line M2N2.", "ocr_zh": "K是三角形ABC边BC上一点，D1、D2是AK延长线上不同的两点，AK延长线表示为AD,BD1延长与AC延长线交于点N1，CD1延长与AB延长线交于点M1，E1为射线AK与辅助线M1N1的交点。BD2延长与AC延长线交于点N2，CD2延长与AB延长线交于点M2，E2为射线AK与辅助线M2N2的交点。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4353.jpg" }, { "index": 4354, "ocr_en": "In triangle ABC, D is the point on the line segment BC, which satisfies the AD vertical BC, takes the point E on the edge AB, and the point F on the edge AC, connecting DE and DF. Auxiliary line: pass A to make the parallel line l of BC, and intersect with the DE extension line and DF extension line respectively at G and H", "ocr_zh": "三角形ABC中，D为线段BC上一点，满足AD垂直BC，取边AB上点E，边AC上点F，连结DE、DF。辅助线：过A 做BC的平行线l，并与DE延长线、DF延长线分别交于G、H", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4354.jpg" }, { "index": 4355, "ocr_en": "A1, B1 and C1 are the inner points of the sides of the triangle ABC BC, CA and AB respectively, and they are connected in pairs, and the positions of A1, B1 and C1 meet the requirements of connecting AA1, CC1 and BB1, and the three line segments intersect at one point. Ga, Gb and Gc are one point in the triangle AB1C1, triangle BC1A1 and triangle CA1B1, respectively. Connect AGa, BGb, CGc.", "ocr_zh": "A1、B1、C1分别是三角形ABC的边BC、CA、AB内一点，将其两两相连，A1、B1、C1位置满足连接AA1,CC1,BB1,三条线段交于一点。Ga，Gb，Gc分别为三角形AB1C1， 三角形BC1A1， 三角形CA1B1内一点。连接AGa,BGb,CGc。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4355.jpg" }, { "index": 4356, "ocr_en": "P is a point within the triangle ABC, connecting AP, BP, CP", "ocr_zh": "P为三角形ABC内一点，连接AP,BP,CP", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4356.jpg" }, { "index": 4357, "ocr_en": "The points D, E, and F are respectively on the edges BC, CA, and AB of the acute triangle ABC (all of which are not endpoints), connecting DE, EF, FD, satisfying BC parallel EF, D1 is a point on the edge BC (different from B, D, C), passing D1 as D1E1 is parallel to DE, D1F1 is parallel to DF, AC and AB are intersected at points E1 and F1 respectively, connecting E1F1, and then making a triangle PBC above BC (on the same side as A), connecting PD1, and connecting PE1 and PF1 with auxiliary lines. CF1,BE1。 Let E1F1 be handed over to EF to K", "ocr_zh": "点D、E、F分别在锐角三角形ABC的边BC、CA、AB上（均不是端点），连结DE,EF,FD,满足BC平行EF，D1是边BC上一点（不同于B、D、C)，过D1作D1E1平行于DE，D1F1平行于DF，分别交AC、AB两边于点E1、F1，连结E1F1，再在BC上方（与A同侧）作三角形PBC，连结PD1，辅助线连结PE1，PF1，CF1,BE1。设E1F1交EF于K", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4357.jpg" }, { "index": 4358, "ocr_en": "In the quadrilateral ABCD, connect AC, BD, angular CAB=30°, angular ABD=26°, angular ACD=13°, angular DBC=51°. Linking AC, BD.", "ocr_zh": "在四边形 ABCD中，连接AC,BD，角CAB=30°,角ABD= 26°，角ACD=13°,角DBC=51°。连结AC,BD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4358.jpg" }, { "index": 4359, "ocr_en": "In the convex pentagonal ABCDE, each vertex is connected in pairs, so that BD and CE intersect at the point F, and connect AF with BE at the point H", "ocr_zh": "在凸五边形ABCDE中，各个顶点两两相连，设BD，CE交于点F，连接AF交BE于点H", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4359.jpg" }, { "index": 4360, "ocr_en": "The points P and Q are the two points on the circumscribed circle of the triangle ABC (different from A, B, C), and the points P are about the lines BC, CA, The symmetry points of AB are U, V, W, respectively, and the connecting QU, QV, QW intersect with the straight lines BC, CA, and AB respectively at the points D, E, F. auxiliary lines: let the slave point P make the perpendicular line to BC, CA, and AB, and the perpendicular feet are X, Y, Z. respectively, and connect PY, PB, PC, PZ, PX, DE, XY, QC, XW", "ocr_zh": "点P、Q是三角形ABC的外接圆上（异于A、B、C)的两点，点P关于直线BC、CA、\nAB的对称点分别U、V、W，连结QU、QV、QW分别与直线BC、CA、AB交于点D、E、F.辅助线：设从点P向BC、CA、AB作垂线，垂足分别为X、Y、Z.连接PY,PB,PC,PZ,PX,DE,XY,QC,XW", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4360.jpg" }, { "index": 4361, "ocr_en": "The intersection points of the three sides of a circle and the triangle ABC BC, CA and AB are D1, D2, E1, E2, F1 and F2 in turn, and are connected in pairs, wherein F1D2, D1E2 and E1F2 are auxiliary lines, that is, dashed lines, and the line segments D1E1 and D2F2 intersect at the point L, and E1F1 and D2E2 intersect Point M, F1D1 and F2E2 intersect at point N and connect AL", "ocr_zh": "一圆与三角形ABC的三边BC、CA、AB的交点依次为D1、D2、E1、E2、F1、F2，并两两相连，其中F1D2,D1E2,E1F2为辅助线，即虚线，线段D1E1与D2F2交于点L，E1F1与D2E2交于\n点M，F1D1与F2E2交于点N，连接AL", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4361.jpg" }, { "index": 4362, "ocr_en": "A point A4 inside the triangle A1A2A3 connects A1A4, A2A4, A3A4. Do auxiliary line A1P, A4P, A3P", "ocr_zh": "三角形A1A2A3内一点A4，连接A1A4,A2A4,A3A4.在三角形的A1A3边外有一点P。做辅助线A1P,A4P,A3P", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4362.jpg" }, { "index": 4363, "ocr_en": "A little A4 inside the triangle A1A2A3 connects A1A4, A2A4, A3A4. Do auxiliary wire connection A1O1, A1O2, A1O3, A2O1, A2O2, A2O3, A3O1, A3O2, A3O3.", "ocr_zh": "三角形A1A2A3内一点A4，连接A1A4,A2A4,A3A4.在三角形的A2A3边外有一点O1,A1A3边外有一点O2,A1A2边外有一点O3。做辅助线连接A1O1,A1O2,A1O3,A2O1,A2O2,A2O3,A3O1,A3O2,A3O3.", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4363.jpg" }, { "index": 4364, "ocr_en": "There is a point I in the acute triangle ABC, which connects to AI, BI, CI and extends to AIa, BIb, CIc, and then connects BIa, CIa. AB, AC are extended.", "ocr_zh": "锐角三角形ABC内有一点I，连接AI,BI,CI并延长为AIa,BIb,CIc,进而连接BIa，CIa。将AB,AC延长。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4364.jpg" }, { "index": 4365, "ocr_en": "There is a little IA on the outside of the BC of the triangle ABC, which is connected with A, B, and C as auxiliary lines, and then the perpendicular line of the straight line where the three sides of the triangle are located is made of the auxiliary line Ia, which intersects with AC at the point F, with BC at the point D, and with AB at the point E.", "ocr_zh": "三角形ABC的BC外侧有一点IA，并与A,B,C相连作为辅助线，再做辅助线Ia与三角形三条边所在直线的垂线，与AC交于点F，与BC交于点D，与AB交于点E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4365.jpg" }, { "index": 4366, "ocr_en": "Make the circumscribed circle of the triangle ABC, the center of the circle is O, connect AO and extend the intersection BC at the point D, and the intersection circle O at the point E. Connect the CO and extend the intersection AB at point F. Connect BO.BO, and take points N and M on OC. Connect BE, CE as auxiliary cables, and ND, MD as auxiliary cables", "ocr_zh": "作三角形ABC的外接圆，圆心为O，连接AO并延长交BC于点D，交圆O于点E。连接CO并延长交AB于点F。连接BO .BO,OC上取点N和M。连接BE,CE作为辅助线，连接ND,MD作为辅助线", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4366.jpg" }, { "index": 4367, "ocr_en": "The inscribed circle of triangle ABC and AB, BC, and CA are tangent to the points C', A', B', respectively, and the vertices of the triangle and the tangent point are connected in pairs to intersect at the point Q, so that AC'=AB'= x, BC'=BA'= y, CA'= CB′= z", "ocr_zh": "三角形ABC的内切圆与AB,BC,CA分别切于点C',A',B'，三角形顶点与切点两两相连交于点Q，设AC'＝AB'= x,BC'=BA'= y，CA'= CB′= z", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4367.jpg" }, { "index": 4368, "ocr_en": "There are two points O, I in the triangle ABC, which are used as straight line OI, and a point P is taken in the line segment OI, and the perpendicular line of the three sides is crossed by the point P, and AB, AC, and BC are intersected at the points F, E, D", "ocr_zh": "三角形ABC内有两点O,I,作直线OI，并且在线段OI内取一点P，过点P作三条边的垂线，分别交AB,AC,BC于点F,E,D", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4368.jpg" }, { "index": 4369, "ocr_en": "There are two points O, I in the triangle ABC as a straight line OI, and a point P is taken in the line segment OI to make an auxiliary line: the point P is used as a perpendicular line of three sides, and AB, AC, and BC are intersected at the points F, E, D. The crossing points O and I also make corresponding trilateral perpendicular lines, and the intersection points are F1, E1, D1; F2,E2,D2", "ocr_zh": "三角形ABC内有两点O,I,作直线OI，并且在线段OI内取一点P，做辅助线：过点P作三条边的垂线，分别交AB,AC,BC于点F,E,D。而过点O和I也作相应三边垂线，交点为F1,E1,D1；F2,E2,D2", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4369.jpg" }, { "index": 4370, "ocr_en": "There are two parallel lines l1 and l2 in the plane, a square ABCD of AB, AD and l1 intersect at points E, F, edge BC, DC and l2 intersect at points G and H. To make the auxiliary line EH, FG intersects at the point O, with the point O as the center of the circle to make a circle tangent to BC and CD to the points N, Q, and to make a circle and AB, AD tangent to the points P and M", "ocr_zh": "平面内有两条平行直线l1和l2,一个正方形ABCD的AB,AD与l1交于点E,F，边BC,DC与l2交于点G,H。做辅助线EH,FG交于点O，以点O为圆心作圆与BC,CD相切于点N,Q，作圆与AB,AD相切于点P,M", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4370.jpg" }, { "index": 4371, "ocr_en": "The point O is the outer center of the acute triangle ABC. Respectively, the midpoint C', A', B' of the triangle ABC three sides AB, BC, AC is the center of the circle to make the circle of the point O, and the intersection points of the three circles are K (circle B' and circle C'), L (circle A' and circle C'), M (circle B' and circle A').", "ocr_zh": "点O是锐角三角形ABC的外心.分别以三角形ABC三边AB,BC,AC的中点C',A',B'为圆心作过点O的圆，这三个圆两两的异于O的交点分别为K（圆B'与圆C'）、L（圆A'与圆C'）、M（圆B'与圆A'）.做辅助线：连接B'C'交OK于点K'，分别连接三个圆心", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4371.jpg" }, { "index": 4372, "ocr_en": "In the triangle ABC, the point A is perpendicular to BC, the point P is perpendicular to AD, the point P is perpendicular to AC, the perpendicular foot is E, the perpendicular foot is perpendicular to AB and the perpendicular foot is F, and the auxiliary line is to connect EF, CP, BP. Take the points O2, O1, B' on CP, BP, BC, and connect AB', FO1, O1O2, EO2", "ocr_zh": "在三角形ABC中过点A作AD垂直于BC，点P在AD上，过点P作PE垂直于AC，垂足为E，作PF垂直于AB 垂足为F，做辅助线：连接EF，CP,BP.在CP,BP,BC上取点O2,O1,B',并连接AB',FO1,O1O2,EO2", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4372.jpg" }, { "index": 4373, "ocr_en": "In triangle ABC, the length of AB is greater than that of AC. Draw the circumcircle of triangle ABC, and draw a tangent line l to this circumcircle through point A. There is a point D on AB such that the length of AD is equal to that of AC. On tangent line l, there is a point F near point B and a point E near point C. Connect DE and DC with dashed lines, and there is a point G on DC. Connect AG with a dashed line and extend it; connect FD with a dashed line and extend it, and this extension intersects the extension of AG at point I₁. Connect BI₁ with a dashed line. There is a point I inside triangle ADG, and point I lies within angle ADE. Segment AG intersects DE at a point, which is designated as I'; connect BI' and CI' with dashed lines. It should be noted that in fact, this point is not marked as I' in the figure; it is only assumed to be I' for the convenience of description.", "ocr_zh": "在三角形 ABC 中，AB 的长度大于 AC 的长度。作三角形 ABC 的外接圆，并过点 A 作该外接圆的切线 l。AB 边上有一点 D，使得 AD 与 AC 的长度相等。切线 l 上，靠近点 B 的位置有一点 F，靠近点 C 的位置有一点 E。用虚线连接 DE 和 DC，DC 上有一点 G。用虚线连接 AG 并延长，用虚线连接 FD 并延长，该延长线与 AG 的延长线相交于点 I₁。用虚线连接 BI₁。三角形 ADG 内部有一点 I，且点 I 位于角 ADE 的内部。线段 AG 与 DE 相交于一点，将该点设为 I'，用虚线连接 BI' 和 CI'。需要注意的是，实际上图中并没有用 I' 来标记该点，这里只是为了便于描述而假设该点为 I'。\n", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4373.jpg" }, { "index": 4374, "ocr_en": "In the acute triangle ABC, there is a point I inside it. Connect BI and AI, then extend them to intersect sides AC and BC at points B₁ and A₁ respectively. Draw a segment PQ through I, intersecting segments AB₁ and BA₁ at points P and Q respectively. Take a point H on segment PI and a point O on segment IQ, and connect AH and BO with dashed lines.", "ocr_zh": "在锐角三角形 ABC 内部有一点 I，连接 BI、AI 并延长，分别与边 AC、BC 相交于点 B₁、A₁。过点 I 作线段 PQ，分别交线段 AB₁、BA₁于点 P、Q。再在线段 PI 上取一点 H，在线段 IQ 上取一点 O，用虚线连接 AH 和 BO。\n", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4374.jpg" }, { "index": 4375, "ocr_en": "In the acute triangle ABC, the length of side BC is greater than side AC, which is greater than side AB. Take a point D on side BC and a point E on side AC respectively, connect AD and BE, which intersect at point K. There are four points H, I, O, G inside the triangle. Draw auxiliary lines: take L on AC, connect IL and IK, connect AI and extend it to intersect BC at point N, connect HO, take a point M between N and D, and connect IM and LM.", "ocr_zh": "在锐角三角形 ABC 中，BC 边的长度大于边 AC 大于边 AB，在边 BC，AC 上分别取一点 D，E，连接 AD,BE 交于点 K，三角形内有四点 H,I,O,G。做辅助线：AC 上取 L，连接 IL,IK，连接 AI 并延长交 BC 于点 N, 连接 HO，在 N,D 之间取一点 M，连接 IM,LM。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4375.jpg" }, { "index": 4376, "ocr_en": "There is a point P inside triangle ABC. Draw perpendicular lines from P to the three sides, intersecting AB, BC, and CA at points F, D, and E respectively, which are the corresponding feet of the perpendiculars. Connect AP, BP, and CP.", "ocr_zh": "三角形 ABC 内有一点 P，过 P 作三边的垂线，分别交 AB、BC、CA 于点 F、D、E，这三点是对应的垂足，连接 AP、BP 和 CP。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4376.jpg" }, { "index": 4377, "ocr_en": "In triangle ABC, AB is equal to AC. A circle is tangent internally to the circumcircle O of triangle ABC at point M, and is tangent to AB and AC at points P and Q respectively. Connect PQ.", "ocr_zh": "在三角形 ABC 中，AB 等于 AC，一个圆内切于三角形 ABC 的外接圆 O 于点 M，且与 AB、AC 分别相切于点 P、Q，连接 PQ。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4377.jpg" }, { "index": 4378, "ocr_en": "In the isosceles triangle ABC, AB is equal to AC. D is the midpoint of AB, and CD is connected. There are two points O and E inside triangle ACD, with point O close to side CD and point E close to side AC. Connect OE.", "ocr_zh": "在等腰三角形 ABC 中，AB 等于 AC，D 为 AB 的中点，连接 CD。三角形 ACD 内部还有两点 O、E，其中 O 点靠近边 CD，E 点靠近边 AC，连接 OE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4378.jpg" }, { "index": 4379, "ocr_en": "In the plane, there are two circles S₁ and S₂. There is a point B on circle S₁, and this point is on the arc on the side farther from circle S₂. Draw a tangent to circle S₁ through point B, and there is a point A on the tangent. There is a point C on the side of circle S₂ close to circle S₁, and C is not on circle S₁. Connect AC, and segment AC intersects circle S₁ at two distinct points, and AC is tangent to circle S₂ at point C. Circle S₂ is tangent to circle S₁ at point D, and D and B are on opposite sides of line AC. Draw the auxiliary common tangent TT' of the two circles through point D, with T and A on the same side of BD. Let E be the midpoint of segment BD, and F be the midpoint of segment CD. There is a point K on the major arc BD, and connect BK, EK, DK, FK, CK with dashed lines.", "ocr_zh": "平面内有两个圆 S₁和 S₂，圆 S₁上有一点 B，且该点在离圆 S₂较远的一侧圆弧上。过点 B 作圆 S₁的切线，切线上有一点 A。圆 S₂上靠近圆 S₁的一侧有一点 C，且 C 不在圆 S₁上，连接 AC，线段 AC 与圆 S₁相交于两个不同的点，并且 AC 与圆 S₂相切于点 C。圆 S₂与圆 S₁相切于点 D，且 D 和 B 在直线 AC 的两侧。过 D 点作两圆的公切线辅助线 TT'，且满足 T 与 A 在 BD 的同侧。线段 BD 的中点为 E，线段 CD 的中点为 F。在优弧 BD 上有一点 K，用虚线连接 BK、EK、DK、FK、CK。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4379.jpg" }, { "index": 4380, "ocr_en": "The incircle I of triangle ABC is tangent to BC and CA at points D and E respectively, and K, L are the midpoints of AB and AC. Segment DE and segment KL intersect at point T. Connect BI with a dashed line. There is a point T' on segment LT, close to point T. Connect IT' with a dashed line, and IT' intersects segment DT at point T''. Connect AT', AI, ID, and ET'.", "ocr_zh": "三角形 ABC 的内切圆 I 与 BC、CA 分别相切于 D、E 两点，K、L 分别为 AB、AC 的中点。线段 DE 和 KL 相交于点 T，用虚线连接 BI，线段 LT 上靠近 T 点的位置有一点 T'，用虚线连接 IT'，IT' 与线段 DT 相交于点 T''，连接 AT'、AI、ID、ET'。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4380.jpg" }, { "index": 4381, "ocr_en": "The incircle I of triangle ABC is tangent to sides BC, CA, and AB at points F, E, and D respectively. Connect EF. The center of the incircle is I. Extend AI and BI to form straight lines, which intersect segment EF at points M and N respectively. Connect ND, MD, and ID. Let O be the midpoint of side AB.", "ocr_zh": "三角形 ABC 的内切圆 I 在边 BC、CA、AB 上的切点分别为 F、E、D，连接 EF，内切圆的圆心为 I，连接 AI 和 BI 并延长成直线，分别与线段 EF 相交于点 M、N。连接 ND、MD、ID，边 AB 的中点为 O。\n", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4381.jpg" }, { "index": 4382, "ocr_en": "In the acute triangle ABC, O is a point inside the triangle. Connect AO and extend it to intersect side BC at point D. There is a point E on side AB and a point F on side AC, such that points A, E, D, and F are concyclic; construct a circumcircle passing through these four points. Take another point E' on side AB and a point F' on side AC, so that points A, E', D, and F' are also concyclic; construct a circumcircle passing through points A, E', D, and F' as well.Draw auxiliary lines: connect DE, DE', DF, and DF'.", "ocr_zh": "在锐角三角形 ABC 中，O 是三角形内的一点，连接 AO 并延长，交边 BC 于点 D。边 AB 上有一点 E，边 AC 上有一点 F，且点 A、E、D、F 四点共圆，过这四点作一个外接圆。再在边 AB 上取一点 E'，边 AC 上取一点 F'，使得点 A、E'、D、F' 也满足四点共圆，过点 A、E'、D、F' 同样作一个外接圆。作辅助线：连接 DE、DE'、DF、DF'。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4382.jpg" }, { "index": 4383, "ocr_en": "In the plane, there is a circle with center O, denoted as circle O. AB is the diameter of circle O. There are two distinct points C and D on arc AB, both on the same side of AB, with C close to point A and D close to point B. Draw tangents to the circle through C and D, and these two tangents intersect at a point E outside the circle. Connect AD and BC, which intersect at point F. Connect EF and extend it to intersect segment OB at point M. Connect AC, OC, OE, and OD with dashed lines.", "ocr_zh": "平面内有一个圆，圆心为 O，记这个圆为圆 O。AB 是圆 O 的直径，圆弧 AB 上有两个不同的点 C 和 D，这两点在 AB 的同一侧，其中 C 靠近点 A，D 靠近点 B。过点 C、D 作圆的切线，两条切线相交于圆外一点 E。连接 AD 和 BC，它们相交于点 F。连接 EF 并延长，与线段 OB 相交于点 M。用虚线连接 AC、OC、OE、OD。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4383.jpg" }, { "index": 4384, "ocr_en": "In triangle ABC, angle BAC is a right angle. Point E lies on the arc BC (not containing point A) of the circumcircle of triangle ABC, and the length of AE is greater than that of EC. Connect EC and extend it to point F, connect AF and extend it, connect BF to intersect the circle at point D, and connect ED. Construct the circumcircle of triangle AEF as an auxiliary line, which intersects the dashed extension of AC at point O'. Connect O'E and O'F.", "ocr_zh": "在三角形 ABC 中，角 BAC 为直角，点 E 位于三角形 ABC 外接圆的弧 BC（不含点 A）上，且 AE 的长度大于 EC 的长度。连接 EC 并延长至点 F，连接 AF 并延长，连接 BF 交圆于点 D，连接 ED。作三角形 AEF 的外接圆作为辅助线，该圆与 AC 的虚线延长线相交于点 O'，连接 O'E、O'F。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4384.jpg" }, { "index": 4385, "ocr_en": "M and N are the midpoints of arcs BC and AC respectively on the circumcircle of acute triangle ABC (where angle A is greater than angle B). Connect MN. Draw PC through point C parallel to MN, intersecting the circle at point P. I is a point inside triangle ABC. Connect PI and extend it to intersect the circle at T. Connect NT, MT, PN, and PM. Connect AM, CI, NC, and CM with dashed lines.", "ocr_zh": "M、N 分别是锐角三角形 ABC（其中角 A 大于角 B）的外接圆上弧 BC、弧 AC 的中点，连接 MN。过点 C 作 PC 平行于 MN，交圆于点 P。I 是三角形 ABC 内的一点，连接 PI 并延长，交圆于 T，连接 NT、MT、PN、PM。用虚线连接 AM、CI、NC、CM。\n\n", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4385.jpg" }, { "index": 4386, "ocr_en": "M and N are the midpoints of arcs BC and AC respectively on the circumcircle of acute triangle ABC (where angle A is greater than angle B). Connect MN. Draw PC through point C parallel to MN, intersecting the circle at point P. I is a point inside triangle ABC. Connect PI and extend it to intersect the circle at T. Connect NT, MT, PN, PM, NC, and CM. There is a point Q on minor arc AT. Connect PQ, QN, MQ, AQ, and BQ with dashed lines. There is a point I₁ inside angle AQP and a point I₂ inside angle MTP; both points are inside triangle ABC and close to side AB, and it is satisfied that points Q, I₁, I₂, and T are concyclic. Construct a circumcircle passing through these four points as an auxiliary line. Connect I₁T and I₂T with dashed lines, where I₁ lies on segment NQ and I₂ lies on segment QM.", "ocr_zh": "M、N 分别是锐角三角形 ABC（其中角 A 大于角 B）的外接圆上弧 BC、弧 AC 的中点，连接 MN。过点 C 作 PC 平行于 MN，交圆于点 P。I 是三角形 ABC 内的一点，连接 PI 并延长，交圆于 T，连接 NT、MT、PN、PM、NC、CM。在劣弧 AT 上有一点 Q，用虚线连接 PQ、QN、MQ、AQ、BQ。角 AQP 内有一点 I₁，角 MTP 内有一点 I₂，这两点都在三角形 ABC 内且靠近边 AB，同时满足 Q、I₁、I₂、T 四点共圆，过这四点作一个外接圆作为辅助线。用虚线连接 I₁T、I₂T，其中 I₁在线段 NQ 上，I₂在线段 QM 上。\n", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4386.jpg" }, { "index": 4387, "ocr_en": "Construct the circumcircle of triangle ABC. There is a point I inside the triangle. Connect AI with a dashed line and extend it to intersect the circumcircle at point D. Then connect BD and CD with dashed lines.", "ocr_zh": "作三角形 ABC 的外接圆，三角形内部有一点 I，用虚线连接 AI 并延长，交外接圆于点 D，再用虚线连接 BD、CD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4387.jpg" }, { "index": 4388, "ocr_en": "There is a circle in the plane with center O, denoted as circle O. There are three points M, K, and C on circle O. Connect KC and MC. Points B and A are on MC and KC respectively, with B close to M and A close to K. Connect BA. L is a point on BA, connect OL and LC. The dashed extension of BA intersects the tangent of circle O at point C at point P. Connect PO with a dashed line, which intersects circle O at point S. Extend the dashed line PO to intersect the other side of the circle at point R. Connect MS and RK with dashed lines.", "ocr_zh": "平面内有一个圆，圆心为 O，记这个圆为圆 O。圆 O 上有三点 M、K、C，连接 KC、MC。点 B、A 分别在 MC、KC 上，且 B 靠近 M，A 靠近 K，连接 BA。L 是 BA 上的一点，连接 OL、LC。BA 的虚线延长线与圆 O 在点 C 处的切线相交于点 P，用虚线连接 PO，交圆 O 于点 S，虚线延长 PO 交圆的另一边于点 R，用虚线连接 MS 和 RK。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4388.jpg" }, { "index": 4389, "ocr_en": "There are six points on a circle, which are A, E, C, F, B, D in counterclockwise order starting from A. Connect CD, CB, FE, FA, ED, and BA. Among them, DE and AB intersect at point X, DC and AF intersect at point Z, BC and EF intersect at point Y. Connect XZ and YZ with dashed lines.", "ocr_zh": "一个圆上有六个点，从 A 开始按逆时针顺序依次为 A、E、C、F、B、D，连接 CD、CB、FE、FA、ED、BA，其中 DE 和 AB 相交于点 X，DC 和 AF 相交于点 Z，BC 和 EF 相交于点 Y，用虚线连接 XZ 和 YZ。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4389.jpg" }, { "index": 4390, "ocr_en": "Construct the incircle of triangle ABC, which is tangent to sides AB and BC at points F and D respectively. Connect AD, which intersects the incircle at another point H. Connect CF, which intersects the incircle at another point K. There is a point I inside the incircle. Connect HF, FD, DK, and KH.", "ocr_zh": "作三角形 ABC 的内切圆，该圆分别与边 AB、BC 相切于点 F、D，连接 AD，AD 交内切圆于另一点 H，连接 CF，CF 交内切圆于另一点 K。内切圆内有一点 I，连接 HF、FD、DK、KH。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4390.jpg" }, { "index": 4391, "ocr_en": "There are two circles in the plane, with a point O₁ inside one circle and a point O₂ inside the other. The circle containing O₁ is denoted as circle O₁, and the other as circle O₂ accordingly. Circle O₁ and circle O₂ are externally tangent at point T. A straight line is tangent to circle O₂ at point X and intersects circle O₁ at points A and B, with point B lying inside segment AX. Connect XT and extend it, and the extension intersects circle O₁ at another point S. C is a point on arc TS that does not contain points A and B. Draw a tangent from point C to circle O₂, with the point of tangency being Y, and segment CY does not intersect segment ST. Connect SC and extend it, connect XY, and the straight line SC intersects XY at point I. Connect BI, AI, AS, TY, TC with dashed lines, and connect AC and BC.", "ocr_zh": "平面内有两个圆，圆内各有一点 O₁和 O₂，将圆内有点 O₁的圆记为圆 O₁，同理将另一个圆记为圆 O₂。圆 O₁与圆 O₂外切于点 T，一条直线与圆 O₂相切于点 X，与圆 O₁相交于点 A、B，且点 B 在线段 AX 的内部。连接 XT 并延长，延长线与圆 O₁相交于另一点 S。C 是不包含点 A、B 的圆弧 TS 上的一点，过点 C 作圆 O₂的切线，切点为 Y，且线段 CY 与线段 ST 不相交。连接 SC 并延长，连接 XY，直线 SC 与 XY 相交于点 I。用虚线连接 BI、AI、AS、TY、TC，连接 AC 和 BC。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4391.jpg" }, { "index": 4392, "ocr_en": "There is a point B on circle S₁. Draw a tangent to circle S₁ through B, and A is a point on this tangent different from B. C is not a point on circle S₁; connect AC, and segment AC intersects circle S₁ at two distinct points. Circle S₂ is tangent to AC at point C and tangent to circle S₁ at point D, with D and B on opposite sides of line AC. Connect BD, CD, and BC.", "ocr_zh": "圆 S₁上有一点 B，过 B 作圆 S₁的切线，A 是这条切线上不同于 B 的一点。C 不是圆 S₁上的点，连接 AC，线段 AC 与圆 S₁相交于两个不同的点。圆 S₂与 AC 相切于点 C，与圆 S₁相切于点 D，且 D 和 B 在直线 AC 的两侧，连接 BD、CD、BC。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4392.jpg" }, { "index": 4393, "ocr_en": "Three circles share a common chord AB, and the minor arc AB of each circle lies on the same side of chord AB. Draw a line passing through point A, and the intersection points of this line with the three circles are X, Y, Z in sequence.", "ocr_zh": "三个圆有一条公共弦 AB，且每个圆的劣弧 AB 都在弦 AB 的同一侧。作一条过点 A 的直线，该直线与三个圆的交点按顺序依次为 X、Y、Z。\n", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4393.jpg" }, { "index": 4394, "ocr_en": "In the acute triangle ABC, the length of AB is greater than that of AC, and M is the midpoint of BC. Take a point P on the extension of BC, connect PA and extend it. Draw a perpendicular line to BC through point M, which intersects the extension of PA at point F. Connect FM (that is, FM is perpendicular to BC with the foot of the perpendicular at M). Draw MK through point M perpendicular to segment AF, with the foot of the perpendicular at point K.", "ocr_zh": "在锐角三角形 ABC 中，AB 的长度大于 AC 的长度，M 是 BC 的中点。在 BC 的延长线上取一点 P，连接 PA 并延长。过点 M 作 BC 的垂线，这条垂线与 PA 的延长线相交于点 F，连接 FM（即 FM 垂直于 BC，垂足为 M）。过点 M 作 MK 垂直于线段 AF，垂足为点 K。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4394.jpg" }, { "index": 4395, "ocr_en": "In triangle ABC, the lengths of AB and AC are equal (i.e., triangle ABC is an isosceles triangle), and D is the midpoint of BC. E is a point outside triangle ABC, close to side AB. Connect CE, DE, and BE, where CE is perpendicular to AB. Construct the circle determined by points A, B, and D. Take M as the midpoint of segment BE, draw a perpendicular line to segment BE through point M, and this perpendicular line intersects the minor arc AD at point F (with M as the foot of the perpendicular). Connect DF.", "ocr_zh": "在三角形 ABC 中，AB 和 AC 的长度相等（即三角形 ABC 为等腰三角形），D 是 BC 的中点。E 是三角形 ABC 外部一点，且靠近边 AB，连接 CE、DE、BE，其中 CE 与 AB 垂直。作由 A、B、D 三点确定的圆，取线段 BE 的中点 M，过点 M 作线段 BE 的垂线，该垂线交劣弧 AD 于点 F（M 为垂足），连接 DF。\n", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4395.jpg" }, { "index": 4396, "ocr_en": "There are two circles in the plane, with a point O₁ inside one circle and a point O₂ inside the other. The circle containing O₁ is denoted as circle O₁, and the other as circle O₂ accordingly, and the two circles are separate (non-intersecting). Connect O₁O₂. There is a point M between segment O₁O₂ and outside both circles. Draw a perpendicular line AM to O₁O₂ through point M, with the foot of the perpendicular being point M. Let D be the midpoint of segment O₁O₂, connect AD with a dashed line, and connect AO₁ and AO₂. Note that point M lies between segment O₂D, and point D is also outside both circles.", "ocr_zh": "平面内有两个圆，圆内各有一点 O₁和 O₂，将圆内有点 O₁的圆记为圆 O₁，同理将另一个圆记为圆 O₂，两圆相离。连接 O₁O₂，在线段 O₁O₂之间且位于两圆外部有一点 M，过点 M 作 O₁O₂的垂线 AM，垂足就是点 M。线段 O₁O₂的中点为 D，用虚线连接 AD，连接 AO₁和 AO₂。注意，点 M 在线段 O₂D 之间，且点 D 也在两圆外部。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4396.jpg" }, { "index": 4397, "ocr_en": "Circle Γ is tangent to the circumcircle of triangle ABC (with angle BAC being obtuse) at point A, intersects side AB at point K, intersects side AC at point M, and intersects side BC. Draw a tangent to circle Γ through point C, with the point of tangency being L (L is close to side BC), connect KL, and intersect side BC at point T.Draw auxiliary lines: draw the common tangent ED of the two circles through point A (D is close to B, E is close to C), and connect KM, AT, AL.", "ocr_zh": "圆 Γ 与三角形 ABC（其中角 BAC 为钝角）的外接圆相切于点 A，与边 AB 交于点 K，与边 AC 交于点 M，且与边 BC 相交。过点 C 作圆 Γ 的切线，切点为 L（L 靠近 BC 边），连接 KL，交边 BC 于点 T。作辅助线：过点 A 作两圆的公切线 ED（D 靠近 B，E 靠近 C），连接 KM、AT、AL。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4397.jpg" }, { "index": 4398, "ocr_en": "In triangle ABC, the length of AB is greater than that of BC. Construct the circumcircle of triangle ABC. Draw the tangent to the circumcircle through point B, which intersects the extension of AC at point P, and extend this tangent to point D. Connect AD and CD. There is another point E outside the circle, where E and C are on opposite sides of BD, and E is symmetric to C with respect to BD. Connect BE, DE, and CE.", "ocr_zh": "在三角形 ABC 中，AB 的长度大于 BC 的长度，作三角形 ABC 的外接圆。过点 B 作该外接圆的切线，交 AC 的延长线于点 P，并将这条切线延长至点 D，连接 AD、CD。圆外还有一点 E，点 E 与点 C 在 BD 的两侧，且点 E 与点 C 关于 BD 对称，连接 BE、DE、CE。\n", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4398.jpg" }, { "index": 4399, "ocr_en": "There are two circles in the plane, with a point O₁ inside one circle and a point O₂ inside the other. The circle containing O₁ is denoted as circle O₁, and the other as circle O₂ accordingly. Circle O₁ and circle O₂ intersect at points D and C. A line passing through D intersects circle O₁ at point A and circle O₂ at point B. There is a point P on arc AD; connect PD and AC, then extend both, and their extensions intersect at point M. There is a point Q on arc BD; connect QD and BC, then extend both, and their extensions intersect at point N. Connect MN. O is a point outside both circles and lies on the same side of lines AB and MN; connect OD.", "ocr_zh": "平面内有两个圆，圆内各有一点 O₁和 O₂，将圆内有点 O₁的圆记为圆 O₁，同理将另一个圆记为圆 O₂。圆 O₁与圆 O₂相交于点 D、C，过点 D 的一条直线交圆 O₁于点 A，交圆 O₂于点 B。弧 AD 上有一点 P，连接 PD 和 AC 并延长，两条延长线相交于点 M；弧 BD 上有一点 Q，连接 QD 和 BC 并延长，两条延长线相交于点 N。连接 MN，O 是两圆外的一点，且位于直线 AB 和 MN 的同侧，连接 OD。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4399.jpg" }, { "index": 4400, "ocr_en": "In the plane, there are two circles of different sizes that are internally tangent at point A. Draw two rays starting from point A, which intersect the two circles at points B₁, C₁ and B₂, C₂ respectively. Point B₁ lies on segment AB₂, and point C₁ lies on segment AC₂. Connect B₁C₁ and B₂C₂.", "ocr_zh": "平面内有两个大小不同的圆，两圆内切于点 A。从点 A 出发作两条射线，这两条射线与两个圆分别交于点 B₁、C₁和 B₂、C₂，且点 B₁在线段 AB₂上，点 C₁在线段 AC₂上，连接 B₁C₁、B₂C₂。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4400.jpg" }, { "index": 4401, "ocr_en": "In the plane, there are two circles of different sizes that are internally tangent at point C. Take three distinct points B, A, D on the smaller circle, connect AB, BC, CD, and DA to form a cyclic quadrilateral ABCD, and connect AC. Extend CB with a dashed line to intersect the larger circle at a point E, and connect AE with a dashed line.", "ocr_zh": "平面内有两个大小不同的圆，两圆内切于点 C。在小圆上取三个不同的点 B、A、D，连接 AB、BC、CD、DA，从而形成圆内接四边形 ABCD，连接 AC。用虚线延长 CB，与大圆交于一点 E，用虚线连接 AE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4401.jpg" }, { "index": 4402, "ocr_en": "Construct a semicircle with diameter AB and center O. Take a point C on the semicircular arc near point B, connect AC. Draw CD through point C perpendicular to AB, with the foot of the perpendicular at point D, and D lies on segment OB. Construct circle O₁, which is tangent to arc BC, segment CD, and segment DB at points E, F, G respectively, with the center of the circle being O₁.Draw auxiliary lines: connect CE, AF, EF, BC, OE, O₁F, and construct the circumcircle of triangle CEF through points C, E, F.", "ocr_zh": "作直径为 AB 的半圆，圆心为 O。在半圆圆弧上靠近点 B 处取一点 C，连接 AC。过点 C 作 CD 垂直于 AB，垂足为点 D，且 D 在线段 OB 上。作圆 O₁，使其分别与弧 BC、线段 CD、线段 DB 相切于点 E、F、G，该圆的圆心为 O₁。作辅助线：连接 CE、AF、EF、BC、OE、O₁F，过 C、E、F 三点作三角形 CEF 的外接圆。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4402.jpg" }, { "index": 4403, "ocr_en": "Two circles O₁ and O₂ with unequal radii intersect at two points M and N. Construct circle O such that circles O₁ and O₂ are internally tangent to circle O at points S and T respectively, where points O, O₁, and O₂ are the centers of their respective circles.Draw auxiliary lines: connect OS, OT, ST, SM; draw the common tangent SP of circle O₁ and circle O, and the common tangent TP of circle O₂ and circle O, with the two common tangents intersecting at point P; connect OP, NP, MP, OM, MT.", "ocr_zh": "两个半径不相等的圆 O₁与圆 O₂相交于两点 M、N，作圆 O，使得圆 O₁和圆 O₂分别与圆 O 内切于点 S、T，点 O、O₁、O₂分别为各自圆的圆心。作辅助线：连接 OS、OT、ST、SM；作圆 O₁与圆 O 的公切线 SP，以及圆 O₂与圆 O 的公切线 TP，两条公切线相交于点 P；连接 OP、NP、MP、OM、MT。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4403.jpg" }, { "index": 4404, "ocr_en": "Construct triangle ABC, then construct the incircle I of triangle ABC with center I, which is tangent to sides AB, AC, and BC at points F, E, and D respectively. Take a point O inside circle I, where O is closer to side BC than I is, and connect IO. Connect DF and extend it to intersect the extension of CA at point P; connect DE and extend it to intersect the extension of BA at point Q. Take midpoints M and N on PE and QF respectively, with M on segment AP and N on segment AQ, then connect MN.", "ocr_zh": "作三角形 ABC，再作三角形 ABC 的内切圆 I，其圆心为点 I，该圆分别与边 AB、AC、BC 相切于点 F、E、D。在圆 I 内取一点 O，且 O 比 I 更靠近边 BC，连接 IO。连接 DF 并延长，使其与 CA 的延长线交于点 P；连接 DE 并延长，使其与 BA 的延长线交于点 Q。在 PE、QF 上各取中点 M、N，其中 M 在线段 AP 上，N 在线段 AQ 上，连接 MN。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4404.jpg" }, { "index": 4405, "ocr_en": "Construct a circle with center O, denoted as circle O. Take two points A and C on circle O, and a point B outside the circle. Connect these three points to form triangle ABC. Sides AB and BC intersect circle O at points K and N respectively; connect KN. Construct the circumcircle of triangle KBN with center O₂, denoted as circle O₂; construct the circumcircle of triangle ABC with center O₁, denoted as circle O₁. Circles O₁ and O₂ intersect at points B and M. Connect BM and extend it to intersect the extension of AC at point P; connect NP and MO.Draw auxiliary lines: connect BO, and construct a circle with BO as the diameter, with center O₃, denoted as circle O₃. Circle O₃ intersects circle O at points X and Y; connect XO₁, O₁Y, YP, and MN. (It is known that angles BKN, PMN, and NCA are equal in measure.)", "ocr_zh": "作一个圆，圆心为 O，记这个圆为圆 O。在圆 O 上取两点 A、C，在圆外取一点 B，将这三点两两相连，构成三角形 ABC。边 AB、BC 分别交圆 O 于点 K 和点 N，连接 KN。作三角形 KBN 的外接圆，其圆心为 O₂，记该圆为圆 O₂；作三角形 ABC 的外接圆，其圆心为 O₁，记该圆为圆 O₁。圆 O₁与圆 O₂相交于 B、M 两点，连接 BM 并延长，与 AC 的延长线相交于点 P，连接 NP、MO。作辅助线：连接 BO，并以 BO 为直径作圆，其圆心为 O₃，记该圆为圆 O₃。圆 O₃与圆 O 相交于点 X、Y，连接 XO₁、O₁Y、YP、MN。角 BKN、角 PMN 和角 NCA 大小相等。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4405.jpg" }, { "index": 4406, "ocr_en": "In the plane, there are two circles, with a point O₁ inside one circle and a point O₂ inside the other. The circle containing O₁ is denoted as circle O₁, and the other as circle O₂ accordingly, and the two circles are separate (non-intersecting). Draw an external common tangent to the two circles, which is tangent to circle O₁ and circle O₂ at points A and C respectively, with points O₁ and O₂ on the same side of line AC; draw an internal common tangent to the two circles, which is tangent to circle O₁ and circle O₂ at points B and D respectively, with points O₁ and O₂ on opposite sides of line BD. Extend DB to intersect AC at point E, and connect CD; connect AB and extend it to intersect CD at point K.Draw auxiliary lines: connect AO₁, EO₁, O₁O₂, BO₁, and EO₂.", "ocr_zh": "平面内有两个圆，圆内各有一点 O₁和 O₂，将圆内有点 O₁的圆记为圆 O₁，同理将另一个圆记为圆 O₂，且两圆相离。作两圆的一条外公切线，与圆 O₁、圆 O₂分别相切于点 A、C，且点 O₁、O₂在直线 AC 的同一侧；再作两圆的一条内公切线，与圆 O₁、圆 O₂分别相切于点 B、D，且点 O₁、O₂在直线 BD 的两侧。延长 DB，交 AC 于点 E，连接 CD；连接 AB 并延长，交 CD 于点 K。作辅助线：连接 AO₁、EO₁、O₁O₂、BO₁、EO₂。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4406.jpg" }, { "index": 4407, "ocr_en": "There are two circles in the plane, with a point O₁ inside one circle and a point O₂ inside the other. The circle containing O₁ is denoted as circle O₁, and the other as circle O₂ accordingly. Circle O₁ and circle O₂ intersect at points A and B; connect AB. P is a point on segment AB; draw chord MK of circle O₁ and chord LN of circle O₂ through point P.", "ocr_zh": "平面内有两个圆，圆内各有一点 O₁和 O₂，将圆内有点 O₁的圆记为圆 O₁，同理将另一个圆记为圆 O₂。圆 O₁与圆 O₂相交于点 A、B，连接 AB。P 是线段 AB 上的一点，过点 P 作圆 O₁的弦 MK 和圆 O₂的弦 LN。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4407.jpg" }, { "index": 4408, "ocr_en": "In the plane, there is a circle with center O. Construct a cyclic quadrilateral ABCD inscribed in the circle. Extend DC and AB to intersect at point E; extend AD and BC to intersect at point F, and connect EF. Draw a tangent to the circle through point E, which is tangent to the minor arc AB at point P; draw a tangent to the circle through point F, which is tangent to the minor arc AD at point Q.", "ocr_zh": "平面内有一个圆，圆心为点 O，在圆内作一个内接四边形 ABCD。延长 DC 和 AB，二者相交于点 E；延长 AD 和 BC，二者相交于点 F，连接 EF。过点 E 作该圆的切线，与劣弧 AB 相切于点 P；过点 F 作该圆的切线，与劣弧 AD 相切于点 Q。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4408.jpg" }, { "index": 4409, "ocr_en": "In the plane, there are three circles of different sizes, with a point O inside one circle, a point O₁ inside another, and a point O₂ inside the third. The circle containing O is denoted as circle O, the circle containing O₁ as circle O₁, and the third circle as circle O₂ accordingly. Draw two parallel lines l₁ and l₂; circle O is tangent to both parallel lines. Circle O₁ is tangent to line l₁ at point A and externally tangent to circle O at point C. Circle O₂ is tangent to line l₂ at point B, externally tangent to circle O at point D, and externally tangent to circle O₁ at point E. Connect CB and AD, and the two segments intersect at point Q.", "ocr_zh": "平面内有三个大小不同的圆，圆内各有一点 O、O₁和 O₂，将圆内有点 O 的圆记为圆 O，圆内有点 O₁的圆记为圆 O₁，同理将第三个圆记为圆 O₂。作两条平行直线 l₁、l₂，圆 O 与这两条平行线都相切；圆 O₁与直线 l₁相切于点 A，且与圆 O 外切于点 C；圆 O₂与直线 l₂相切于点 B，与圆 O 外切于点 D，且与圆 O₁外切于点 E。连接 CB、AD，两条线段相交于点 Q。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4409.jpg" }, { "index": 4410, "ocr_en": "In triangle ABC, take midpoints E and F of AB and AC respectively, and connect EF. Take a point M on segment AE and a point N on segment CF, then connect CM, BN, and MN. MN intersects EF at point P, and connect AP. BN intersects MC at point H. There is a point O inside triangle BHM, and O and H are on the same side of segment EF. Mark point O with the Chinese character \"外心\" (circumcenter), and connect HO.", "ocr_zh": "在三角形 ABC 中，取 AB、AC 的中点 E、F，连接 EF。在线段 AE、CF 上各取一点 M、N，连接 CM、BN、MN。MN 与 EF 相交于点 P，连接 AP。BN 与 MC 相交于点 H。在三角形 BHM 内部有一点 O，且 O 和 H 在线段 EF 的同侧，用汉字标记 O 点为 “外心”，连接 HO。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4410.jpg" }, { "index": 4411, "ocr_en": "Inside the convex quadrilateral ABCD, there is a circle tangent to AB and BC at points G and H respectively. Connect AC, and AC intersects this circle at points E and F. Draw another circle inside angle ADC, which intersects side AC and is tangent to the extensions of DA and DC at points J and K respectively.", "ocr_zh": "在凸四边形 ABCD 内部有一个圆，该圆与 AB、BC 分别相切于点 G、H。连接 AC，AC 与这个圆相交于 E、F 两点。在角 ADC 的内部再作一个圆，这个圆与 AC 边相交，且与 DA、DC 的延长线分别相切于点 J、K。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4411.jpg" }, { "index": 4412, "ocr_en": "Construct triangle ABC, with points D, E, and F lying on sides BC, AC, and AB respectively. Connect DE, EF, and DF. On segments BF, BD, and CE, there are points F₀, D₀, and E₀ respectively; connect F₀D₀, D₀E₀, and E₀F₀.Next, reflect triangle ABC over axis AB to obtain triangle ABC₁; then reflect triangle ABC₁ over axis BC₁ to obtain triangle A₁BC₁; reflect over axis A₁C₁ to obtain triangle A₁B₁C₁; then reflect triangle A₁B₁C₁ over axis A₁B₁ to obtain triangle A₁B₁C₂; finally, reflect triangle A₁B₁C₂ over axis B₁C₂ to obtain triangle A₂B₁C₂.Similarly, perform the same reflection process on triangle DEF inside triangle ABC. Let D₁ be the point on side BC₁ of triangle ABC₁ corresponding to point D; E₁ be the corresponding point on side A₁C₁ of triangle A₁BC₁; F₁ be the corresponding point on side A₁B₁ of triangle A₁B₁C₁; D' be the corresponding point on side B₁C₂ of triangle A₁B₁C₂; E' be the point on side A₂C₂ of triangle A₂B₁C₂ corresponding to point E; and E₀' be the point corresponding to E₀.Connect EE' and F₀E₀' with dashed lines. In the entire figure, the lengths of segments EF and E₁F₁ are both d; the lengths of segments DE, D₁E₁, and D'E' are all f; the lengths of segments DF, D₁F, and D'F are all e.Note: D₁, E₁, F₁, and D' are points set for convenience of description and are not labeled with letters in the actual figure.", "ocr_zh": "作三角形 ABC，点 D、E、F 分别在边 BC、AC、AB 上，连接 DE、EF、DF。在线段 BF、BD、CE 上分别取一点 F₀、D₀、E₀，连接 F₀D₀、D₀E₀、E₀F₀。接着，以 AB 为轴将三角形 ABC 反射得到三角形 ABC₁；再以 BC₁为轴将三角形 ABC₁反射得到三角形 A₁BC₁；再以 A₁C₁为反射轴反射得到三角形 A₁B₁C₁；然后以 A₁B₁为轴将三角形 A₁B₁C₁反射得到三角形 A₁B₁C₂；最后以 B₁C₂为轴将三角形 A₁B₁C₂反射得到三角形 A₂B₁C₂。同样地，对三角形 ABC 内的三角形 DEF 执行相同的反射过程。设三角形 ABC₁的边 BC₁上与点 D 对应的点为 D₁，三角形 A₁BC₁的 A₁C₁边上与对应点为 E₁，三角形 A₁B₁C₁的 A₁B₁上对应的点为 F₁，三角形 A₁B₁C₂的 B₁C₂边上对应的点为 D'，三角形 A₂B₁C₂的 A₂C₂边上与点 E 对应的点为 E'，与 E₀对应的点为 E₀'。用虚线连接 EE' 和 F₀E₀'。在整个图形中，线段 EF、E₁F₁的长度均为 d；线段 DE、D₁E₁、D'E' 的长度均为 f；线段 DF、D₁F、D'F 的长度均为 e。注意：D₁、E₁、F₁、D' 均为便于描述所设定的点，实际图形中并未用字母标注这些点。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4412.jpg" }, { "index": 4413, "ocr_en": "Construct hexagon AC₁BA₁CB₁, and connect AC, AB, A₁C₁, BC. There is a point A₁' outside the hexagon, where A₁' and C₁ are on the same side of line AB. Connect C₁A₁', AA₁', and BA₁' with dashed lines.", "ocr_zh": "作六边形 AC₁BA₁CB₁，连接 AC、AB、A₁C₁、BC。在六边形的外侧有一点 A₁'，A₁' 与 C₁在直线 AB 的同侧，用虚线连接 C₁A₁'、AA₁'、BA₁'。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4413.jpg" }, { "index": 4414, "ocr_en": "Construct parallelogram ABCD, with P being a point inside it. Connect P to each vertex of the parallelogram, where the measure of ∠PAB is equal to that of ∠PCB. There is a point P' outside the parallelogram, with P and P' on opposite sides of line CD. Connect PP' with a dashed line, and segment PP' intersects segment CD; connect DP' and CP' with dashed lines.Denote ∠BAP as ∠1, ∠PCB as ∠2, ∠ABP as ∠3, ∠ADP as ∠4, ∠CDP' as ∠5, ∠DCP' as ∠6, ∠DPP' as ∠7, and ∠CPP' as ∠8.", "ocr_zh": "作平行四边形 ABCD，P 为其内部一点，连接 P 与平行四边形的各个顶点，其中角PAB 与角PCB 大小相等。在平行四边形外部有一点 P'，P 和 P' 分别位于直线 CD 的两侧，用虚线连接 PP'，线段 PP' 与线段 CD 相交，再用虚线连接 DP' 和 CP'。记角BAP为角1，角PCB为角2，角ABP为角3，角ADP为角4，角CDP'为角5，角DCP'为角6，角DPP'为角7，角CPP'为角8。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4414.jpg" }, { "index": 4415, "ocr_en": "There are three non-collinear points A, B, and C in the plane. Connect AC and AB, where ∠CAB is an obtuse angle. Construct an equilateral triangle ABD with AB as a side, such that points C and D are on the same side of line AB. Let E be the symmetric point of D with respect to line AB; connect AE and BE, and it can be seen that triangle ABE is also an equilateral triangle. Construct an equilateral triangle CAF with AC as a side, where points F and D are on the same side of line AC; connect DF and CE. Construct an equilateral triangle DFG with DF as a side, with G inside triangle AFD, and connect CG with a dashed line. Construct an equilateral triangle ECH with CE as a side, where H and A are on opposite sides of line CE (while A and B are on the same side of line CE), and connect HG. Construct an equilateral triangle GHI with HG as a side, where A and E are inside this triangle, and C and I are on opposite sides of line HG; connect EI with a dashed line.All the triangles in the figure, namely ABD, BAE, CAF, DFG, ECH, and GHI, are positively oriented equilateral triangles.", "ocr_zh": "平面内有三个不共线的点 A、B、C，连接 AC 和 AB，其中角CAB 为钝角。以 AB 为边作一个等边三角形 ABD，且点 C 与点 D 在直线 AB 的同侧；E 为点 D 关于直线 AB 的对称点，连接 AE、BE，可知三角形 ABE 也是等边三角形。以 AC 为边作一个等边三角形 CAF，且点 F 与点 D 在直线 AC 的同侧，连接 DF 和 CE。以 DF 为边作一个等边三角形 DFG，且 G 在三角形 AFD 内部，用虚线连接 CG。以 CE 为边作一个等边三角形 ECH，且 H 与 A 在直线 CE 的两侧（A 和 B 在直线 CE 的同侧），连接 HG。以 HG 为边作一个等边三角形 GHI，且 A 和 E 在该三角形内部，C 和 I 在直线 HG 的两侧，用虚线连接 EI。图中的三角形 ABD、BAE、CAF、DFG、ECH、GHI 均为正定向等边三角形。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4415.jpg" }, { "index": 4416, "ocr_en": "Construct an isosceles triangle ABC with AC equal to AB. Construct the circumcircle of triangle ABC, with its center being O, denoted as circle O, and connect OA and OB with dashed lines. Take a point N on the minor arc AC, connect BN, and BN intersects side AC at point D, with O and A on the same side of line BN. Take a point E on side AB, connect EO and extend it; the extension intersects segment BD, and EO is perpendicular to BD. Draw a perpendicular line to EO through point E, which intersects segment AD at point F, with E being the foot of the perpendicular. Extend AO with a dashed line, and the extension intersects segment BD at point D', where D' and D are on opposite sides of line EO; connect ED' with a dashed line.", "ocr_zh": "作等腰三角形 ABC，其中 AC 等于 AB。作三角形 ABC 的外接圆，其圆心为 O，记该圆为圆 O，用虚线连接 OA、OB。在劣弧 AC 上取一点 N，连接 BN，BN 与 AC 边相交于点 D，且 O 和 A 在直线 BN 的同侧。在 AB 边上取一点 E，连接 EO 并延长，延长线与线段 BD 相交，且 EO 与 BD 垂直。过点 E 作 EO 的垂线，该垂线与线段 AD 相交于点 F，垂足为 E。用虚线延长 AO，延长线与线段 BD 相交于点 D'，且 D' 和 D 在直线 EO 的两侧，用虚线连接 ED'。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4416.jpg" }, { "index": 4417, "ocr_en": "Construct triangle ABC and its circumcircle. Draw a circle passing through points A and C, with its center being O, thus denoted as circle O. Circle O intersects sides AB and BC at points K and N respectively; connect KN. Construct the circumcircle of triangle KBN, which intersects the circumcircle of triangle ABC at point M; connect BM and OM. Draw points C' and K', which are the symmetric points of C and K with respect to OM.Draw auxiliary lines: connect CC', KK', CM, C'K, KM, and CK'.", "ocr_zh": "作三角形 ABC 及其外接圆。作一个经过点 A、C 的圆，其圆心为 O，因此记该圆为圆 O。圆 O 与边 AB、BC 分别交于点 K、N，连接 KN。作三角形 KBN 的外接圆，该圆与三角形 ABC 的外接圆相交于点 M，连接 BM、OM。作点 C、K 关于 OM 的对称点 C'、K'。作辅助线：连接 CC'、KK'、CM、C'K、KM、CK'。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4417.jpg" }, { "index": 4418, "ocr_en": "In the plane, there are two circles, with a point O₁ inside one circle and a point O₂ inside the other. The circle containing O₁ is denoted as circle O₁, and the other as circle O₂ accordingly. Circle O₁ and circle O₂ intersect at points A and B. Draw an arbitrary secant through point A, which intersects circle O₁ and circle O₂ at points P and Q respectively. Draw tangents to the two circles through points P and Q respectively, and the two tangents intersect at point R. Construct another circle passing through points O₁, O₂, and B; connect RB, which intersects this circle at point S. Connect and extend PO₁ and QO₂, and the two extensions intersect at point C.Draw auxiliary lines: connect PB, BO₁, RC, BO₂, BQ, CS, and SO₂.", "ocr_zh": "平面内有两个圆，圆内各有一点 O₁和 O₂，将圆内有点 O₁的圆记为圆 O₁，同理将另一个圆记为圆 O₂。圆 O₁与圆 O₂相交于点 A、B，过点 A 作任意一条割线，与圆 O₁、圆 O₂分别交于点 P、Q。分别过点 P、Q 作两圆的切线，两条切线相交于点 R。过点 O₁、O₂、B 三点再作一个圆，连接 RB，RB 交这个圆于点 S。连接并延长 PO₁、QO₂，两条延长线相交于点 C。作辅助线：连接 PB、BO₁、RC、BO₂、BQ、CS、SO₂", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4418.jpg" }, { "index": 4419, "ocr_en": "In the plane, there is a convex quadrilateral ABCD; connect AC and BD. Construct a circumcircle passing through points A, B, and C, with its center being O, and AB is the diameter of this circle, so O lies on segment AB. Segment BD intersects the minor arc AC of circle O at point P. Take a point E on the semicircular arc AB, where E and C are on opposite sides of line AB; connect AE and CE.", "ocr_zh": "平面内有一个凸四边形 ABCD，连接 AC 和 BD。过 A、B、C 三点作一个外接圆，其圆心为 O，且 AB 为该圆的直径，因此 O 在线段 AB 上。线段 BD 与圆 O 的劣弧 AC 相交于点 P。在半圆弧 AB 上取一点 E，且 E 和 C 在直线 AB 的两侧，连接 AE 和 CE。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4419.jpg" }, { "index": 4420, "ocr_en": "Construct an acute triangle ABC. Draw the perpendicular from point A to side BC, take a point F on this perpendicular line, connect CF and BF, then extend them respectively: let the extension of CF intersect side AB at point E, and the extension of BF intersect side AC at point D.", "ocr_zh": "作锐角三角形 ABC，过点 A 作 BC 边的垂线，在这条垂线上取一点 F，连接 CF、BF 并分别延长，使 CF 的延长线交 AB 边于点 E，BF 的延长线交 AC 边于点 D。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4420.jpg" }, { "index": 4421, "ocr_en": "Construct an acute triangle ABC. Take a point O inside the triangle, and connect O to each vertex of the triangle (i.e., connect OA, OB, OC) such that the measures of ∠BAO, ∠ACO, and ∠CBO are equal, each being α.", "ocr_zh": "作锐角三角形 ABC，在三角形内部取一点 O，连接 O 与三角形的各个顶点（即连接 OA、OB、OC），使得角BAO、角ACO 和角CBO 的大小相等，且均为 α。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4421.jpg" }, { "index": 4422, "ocr_en": ",Construct triangle A₁B₁C₁ and draw its incircle with center O. The incircle is tangent to sides B₁C₁, A₁C₁, and A₁B₁ at points A, B, and C respectively; connect AC, CB, and AB. Point O lies inside triangle ABC, which is an acute triangle. Connect CO, BO, and AO, then extend them respectively: the extension of CO intersects side AB at point F, the extension of BO intersects side CA at point E, and the extension of AO intersects side CB at point D.", "ocr_zh": "作三角形 A₁B₁C₁，作它的内切圆，圆心为点 O。此内切圆与边 B₁C₁、A₁C₁、A₁B₁分别相切于点 A、B、C，连接 AC、CB、AB。点 O 也在三角形 ABC 的内部，且三角形 ABC 是锐角三角形。连接 CO、BO、AO 并分别延长，使 CO 的延长线交边 AB 于点 F，BO 的延长线交边 CA 于点 E，AO 的延长线交边 CB 于点 D。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4422.jpg" }, { "index": 4423, "ocr_en": "There is a triangle ABC in the plane. Take three points E, F, and D inside the triangle, where D is close to side BC, E is close to side AC, and F is close to side AB. Connect EF, FD, and DE, and triangle EFD is an equilateral triangle. Then connect AE, AF, BF, BD, CD, and CE respectively.", "ocr_zh": "平面内有三角形 ABC，在三角形内部取三点 E、F、D，其中 D 靠近边 BC，E 靠近边 AC，F 靠近边 AB。连接 EF、FD、DE，且三角形 EFD 是正三角形。再分别连接 AE、AF、BF、BD、CD、CE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4423.jpg" }, { "index": 4424, "ocr_en": "Construct an acute triangle ABC. Take a point P inside it, connect AP, BP, and CP, then extend them respectively: let the extension of AP intersect side BC at point D, the extension of BP intersect side CA at point E, and the extension of CP intersect side AB at point F. Then connect DE, EF, and DF.", "ocr_zh": "作锐角三角形 ABC，在其内部取一点 P，连接 AP、BP、CP 并分别延长，使 AP 的延长线交边 BC 于点 D，BP 的延长线交边 CA 于点 E，CP 的延长线交边 AB 于点 F，再连接 DE、EF、DF。\n", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4424.jpg" }, { "index": 4425, "ocr_en": "There are two points I and O inside triangle ABC. Connect IB and OI, and connect BO with a dashed line. Points O and A are on opposite sides of line BI, and O is close to side BC. Denote the length of BC as a, the length of AC as b, and the length of AB as c.", "ocr_zh": "在三角形 ABC 内部有两点 I 和 O，连接 IB 和 OI，用虚线连接 BO。点 O 与点 A 分别位于直线 BI 的两侧，且 O 靠近边 BC。记 BC 的长度为 a，AC 的长度为 b，AB 的长度为 c。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4425.jpg" }, { "index": 4426, "ocr_en": "Construct a circle with AB as its diameter. Draw the tangent line CD of the circle through point A, where the length of AC is less than that of AD. Connect CB and BD, which intersect the circle at points P and Q respectively. There is a point R on one side of line CD, with R and B on opposite sides of line CD. Connect PR, QR, and BR; among them, segment BR intersects the minor arc AQ of the circle at point T and intersects segment AD at point M.Draw auxiliary lines: connect PA, PM, QA, and QM.", "ocr_zh": "作直径为 AB 的圆，过点 A 作该圆的切线 CD，且 AC 的长度小于 AD 的长度。连接 CB、BD，它们分别与圆相交于点 P、Q。在直线 CD 的一侧有一点 R，且 R 与 B 在直线 CD 的两侧，连接 PR、QR、BR，其中线段 BR 与圆的劣弧 AQ 相交于点 T，与线段 AD 相交于点 M。作辅助线：连接 PA、PM、QA、QM。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4426.jpg" }, { "index": 4427, "ocr_en": "Construct an acute triangle ABC, and draw its circumcircle passing through the three vertices of the triangle. Draw CD perpendicular to AB through point C, with D being the foot of the perpendicular. Let M be the midpoint of AB, and M lies on segment AD. Draw a line through M, which intersects AC at point K (where K is close to A) and the extension of CB at point L. There is a point S inside triangle ACD; connect MS and DS.Draw auxiliary lines: connect CS and extend it to intersect the circumcircle at point E, then connect EM.", "ocr_zh": "作锐角三角形 ABC，并过三角形的三个顶点作其外接圆。过点 C 作 CD 垂直于 AB，垂足为点 D。M 为 AB 的中点，且 M 在线段 AD 上。过 M 作一条直线，交 AC 于点 K（点 K 靠近点 A），交 CB 的延长线于点 L。在三角形 ACD 内部有一点 S，连接 MS、DS。作辅助线：连接 CS 并延长，使其交外接圆于点 E，连接 EM。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4427.jpg" }, { "index": 4428, "ocr_en": "Construct an acute triangle ABC and its incircle with center I. The incircle is tangent to sides AB and AC at points K and L respectively; connect KL. Take a point D on segment LC, connect BD, and connect IC with a dashed line. Take a point J on IC, where J is inside triangle BCD and outside the incircle; connect AJ.Draw auxiliary lines: connect KI, LI, AI, BI, and BJ; segment AI intersects KL at point P.", "ocr_zh": "作锐角三角形 ABC 及其内切圆，圆心为 I。该内切圆与边 AB、AC 分别相切于点 K、L，连接 KL。在线段 LC 上取一点 D，连接 BD，用虚线连接 IC。在 IC 上取一点 J，点 J 既在三角形 BCD 内部，又位于内切圆外部，连接 AJ。作辅助线：连接 KI、LI、AI、BI、BJ，线段 AI 与 KL 相交于点 P。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4428.jpg" }, { "index": 4429, "ocr_en": "Construct a convex quadrilateral ABFD. Extend BF and DF such that the extension of BF intersects the extension of AD at point E, and the extension of DF intersects the extension of AB at point C.Draw an auxiliary line: connect AF.Denote ∠CAF as α, ∠FAD as β, ∠BFA as γ, and ∠DFA as θ.", "ocr_zh": "作凸四边形 ABFD，延长 BF、DF，使 BF 的延长线与 AD 的延长线相交于点 E，DF 的延长线与 AB 的延长线相交于点 C。做辅助线：连接AF。记角CAF为α，角FAD为β，角BFA为γ，角DFA为θ。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4429.jpg" }, { "index": 4430, "ocr_en": "Construct an acute triangle ABC and its circumcircle with center I, where the lengths of sides AC and BC are not equal. Draw a dashed line AH through point A, perpendicular to BC; H is not the foot of the perpendicular but a point on this perpendicular line. Draw a solid line CH and extend it to intersect the circumcircle at point D. Take a point L on the minor arc AD, and connect CL and LD. Take a point on CL (here, this point is assumed to be E for the convenience of describing its position, though it is not actually labeled with a letter in the figure), and connect ED and HE. HE is perpendicular to CL, with the foot of the perpendicular being this point; ED is perpendicular to LD, with the foot of the perpendicular being D. Connect EA with a dashed line.", "ocr_zh": "作锐角三角形 ABC 及其外接圆，圆心为 I，且边 AC 与 BC 的长度不相等。过点 A 作一条虚线 AH 垂直于 BC，H 不是垂足，而是这条垂线上的一点。作实线 CH 并延长，使其交外接圆于点 D。在劣弧 AD 上取一点 L，连接 CL、LD。在 CL 上取一个点（此处假设该点为 E，仅为方便描述其位置，图中实际并未用字母标记该点），连接 ED、HE，其中 HE 与 CL 垂直，垂足为该点；ED 与 LD 垂直，垂足为 D。用虚线连接 EA。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4430.jpg" }, { "index": 4431, "ocr_en": "Construct an acute triangle ABC and its circumcircle. Take two points E and F on side BC, where E is close to point B and F is close to point C. Connect AF, connect AE and extend it so that the extension intersects the circumcircle at point D. Take points M and N on sides AB and AC respectively, connect FM, FN, DM, DN, and connect BD and CD with dashed lines.", "ocr_zh": "作锐角三角形 ABC 及其外接圆。在 BC 边上取两点 E、F，其中 E 靠近点 B，F 靠近点 C。连接 AF，连接 AE 并延长，使延长线交外接圆于点 D。在边 AB、AC 上分别取点 M、N，连接 FM、FN、DM、DN，用虚线连接 BD、CD。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4431.jpg" }, { "index": 4432, "ocr_en": "Construct an acute triangle ACF. Take a point B on side AC near point C, and a point D on side CF. Connect BD and extend it to intersect the extension of AF at point E. Construct the circumcircle of triangle CBD and the circumcircle of triangle DEF; these two circles intersect at points D and M.Draw auxiliary lines: Take a point P on segment CB and a point S on segment EF. Connect PM, SM, and PS, where segment PS intersects segment CF and BE at points Q and R respectively. Connect MQ and MR.", "ocr_zh": "作锐角三角形 ACF，在边 AC 上靠近点 C 的位置取一点 B，在边 CF 上取一点 D。连接 BD 并延长，与 AF 的延长线相交于点 E。作三角形 CBD 的外接圆和三角形 DEF 的外接圆，这两个圆相交于点 D 和点 M。作辅助线：在线段 CB 上取一点 P，在线段 EF 上取一点 S，连接 PM、SM、PS，其中线段 PS 分别与线段 CF 和 BE 相交于点 Q、R，连接 MQ、MR。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4432.jpg" }, { "index": 4433, "ocr_en": "Construct triangle ACE. Take points B and F on sides AC and AE respectively, and connect BF. Connect BE and CF, and these two line segments intersect at point D; connect AD.", "ocr_zh": "作三角形 ACE，在边 AC、AE 上分别取点 B、F，连接 BF。连接 BE、CF，这两条线段相交于点 D，连接 AD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4433.jpg" }, { "index": 4434, "ocr_en": "Construct triangle AQR. Take a point C inside the triangle, connect RC, QC, and AC, then extend them respectively: let the extension of RC intersect side AQ at point B, the extension of QC intersect side AR at point D, and the extension of AC intersect side QR at point X. Connect BD and extend it to intersect the extension of QR at point Y, and segment BD intersects segment AC at point P.", "ocr_zh": "作三角形 AQR，在三角形内部取一点 C，连接 RC、QC、AC 并分别延长，使 RC 的延长线交边 AQ 于点 B，QC 的延长线交边 AR 于点 D，AC 的延长线交边 QR 于点 X。连接 BD 并延长，与 QR 的延长线相交于点 Y，且线段 BD 与线段 AC 相交于点 P。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4434.jpg" }, { "index": 4435, "ocr_en": "Construct triangle ACE. Take points B and F on sides AC and AE respectively, and connect BF. Connect CF and EB, and these two line segments intersect at point D. Take points H and G on segments DE and CD respectively; connect GH, AG, and AH, where segments AG and AH intersect segment BF at points P and Q respectively. Connect CH and HF with dashed lines.", "ocr_zh": "作三角形 ACE，在边 AC、AE 上分别取一点 B、F，连接 BF。连接 CF、EB，这两条线段相交于点 D。在线段 DE、CD 上分别取点 H、G，连接 GH、AG 和 AH，其中线段 AG 和 AH 分别与线段 BF 相交于点 P、Q，用虚线连接 CH、HF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4435.jpg" }, { "index": 4436, "ocr_en": "There is a line segment AD in the plane. On segment AD, there are points C and B in sequence from A to D, with C close to A and B close to D. Take a point P outside line AD, connect PA, PC, PB, and PD, and denote ∠APC as α and ∠CPB as β.", "ocr_zh": "平面内有一线段 AD，在线段 AD 上从 A 到 D 依次有 C、B 两点，其中 C 靠近 A，B 靠近 D。在直线 AD 外取一点 P，连接 PA、PC、PB、PD，记角APC 为 α，角CPB 为 β。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4436.jpg" }, { "index": 4437, "ocr_en": "Construct a straight line L, and take points A, B, C, D in sequence on line L. Take a point O outside line L, connect OA, OB, OC, and OD, where ∠AOB is denoted as α, ∠BOC as β, and ∠COD as γ.", "ocr_zh": "作一条直线 L，在直线 L 上按顺序取点 A、B、C、D。在直线 L 外取一点 O，连接 OA、OB、OC、OD，其中记角AOB 为 α，角BOC 为 β，角COD 为 γ。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4437.jpg" }, { "index": 4438, "ocr_en": "Draw four horizontal parallel lines, and take points on each line from left to right in sequence. For ease of description, these four lines are referred to as Line 1, Line 2, Line 3, and Line 4 from top to bottom (no numbers are actually marked): On Line 1, the points from left to right are B, C, A, O, D in sequence; On Line 2, the points from left to right are A, C, B, O, D in sequence; On Line 3, the points from left to right are C, O, A, D, B in sequence; On Line 4, the points from left to right are C, O, B, D, A in sequence. It should be noted that in the four lines, point O is the midpoint of segment CD, and all points lie on their respective corresponding lines.", "ocr_zh": "作四条水平平行直线，在每条直线上从左到右依次取点。为便于描述，将这四条直线从上到下依次称为直线 1、直线 2、直线 3、直线 4（这里标数字只是便于描述，实际未标注数字）：直线 1 上从左至右依次为点 B、C、A、O、D；直线 2 上从左至右依次为点 A、C、B、O、D；直线 3 上从左至右依次为点 C、O、A、D、B；直线 4 上从左至右依次为点 C、O、B、D、A。需注意：四条直线中，点 O 均为线段 CD 的中点，且所有点均在各自对应的直线上。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4438.jpg" }, { "index": 4439, "ocr_en": "There is a triangle OA'D' in the plane. Denote the line where side A'D' lies as l₂. There is a point A on side OA' near point A', and a point D on side OD' near point D'. Connect AD, denote the line where AD lies as l₁, and l₁ is not parallel to l₂. There are points B' and C' in sequence from A' to D' on side A'D'. Connect OB' and OC', which intersect segment AD at points B and C respectively.", "ocr_zh": "平面内存在一个三角形 OA'D'，记边 A'D' 所在的直线为 l₂，在边 OA' 上靠近 A' 点处存在一点 A，在边 OD' 上靠近 D' 处存在一点 D，连接 AD，记 AD 所在的直线为 l₁，且 l₁与 l₂不平行，在边 A'D' 上从 A' 到 D' 依次存在点 B' 和 C'，连接 OB' 和 OC'，分别与线段 AD 相交于点 B 和 C。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4439.jpg" }, { "index": 4440, "ocr_en": "Construct triangle AOB, draw a line through point O parallel to AB, let C be the midpoint of side AB, and connect OC.", "ocr_zh": "作三角形 AOB，过点 O 作一条直线平行于 AB，边 AB 的中点为 C，连接 OC。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4440.jpg" }, { "index": 4441, "ocr_en": "There is a point in the plane, from which four rays are drawn outward, denoted as l₁, l₂, l₃, l₄ in counterclockwise order, and the angle between rays l₁ and l₄ is less than 180 degrees.", "ocr_zh": "平面内存在一点，从该点向外引出四条射线，按照逆时针的顺序分别记为 l₁、l₂、l₃、l₄，且射线 l₁与 l₄所夹的角小于 180 度。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4441.jpg" }, { "index": 4442, "ocr_en": "There is a circle in the plane with center O, and a point P outside the circle. Draw two tangents from point P to the circle, which are tangent to the circle at points A and B respectively, and connect AB. M is the midpoint of segment AP, N is the midpoint of segment AB. Connect MN and extend it to intersect the circle at point C, with N between M and C. Connect PC, which intersects the circle at point D and intersects segment NB at point E. Connect ND and extend it to intersect segment PB at point Q.", "ocr_zh": "平面内有一个圆，圆心是点 O，圆外有一点 P。过点 P 作圆的两条切线，分别与圆相切于点 A 和点 B，连接 AB。M 是线段 AP 的中点，N 是线段 AB 的中点，连接 MN 并延长，交圆于点 C，且 N 在 M 与 C 之间。连接 PC，与圆相交于点 D，与线段 NB 相交于点 E，连接 ND 并延长，交线段 PB 于点 Q。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4442.jpg" }, { "index": 4443, "ocr_en": "AB is a chord of a circle, with O being a point on AB. Point C is on the circle. Connect CO and extend it to intersect the circle at D. Point E is on the circle. Connect EO and extend it to intersect the circle at F. Connect CF and let it intersect AB at G. Connect DE and let it intersect AB at H. Draw auxiliary lines AD, AF, BF, and BD.", "ocr_zh": "AB为圆上的弦，O为AB上一点，C为圆上的点，连接CO并延长交圆于D，E为圆上的点，连接EO并延长交圆于F，连接CF与AB交于G，连接DE与AB交于H，做辅助线AD、AF、BF、BD。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4443.jpg" }, { "index": 4444, "ocr_en": "In triangle BCP, construct the circumcircle O of triangle BCP. Draw a tangent from point P to circle O, which intersects the extension of CB at point A. Let H be a point on BP. Connect AH and let it intersect CP at F. Connect CH and let it intersect AP at D. Connect BD and let it intersect AH at G. Connect BF, where BD is perpendicular to BF. Extend BF to intersect circle O at E, and connect PE.", "ocr_zh": "三角形BCP，做三角形BCP的外接圆O，过P做圆O的切线与CB的延长线交于A，H为BP一点，连接AH与CP交于F，连接CH与AP交于D，连接BD与AH交于G，连接BF，此时BD与BF垂直，延长BF与圆O交于E，连接PE。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4444.jpg" }, { "index": 4445, "ocr_en": "In triangle ABC, draw a perpendicular from point A to BC with the foot at D, a perpendicular from point B to AC with the foot at E, and a perpendicular from point C to AB with the foot at F. The lines AD, BE, and CF intersect at H. Extend FE to meet the extension of BC at point P. Draw a line through D parallel to FP, intersecting AB at R and the extension of AC at Q. Construct the auxiliary circumarc of the triangle formed by points R, Q, and P.", "ocr_zh": "三角形ABC，过A点做BC垂线垂足为D，过B点做AC垂线垂足为E，过C点做AB垂线垂足为F，且AD、BE、CF交于H，连接FE并延长与BC的延长线交于P，过D做FP的平行线与AB交于R、与AC延长线交于Q，做以R、Q、P三点所构成的三角形的辅助外接圆弧。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4445.jpg" }, { "index": 4446, "ocr_en": "In triangle AEF, let O be a point on EF. Connect AO. Let B be a point on AE, and connect OB and BF. Let D be a point on AF, and connect BD, ED, and OD. Let BF and ED intersect at C. Connect AC and let it intersect BD at P. Connect OP. Draw auxiliary extensions of DB and FE to intersect at R. Draw an auxiliary extension of AC to intersect EF at Q.", "ocr_zh": "三角形AEF，O为EF上的点，连接AO，B为AE上的点，连接OB、BF，D为AF的点，连接BD、ED、OD，BF与ED交于C，连接AC与BD交于P，连接OP，做DB和FE的辅助延长线交于R，做AC的辅助延长线与EF交于Q。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4446.jpg" }, { "index": 4447, "ocr_en": "In triangle ABC, construct the auxiliary incircle of triangle ABC, which is tangent to AB, AC, and BC at points F, E, and D respectively. Draw auxiliary lines AD, BE, and CF. Connect DF and extend it to intersect the extension of CA at point P. Let M be a point on AP.", "ocr_zh": "三角形ABC，做三角形ABC的辅助内切圆，与AB、AC、BC的切点分别为F、E、D，做辅助线AD、BE、CF，连接DF并延长与CA的延长线交于P，M为AP上的点。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4447.jpg" }, { "index": 4448, "ocr_en": "In triangle ACE, let N be a point on CE, and F be a point on AE. Let B be a point on AC. Connect AN and CF, with their intersection at H. Connect BE, which intersects CF at D and AN at G. Connect BF, which intersects AN at M. Draw auxiliary extensions of BF and CE to intersect at I. Draw auxiliary line ID and extend it to intersect AN at J.", "ocr_zh": "三角形ACE，N为CE上的点，F为AE上的点，B为AC上的点，连接AN、CF，AN与CF交于H，连接BE，BE与CF交于D、与AN交于G，连接BF与AN交于M，做BF与CE的辅助延长线交于I，做辅助线ID并延长与AN交于J。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4448.jpg" }, { "index": 4449, "ocr_en": "In triangle AQR, point B is on AQ, point D is on AR, and point G is on QR. Connect BD, and connect BR and QD which intersect at point C. Point P is on BD, point M is on AB. Connect MP and extend it to intersect QD at N and QR at G. Connect AP and draw an auxiliary extended line of AP to intersect QR at S. Draw the auxiliary line PQ.", "ocr_zh": "三角形AQR，B为AQ上的点，D为AR上的点，G为QR上的点，连接BD，连接BR、QD并交于C，P为BD上的点，M为AB上的点，连接MP并延长与QD交于N、与QR交于G，连接AP并做AP的辅助延长线交QR于S，做辅助线PQ。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4449.jpg" }, { "index": 4450, "ocr_en": "In triangle MBC, point A is on BM, point D is on CM, and point L is on BC. Connect ML and AC, which intersect at point H'. Connect AD, which intersects ML at point H. Connect BD, which intersects ML at point L'.", "ocr_zh": "三角形MBC，A为BM上的点，D为CM上的点，L为BC上的点，连接ML、AC，ML与AC并交于H'，连接AD，AD与ML交于H，连接BD，BD与ML交于L'。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4450.jpg" }, { "index": 4451, "ocr_en": "Points E and F lie on line l, with A being the midpoint of EF. Draw a semicircle A with A as the center and EF as the diameter. Let D be a point on semicircle A; connect DF, ED, and AD. Draw a circle tangent to line l, with A as the point of tangency. Extend AD to intersect this circle at B. Let C be a point on semicircle A; connect AB, AC, and BC. Draw the auxiliary extension of CB to point P, and draw the auxiliary line AP. Let I be a point on DE; draw the auxiliary line AI and its auxiliary extension, which intersects BC at M and the circle at A' (the second point A). Draw the auxiliary lines A'B, BI, and CI. Extend AA' to IA, connect F to IA, and draw the auxiliary lines PIA and CIA.", "ocr_zh": "E、F为直线l上的点，A为EF的中点，以A为圆心、EF为直径画半圆A，D为半圆A上的点，连接DF、ED、AD，做与l相切的圆，切点为A，延长AD与圆交于B，C为半圆A上的点，连接AB、AC、BC，做CB的辅助延长线于P，做辅助线AP，I为DE上的点，做辅助线AI并作AI的辅助延长线于BC交于M、交圆于A'（第二个A点），做辅助线A'B、BI、CI，延长AA'于IA，连接FIA，做辅助线PIA、CIA。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4451.jpg" }, { "index": 4452, "ocr_en": "In triangle ABC, point D is on BC, and point E is on AC. Connect AD and BE, which intersect at point M. Connect CM and extend it to intersect AB at point F. Connect EF and DF. Draw the auxiliary extended lines of BC and FE, which intersect at point G. Point S is on BF. Draw the auxiliary lines MS and DS.", "ocr_zh": "三角形ABC，D为BC上的点，E为AC上的点，连接AD和BE，AD和BE交于M，连接CM并延长交AB于F，连接EF、DF，做BC和FE的辅助延长线交于G，S为BF上的点，做辅助线MS、DS。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4452.jpg" }, { "index": 4453, "ocr_en": "In triangle ABC, which is an inscribed triangle of circle O, point D is on BC. Connect AD and OD. Point IA is outside circle O. Draw the auxiliary line AIA, which intersects OD at I, BC at E, and circle O at M. Draw the auxiliary lines FIA and OM.", "ocr_zh": "三角形ABC为圆O的内接三角形，D为BC上的点，连接AD、OD，IA为圆O外的点，做辅助线AIA与OD交于I、与BC交于E、与圆O交于M，做辅助线FIA、OM。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4453.jpg" }, { "index": 4454, "ocr_en": "Left diagram: Point O is outside the auxiliary circle, and points A and B are on the auxiliary circle. Connect OA and extend it to intersect the auxiliary circle at point A', and connect OB to intersect the auxiliary circle at point B'.Right diagram: Points A, B, A', and B' are on the auxiliary circle. Connect AA' and BB', and their intersection is point O. Draw the auxiliary line AB.", "ocr_zh": "左图：O为辅助圆外的一点，A、B为辅助圆上的点，连接OA并延长与辅助圆交于A'，连接OB与辅助圆交于B'。右图：A、B、A'、B'为辅助圆上的点，连接AA'和BB'，AA'和BB'交于O，做辅助线AB。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4454.jpg" }, { "index": 4455, "ocr_en": "Point A is outside circle O. Through A, two secant lines of circle O are drawn, intersecting the circle at Q, P and S, R respectively. Connect SQ and RP, extend QS and PR to intersect at point B. Connect AB, connect SP and QR, with SP and QR intersecting at point C. Connect AC. BM intersects SR at point N, and extend BC to intersect PQ at point M.", "ocr_zh": "A为圆O外一点，过A做圆O的两条割线分别交圆于Q、P和S、R，连接SQ、RP，延长QS、PR交于B，连接AB，连接SP和QR，SP和QR交于C，连接AC，BM与SR交于N，延长BC交PQ于M。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4455.jpg" }, { "index": 4456, "ocr_en": "Circle O. Draw the first straight line through O, with point A on the first line. Draw the second straight line through O, with point B on the second line. Draw a perpendicular from A to the second line, intersecting the second line at point B'. Draw a perpendicular from B to the first line, intersecting the first line at point A'.", "ocr_zh": "圆O，过O做第一条直线，A在第一条直线上，过O做第二条直线，B在第二条直线上，过A做第二条直线的垂线与第二条直线交于B'，过B做第一条直线的垂线与第一条直线交于A'。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4456.jpg" }, { "index": 4457, "ocr_en": "AB is the diameter of circle O, and P is a point on circle O. Connect BP, extend AB to E. From E, draw a tangent to circle O with the tangent point D. From D, draw a perpendicular to AB with the foot of the perpendicular at C. Connect PC and EP, and draw the auxiliary lines AD, AP, and BD.", "ocr_zh": "AB为圆O的直径，P为圆O上的点，连接BP，延长AB于E，过E做圆O的切线切点为D，过D做AB的垂线垂足为C，连接PC、EP，做辅助线AD、AP、BD。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4457.jpg" }, { "index": 4458, "ocr_en": "In quadrilateral ABCD, connect AC and BD. Extend AB to B', extend AC to C', and extend AD to D'. Draw the auxiliary lines BB', CC', DD', and BD'.", "ocr_zh": "四边形ABCD，连接AC、BD，延长AB到B'，延长AC到C'，延长AD到D'，做辅助线BB'、CC'、DD'、BD'。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4458.jpg" }, { "index": 4459, "ocr_en": "AB is the diameter of a circle, and Q is a point on the circle. Draw a perpendicular from Q to AB with the foot of the perpendicular at H. With Q as the center and QH as the radius, draw a circle intersecting the first circle at points C and D. Connect CD.", "ocr_zh": "AB是圆的直径，Q为圆上的点，过Q做AB的垂线垂足为H，以Q为圆心、QH为半径画圆与第一个圆交于C、D，连接CD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4459.jpg" }, { "index": 4460, "ocr_en": "Circle Q. A secant of circle Q intersects circle Q at points C' and D'. Point H is on circle Q. Connect QH, and draw a circle with QH as the diameter.", "ocr_zh": "圆Q，做圆Q的割线与圆Q交于C'、D'，H为圆Q上的点，连接QH，以QH为直径画圆。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4460.jpg" }, { "index": 4461, "ocr_en": "In triangle BCD, draw a perpendicular from B to CD with the foot of the perpendicular at A. Let ω be the midpoint of AB. Draw a circle with ω as the center and AB as the diameter, intersecting BC at P and BD at Q. Connect AP and AQ. Draw tangents to circle ω at points P and Q respectively, and let their intersection be R. Connect BR, which intersects circle ω at T and CD at M. Connect PT and QT. Draw an auxiliary tangent l' to circle ω at point B.", "ocr_zh": "三角形BCD，过B做CD的垂线垂足为A，ω为AB的中点，以ω为圆心、AB为直径画圆与BC交于P、与BD交于Q，连接AP、AQ，分别以P、Q为切点做圆ω的切线交于R，连接BR，BR与圆ω交于T、与CD交于M。连接PT、QT，以B为切点做圆ω的辅助切线l'。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4461.jpg" }, { "index": 4462, "ocr_en": "Points A, B, and C are on circle O. Connect AC and AB. Draw tangents to circle O at points A and B respectively, and let them intersect at point X. Connect CX, which intersects AB at point G and circle O at point D. Connect BD and extend it to intersect the extension of CA at point F, with CF perpendicular to BF. Point E is on DX. Connect BE. Points H' and B coincide.", "ocr_zh": "A、B、C为圆O上的点，连接AC、AB，分别以A、B为切点做圆O的切线并交于X，连接CX，CX与AB交于G、与圆O交于D，连接BD并延长与CA的延长线交于F，且CF垂直与BF，E为DX上的点，连接BE，H'与B两点重合。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4462.jpg" }, { "index": 4463, "ocr_en": "In quadrilateral ABCD, connect AC and BD, which intersect at O, forming triangles AOB, BOC, COD, and AOD. Construct the circumcircles of triangles AOB, BOC, COD, and AOD respectively. The circumcircle of triangle AOD and the circumcircle of triangle BOC intersect at M. Connect OM and extend it to intersect the circumcircle of triangle AOB at T. Extend MO to intersect the circumcircle of triangle COD at S.", "ocr_zh": "四边形ABCD，连接AC、BD，AC和BD交于O，构成三角形AOB、三角形BOC、三角形COD、三角形AOD，分别做三角形AOB、三角形BOC、三角形COD、三角形AOD的外接圆，三角形AOD的外接圆与三角形BOC的外接圆交于M，连接OM并延长与三角形AOB的外接圆交于T，延长MO与三角形COD的外接圆交于S。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4463.jpg" }, { "index": 4464, "ocr_en": "In triangle ABC, circle I is the incircle of triangle ABC. The points where circle I is tangent to BC, AC, and AB are D, E, and F respectively. Connect AD and let it intersect circle I at X. Connect BX and let it intersect circle I at P. Connect CX and let it intersect circle I at Q. Draw auxiliary lines DQ, QE, XE, and draw auxiliary line DE which intersects CX at R.", "ocr_zh": "三角形ABC，圆I为三角形ABC的内切圆，圆I与BC的切点为D、与AC的切点为E，与AB的切点为F，连接AD与圆I交于X，连接BX与圆I交于P，连接CX与圆I交于Q，做辅助线DQ、QE、XE，做辅助线DE并与CX交于R。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4464.jpg" }, { "index": 4465, "ocr_en": "In a triangle, the intersection point of the three medians is O. One of the medians is M'T' (the first T'), where M' is a vertex of the triangle, and T' (the first T') is the midpoint of the side opposite to M'. Connecting the midpoints of the two sides other than the side with T' (the first T') forms a midline, which intersects M'T' (the first T') at T' (the second T'). S' (the first S') coincides with T' (the first T'), and S' (the second S') coincides with T' (the second T').", "ocr_zh": "三角形三条中线交点为O，其中一条中线为M'T'（第一个T'），其中M'为三角形的顶点，T'（第一个T'）为M'对边的中点，连接除T'（第一个T'）外的两条边的中点构成一条中位线，该中位线与M'T'（第一个T'）交于T'（第二个T'），S'（第一个S'）与T'（第一个T'）重合，S'（第二个S'）与T'（第二个T'）重合。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4465.jpg" }, { "index": 4466, "ocr_en": "In quadrilateral ABCD, its incircle is tangent to AB at A₁, to BC at B₁, to CD at C₁, and to AD at D₁. Connect A₁B₁, B₁C₁, C₁D₁, and D₁A₁. Let E be the midpoint of A₁B₁, F the midpoint of B₁C₁, G the midpoint of C₁D₁, and H the midpoint of A₁D₁. Connect EF, FG, GH, and HE. Draw auxiliary lines AC, BE, and DG, and extend BE and DG such that they both intersect at a point on AC.", "ocr_zh": "四边形ABCD，做四边形ABCD的内切圆与AB切于A1、与BC切于B1、与CD切于C1，与AD切于D1，连接A1B1、B1C1、C1D1、D1A1，E为A1B1中点，F为B1C1中点，G为C1D1中点，H为A1D1中点，连接EF、FG、GH、HE，做辅助线AC、BE、DG，延长BE、DG并同时交于AC上的一点。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4466.jpg" }, { "index": 4467, "ocr_en": "Two circles intersect at C₁' and C₂'. A line through C₁' intersects the left circle at D₁' and the right circle at B₁'. Let D₂' be a point on the left circle and B₂' a point on the right circle. Draw lines D₁'D₂' and B₁'B₂', which intersect at A₂'. Construct the circumcircle of the quadrilateral formed by A₂', B₂', C₂', and D₂'.", "ocr_zh": "两圆交于C1'和C2'，过C1'做直线与左圆交于D1'、与右圆交于B1'，D2'为左圆上的点，B2'为右圆上的点，做直线D1'D2'和B1'B2'并交于A2'，做以A2'、B2'、C2'、D2'做构成的四边形的辅助外接圆。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4467.jpg" }, { "index": 4468, "ocr_en": "Construct the circumcircle of quadrilateral ABCD, connect AC. Let P be a point on AC, E a point on AB, G a point on CD, and F a point on BC. Connect DP, BP, PF, PE, PG, EG, GF, and EF.", "ocr_zh": "做四边形ABCD的外接圆，连接AC，P为AC上的点，E为AB上的点，G为CD上的点，F为BC上的点，连接DP、BP、PF、PE、PG、EG、GF、EF。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4468.jpg" }, { "index": 4469, "ocr_en": "Construct the incircle of quadrilateral ABCD, which is tangent to AB at K, to BC at L, to CD at M, and to DA at N. Connect KM, NL, and AC, and the three lines intersect at the same point G (G').", "ocr_zh": "做四边形ABCD的内接圆，与AB的切点为K、与BC的切点为L、与CD的切点为M、与DA的切点为N，连接KM、NL、AC且三条线同时交于G(G')。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4469.jpg" }, { "index": 4470, "ocr_en": "Construct the incircle of quadrilateral ABCD, which is tangent to AB at K, to BC at L, to CD at M, and to DA at N. Connect KM and NL, and let them intersect at E. Connect KN, NM, ML, and LK. Let P be a point on NL, R a point on KM, A' a point on KN, B' a point on KL, C' a point on LM, and D' a point on MN.", "ocr_zh": "做四边形ABCD的内接圆，与AB的切点为K、与BC的切点为L、与CD的切点为M、与DA的切点为N，连接KM和NL并交于E，连接KN、NM、ML、LK，P为NL上的点，R为KM上的点，A'为KN上的点，B'为KL上的点，C'为LM上的点，D'为MN上的点。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4470.jpg" }, { "index": 4471, "ocr_en": "Construct the circumcircle of quadrilateral ABCD, connect AC. Let P be a point on AC, E a point on AB, G a point on CD, and F a point on BC. Connect DP, BP, PF, PE, PG, EG, GF, and EF. Draw the auxiliary line SP, and draw the auxiliary extended lines of BA, CD, and SP such that the three extended lines intersect at R simultaneously.", "ocr_zh": "做四边形ABCD的外接圆，连接AC，P为AC上的点，E为AB上的点，G为CD上的点，F为BC上的点，连接DP、BP、PF、PE、PG、EG、GF、EF，做辅助线SP，做BA的辅助延长线、CD的辅助延长线、SP的辅助延长线且三条线同时交于R。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4471.jpg" }, { "index": 4472, "ocr_en": "Point C is on AB. Construct circles with AB, AC, and BC as diameters respectively. Construct circle O such that it is internally tangent to the circle with AB as the diameter and externally tangent to the other two circles.", "ocr_zh": "C为AB上的点，分别以AB、AC、BC为直径做圆，做圆O使其与以AB为直径的圆内切且与另外两个圆外切。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4472.jpg" }, { "index": 4473, "ocr_en": "Arc AB. Connect AB. Construct circles O₁ and O₂ that are externally tangent to each other, and both tangent to arc AB and segment AB. The point of tangency between circle O₁ and circle O₂ is P. Draw a line through P that is tangent to both circle O₁ and circle O₂, intersecting arc AB at K and segment AB at L.", "ocr_zh": "圆弧AB，连接AB，做相互外切的圆O1和圆O2且圆O1和圆O2均与圆弧AB、线段AB相切，圆O1与圆O2切点为P，过P做直线与圆O1和圆O2相切且与圆弧AB交于K、与线段AB交于L。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4473.jpg" }, { "index": 4474, "ocr_en": "In triangle ABC, let D be the midpoint of BC, E the midpoint of AC, and F the midpoint of AB. Connect CF, BE, and AD, which intersect at G. Draw the auxiliary line DF.", "ocr_zh": "三角形ABC，D为BC中点、E为AC中点、F为AB中点，连接CF、BE、AD并交于G，做辅助线DF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4474.jpg" }, { "index": 4475, "ocr_en": "In parallelogram ABCD, let P be a point inside the parallelogram. Connect AP, BP, CP, and DP. Let P' be a point outside the parallelogram. Draw auxiliary lines PP', P'C, and P'B.", "ocr_zh": "平行四边形ABCD，P为平行四边形ABCD内的点，连接AP、BP、CP、DP，P'为平行四边形ABCD外的一点，做辅助线PP'，P'C、P'B。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4475.jpg" }, { "index": 4476, "ocr_en": "Let O be the center of a regular hexagon, and let A be a point on the regular hexagon. Connect AO and extend it to intersect the regular hexagon at B. Let M be a point on the regular hexagon, and connect BM, AM, and OM.", "ocr_zh": "O为正六边形的中心，A为正六边形上的点，连接AO并延长与正六边形交于B，M为正六边形上的点，连接BM、AM、OM。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4476.jpg" }, { "index": 4477, "ocr_en": "Construct the circumcircle O of quadrilateral ABCD, and connect OA, OB, OC, OD, AC, and BD.", "ocr_zh": "做四边形ABCD的外接圆O，连接OA、OB、OC、OD、AC、BD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4477.jpg" }, { "index": 4478, "ocr_en": "Auxiliary equilateral triangle A₁B₁C₁, with points P and O inside the equilateral triangle A₁B₁C₁. Draw a perpendicular line from O to B₁C₁ with the foot of the perpendicular being A, draw a perpendicular line from O to A₁C₁ with the foot of the perpendicular being B, and draw a perpendicular line from O to A₁B₁ with the foot of the perpendicular being C. Connect AB, AC, BC, PA, PB, and PC.", "ocr_zh": "辅助正三角形A1B1C1，P、O在正三角形A1B1C1内，过O做B1C1的垂线垂足为A，过O做A1C1的垂线垂足为B，过O做A1B1的垂线垂足为C，连接AB、AC、BC、PA、PB、PC。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4478.jpg" }, { "index": 4479, "ocr_en": "In triangle ABC, point P is on BC, point Q is on AC, and point R is on AB. Connect PQ, PR, and QR. Draw an auxiliary perpendicular line from R to BC with the foot of the perpendicular being M, and draw an auxiliary perpendicular line from Q to BC with the foot of the perpendicular being N.", "ocr_zh": "三角形ABC，P为BC上的点，Q为AC上的点，R为AB上的点，连接PQ、PR、QR，过R做BC的辅助垂线垂足为M，过Q做BC的辅助垂线垂足为N。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4479.jpg" }, { "index": 4480, "ocr_en": "In triangle ABC, point P lies inside the triangle. Draw a perpendicular line from P to BC, with the foot of the perpendicular being D. Draw a perpendicular line from P to AC, with the foot of the perpendicular being E. Draw a perpendicular line from P to AB, with the foot of the perpendicular being F. Connect PA, PB, and PC. Let M be a point on AB. Draw an auxiliary straight line through P, extend it to intersect AB at M and AC at N.", "ocr_zh": "三角形ABC，P在三角形ABC内，过P做BC的垂线垂足为D，过P做AC的垂线垂足为E，过P做AB的垂线垂足为F，连接PA、PB、PC，M为AB上的点，过P做辅助直线并延长交AB于M、交AC于N。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4480.jpg" }, { "index": 4481, "ocr_en": "Circle O is the circumcircle of triangle ABC. Points G, H and M are all inside triangle ABC. Draw auxiliary lines OG, GH, OM, MH and GM.", "ocr_zh": "圆O是三角形ABC的外接圆，G、H、M均在三角形ABC内，做辅助线OG、GH、OM、MH、GM。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4481.jpg" }, { "index": 4482, "ocr_en": "Quadrilateral ABCD, with diagonals BD and AC connected. AB = a, BC = b, CD = c, AD = d.", "ocr_zh": "四边形ABCD，连接BD、AC，AB=a，BC=b，CD=c，AD=d。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4482.jpg" }, { "index": 4483, "ocr_en": "Triangle ABC, with D being the midpoint of AC. Connect BD, and let F be a point on BD. Connect AF, then connect CF and extend it to intersect AB at E.", "ocr_zh": "三角形ABC，D为AC中点，连接BD，F为BD上的点，连接AF，连接CF并延长与AB交于E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4483.jpg" }, { "index": 4484, "ocr_en": "Triangle ABC, with the number 1 outside triangle ABC. D is the midpoint of BC; connect AD. P is a point on AD; connect CP and extend it to intersect AB at F, and connect BP and extend it to intersect AC at E.", "ocr_zh": "三角形ABC，三角形ABC外有序号1，D为BC中点，连接AD，P为AD上的点，连接CP并延长与AB交于F，连接BP并延长与AC交于E。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4484.jpg" }, { "index": 4485, "ocr_en": "In triangle ABC, draw a perpendicular line from A to BC, with the foot of the perpendicular being H. Draw circles with AB and AC as diameters respectively. E and E' are points on the circle with AB as the diameter. D is a point on BH; connect ED, and draw auxiliary lines E'A, E'D, and E'H. F is a point on the circle with AC as the diameter; draw auxiliary lines FH and AF.", "ocr_zh": "三角形ABC，过A做BC的垂线垂足为H，分别以AB、AC为直径做圆，E、E'为以AB为直径的圆上的点，D为BH上的点，连接ED，做辅助线E'A、E'D、E'H，F为以AC为直径的圆上的点，做辅助线FH、AF。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4485.jpg" }, { "index": 4486, "ocr_en": "In triangle ABC, D is a point inside the triangle. Connect AD, CD, and BD. Angle CAD is equal to 30°, angle ACD is equal to 30°, and angle ABD is equal to 60°. E is a point on BC, and F is a point on AC. Connect EF. Draw an auxiliary perpendicular line from D to AC, with the foot of the perpendicular being M, and draw the auxiliary line EM. Draw a perpendicular line from F to CD, with the foot of the perpendicular being N, and draw the auxiliary line EN. These form the pentagon DENFM. Draw an auxiliary circumcircle of the pentagon DENFM.", "ocr_zh": "三角形ABC，D为三角形内的一点，连接AD、CD、BD，角CAD等于30°，角ACD等于30°，角ABD等于60°，E为BC上的点，F为AC上的点，连接EF，过D做AC的辅助垂线垂足为M，做辅助线EM，过F做CD的垂线垂足为N，做辅助线EN，构成五边形DENFM，做无比啊逆行DENFM的辅助外接圆。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4486.jpg" }, { "index": 4487, "ocr_en": "Hexagon ARBPCQ, where R is at the origin of the coordinate system, and A is on the x-axis. Connect AC, AB, BC, QP, RP, and RQ.", "ocr_zh": "六边形ARBPCQ，R在坐标原点，A在x轴上，连接AC、AB、BC、QP、RP、RQ。", "class": "Analytic Geometry", "difficulty": 1, "image_path": "fig_4487.jpg" }, { "index": 4488, "ocr_en": "Triangle AEF, with B being a point on AE. Draw a line through B parallel to EF, intersecting AF at D. Connect BF and DE; DE and BF intersect at C. Connect AC, which intersects BD at Q. P is a point on AC, and R is a point on EF.", "ocr_zh": "三角形AEF，B为AE上的点，过B做EF的平行线交AF于D，连接BF、DE，DE和BF交于C，连接AC并与BD交于Q，P为AC上的点，R为EF上的点。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4488.jpg" }, { "index": 4489, "ocr_en": "Draw the circumcircle of quadrilateral \\(ABCD\\). Connect \\(AC\\) and \\(BD\\), and let them intersect at \\(S\\). Draw a perpendicular line from \\(S\\) to \\(AB\\) with the foot of the perpendicular being \\(E\\), and draw a perpendicular line from \\(S\\) to \\(CD\\) with the foot of the perpendicular being \\(F\\). Connect \\(EF\\). Let \\(M\\) be the midpoint of \\(AD\\), and draw a straight line through \\(M\\) that is perpendicular to \\(EF\\).", "ocr_zh": "做四边形ABCD的外接圆，连接AC、BD且AC与BD交于S，过S做AB的垂线垂足为E，过S做CD的垂线垂足为F，连接EF，M为AD中点，过M做直线使该直线垂直EF。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4489.jpg" }, { "index": 4490, "ocr_en": "Quadrilateral ABCD, with DC lying on the x-axis. Connect AC and BD, and let them intersect at E. The intersection point of AB and the y-axis is F. Draw the straight line EF, and connect DF and CF. Both O1 and O2 are points inside quadrilateral ABCD; connect O1O2.", "ocr_zh": "四边形ABCD，DC在x轴上，连接AC和BD且AC与BD交于E，AB与y轴的交点为F，做直线EF，连接DF、CF，O1、O2均为四边形ABCD内的点，连接O1O2。", "class": "Analytic Geometry", "difficulty": 3, "image_path": "fig_4490.jpg" }, { "index": 4491, "ocr_en": "M is on the y-axis. Draw a circle M with M as the center, and construct an inscribed triangle of circle M. A is the intersection point of circle M and the positive semi-axis of the y-axis. Draw a circle K that is tangent to AB, AC, and circle M, with P being the point of tangency with AB, Q being the point of tangency with AC, and D being the point of tangency with circle M. D is the intersection point of circle O and the negative semi-axis of the y-axis. Draw the auxiliary line OP.", "ocr_zh": "M在y轴上，以M为圆心做圆M，做圆M的内接三角形，A为圆M和y轴正半轴的交点，做圆K使其与AB、AC和圆M相切，与AB的切点为P、与AC的切点为Q，与圆M的切点为D，D为圆O与y轴负半轴的交点，做辅助线OP。", "class": "Analytic Geometry", "difficulty": 3, "image_path": "fig_4491.jpg" }, { "index": 4492, "ocr_en": "Triangle ABM, with M at the origin of the coordinate axes. Draw the circumcircle O of triangle ABM, where O lies on the x-axis. O₁ is a point on OM; draw circle O₁ with O₁ as the center and O₁M as the radius, intersecting AM and BM at C and D respectively. Connect CD. O₂ is a point on circle O₁; draw circle O₂ that is tangent to circle O and CD, with N being the point of tangency with circle O. Draw auxiliary lines O₂M, O₂N, O₂O, and O₂O₁.", "ocr_zh": "三角形ABM，M在坐标轴原点，做三角形ABM的外接圆O且O在x轴上，O1为OM上的点，以O1为圆心、O1M为半径做圆O1与AM、BM分别交于C、D，连接CD，O2在圆O1上，做圆O2使其与圆O、CD相切，与圆O的切点为N，做辅助线O2M、O2N、O2O、O2O1.", "class": "Analytic Geometry", "difficulty": 3, "image_path": "fig_4492.jpg" }, { "index": 4493, "ocr_en": "In square ABCD, there is an interior point P. Connecting P to vertices A, B, C, and D forms triangles PAB, PBC, PCD, and PDA, with areas denoted as S1, S2, S3, and S4 respectively.", "ocr_zh": "正方形ABCD内存在一点P，作辅助线PA、PB、PC、PD，得到三角形PAB、三角形PBC、三角形PCD和三角形PDA，它们的面积分别为S1、S2、S3和S4。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4493.jpg" }, { "index": 4494, "ocr_en": "Construct squares BCNM and CDPQ externally on the sides BC and CD of parallelogram ABCD", "ocr_zh": "在平行四边形ABCD两边BC、CD向外分别作正方形BCNM、CDPQ.", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4494.jpg" }, { "index": 4495, "ocr_en": "Quadrilateral ABCD is inscribed in circle G. Diagonals AC and BD are drawn, with BD passing through the center G. Points E and F lie above diagonal BD on AC. Connect BF, DF, BE, and DE.", "ocr_zh": "四边形ABCD为圆G的内接四边形，连接对角线AC和BD，其中BD经过圆心点G。点E、F为对角线AC上，对角线BD上方的两点，连接BF、DF、BE和DE。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4495.jpg" }, { "index": 4496, "ocr_en": "ABCD is a rectangle with AB > BC. Diagonals AC and BD intersect at point F (or G). On diagonal AC, place two auxiliary points on either side of F, each connected to the nearer of points B or D to form two auxiliary lines.", "ocr_zh": "四边形ABCD为矩形，且AB大于BC，连接对角线AC和BD交于点F(G)。在对角线AC上，点F两侧分别作一辅助点，辅助点就近连接点B或点D，得到两条辅助线。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4496.jpg" }, { "index": 4497, "ocr_en": "In triangle ABC, let G be the midpoint of AC and connect BG. Choose a point F on segment AB, connect CF intersecting BG at point H, then connect BH. Choose a point Z on segment CG, connect FZ intersecting BG at point O. Extend AB to point P, connect HP intersecting BC at point X, then connect XF. Choose a point Y on segment BF.", "ocr_zh": "在三角形ABC中，点G为AC中点，连接BG。取线段AB上一点F，连接CF交BG于点H，连接BH；取线段CG上一点Z，连接FZ交BG于点O。作AB的延长线AP，连接HP交BC于点X，连接XF。取线段BF上一点作点Y。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4497.jpg" }, { "index": 4498, "ocr_en": "Let point F lie on side AB of triangle ABC, and connect CF. Choose a point O inside triangle ACF, and draw an auxiliary line OE perpendicular to AF, with E as the foot of the perpendicular. Connect OF. Select a point H on segment CF, and on the extension of AB beyond B, choose point P such that ∠APH = ∠HAP. Let line segment HP intersect BC at point X. Draw XL perpendicular to AB with foot L, and auxiliary line XN perpendicular to CF with foot N.", "ocr_zh": "点F为三角形ABC边AB上一点，连接CF。在三角形ACF中取一点O，作辅助线OE垂直于AF，垂足为点E，连接OF。在线段CF上取一点H，连接线段AB延长线AP上点P，使得角APH等于角HAP，线段HP交BC于点X。作辅助线XL垂直于AB，垂足为点L，XN垂直于CF，垂足为点N。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4498.jpg" }, { "index": 4499, "ocr_en": "Triangle ABC has interior angles α, β, and γ. Let point O be the intersection point of its three internal angle bisectors. Connect OA, OB, and OC. Let ∠BOC = φ.", "ocr_zh": "三角形ABC三个内角分别为α、β和γ，点O为其三条内角平分线交点。连接OA、OB、OC，角BOC为φ。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4499.jpg" }, { "index": 4500, "ocr_en": "In parallelogram ABCD, points X, Y, and Z lie on sides AB, BC, and CD respectively. Connect XY and AZ such that ∠BYX = ∠ZAD. Connect diagonal BD. Let BD intersect XY at point K and AZ at point L.In diagram (a), connect ZY and let it intersect BD at point M₁.In diagram (b), connect XZ and let it intersect BD at point M₂.", "ocr_zh": "在平行四边形ABCD中，点X、Y、Z分别在边AB、BC、CD上，连接XY、AZ，使得角BYX等于角ZAD。连接对角线BD，BD交XY于点K，交AZ于点L。a图中连接ZY交BD于点M1，b图中连接XZ交BD于点M2。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4500.jpg" }, { "index": 4501, "ocr_en": "Let semicircle Γ have diameter AB, with midpoint M as its center. Let semicircle Γ₁ have diameter BM. Points X and Y lie on semicircle Γ₁, with X closer to M and Y closer to B. Draw lines BX and MY, intersecting at point D. Line MY intersects semicircle Γ at point C.Let point Z be the midpoint of arc BY (on semicircle Γ₁). Draw auxiliary lines MZ and BZ. Extend BZ to intersect semicircle Γ at point E, such that point E coincides with point C.", "ocr_zh": "半圆Г以AB为直径，AB中点M为圆心；半圆Г1以BM为直径。点X、Y为半圆Г1上两点，点X靠近点M，点Y靠近点B，作直线连接BX、MY交于点D，直线MY交半圆Г于点C。点Z为圆弧BY的中点，作辅助线MZ和BZ，延长辅助线BZ交半圆Г于点E，点E和C重合。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4501.jpg" }, { "index": 4502, "ocr_en": "In isosceles triangle ABC with AB = AC, let M be the midpoint of side BC. Connect AM and extend it to point D. Extend AC to point F, and connect DF such that DF is parallel to BC. Connect BD and CD.Choose a point E on side AB and connect DE such that ∠BDE = ∠CDF. Draw auxiliary line EF, which intersects BC at point P′. Point P coincides with point P′.", "ocr_zh": "在等腰三角形ABC中，AB等于AC，点M为BC边上的中点，连接AM并延长至点D。延长AC至点F，连接DF使得DF平行于BC，连接BD、CD。在AB边上取一点E，连接DE，使得角BDE等于角CDF。连接辅助线EF，EF交BC于点P'，点P与点P'重合。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4502.jpg" }, { "index": 4503, "ocr_en": "Two circles O1 and O2 intersect at points A and B, with the diameter of O1 slightly larger than that of O2. Connect chord AB. Points C and D lie on circle O1, while points E and F lie on circle O2. Quadrilaterals ABCD and ABEF are cyclic in O1 and O2 respectively, and the angle CAD equals the angle CBD, which equals the angle EBF, which equals the angle EAF.Connect diagonals AD, BC, AE, and BF. Points M and N lie on segments BC and BD respectively. Connect AM, AN, and MN such that the angle MAN equals the angle CAD, and the angle CAM equals the angle DAN.", "ocr_zh": "圆O1交圆O2于点A、B，圆O1直径略大于圆O2。连接辅助线AB，点C、D在圆O1上，点E、F在圆O2上，四边形ABCD和四边形ABEF分别为圆O1和圆O2的内接四边形，且角CAD等于角CBD等于角EBF等于角EAF。连接对角线AD、BC、AE、BF，点M、N分别为BC、BD上一点，连接AM、AN、MN，使得角MAN等于角CAD，角CAM等于角DAN。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4503.jpg" }, { "index": 4504, "ocr_en": "In quadrilateral PRSU, side PR is parallel to SU. Connect diagonals PU and RS, which intersect at point D. Point T lies on SU, and lines PT and PU are trisectors of angle SPR.Choose a point E on PT and draw auxiliary line DE such that DE is parallel to PR (and SU). Select points F and G on segments PD and SR respectively. Connect EF and EG so that EF is parallel to DG and EG is parallel to DF.Connect SE and extend it to intersect PR at point Q. Connect FG and extend it to intersect PR and SU at points K and L respectively.", "ocr_zh": "在四边形PRSU中，PR平行于SU，连接对角线PU和RS交于点D。点T为SU上一点，连接PT，PT、PU分别为角SPR三等分线。在PT上取一点E，连接辅助线DE，使得DE平行于PR平行于SU。分别在PD、SR上取一点F、G，连接EF、EG，使得EF平行于DG，EG平行于DF。连接SE并延长，交PR于点Q；连接FG并延长，分别交PR、SU于点K、L。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4504.jpg" }, { "index": 4505, "ocr_en": "Two circles O1 and O2 intersect at points C and D, with the diameter of O1 slightly larger than that of O2. Connect chord CD. Points P and A lie on circle O1, and the arc PA equals the arc PD. Connect AC and draw auxiliary lines PA, PD, PC, and AD. Lines AD and PC intersect at point E.Extend PD to meet circle O2 again at point B, then connect BC. Point Q lies on circle O2. Connect PQ such that PQ is perpendicular to CD in space, without a foot of perpendicular on CD. Draw auxiliary line BQ and CQ, which intersects BD at point F.", "ocr_zh": "圆O1交圆O2于C、D两点，圆O1直径略大于圆O2,连接CD。点P和点A为圆O1上两点，且圆弧PA等于圆弧PD，连接AC，作辅助线连接PA、PD、PC、AD，AD交PC于点E。作辅助线延长PD交圆O2于点B，连接BC。Q为圆O2上一点，连接PQ，有平面上PQ垂直于CD，无垂足。作辅助线连接BQ及CQ交BD于点F。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4505.jpg" }, { "index": 4506, "ocr_en": "Parallelogram ABCD is an inscribed parallelogram within quadrilateral PQRS, with points A, B, C, and D lying on sides PQ, QR, RS, and SP respectively. In parallelogram ABCD, BC is an auxiliary line. Point M is the midpoint of side CD. Connect AM and MR, then extend AM to intersect BC at point N, with point S lying on line AN. Point H lies on AM. Connect BH and CH, where CN equals AD, which equals BC, and also equals CH.", "ocr_zh": "平行四边形ABCD为四边形PQRS的内接平行四边形，点A、B、C、D分别在边PQ、QR、RS、SP上。在平行四边形ABCD中，BC为辅助线，点M为CD边中点，连接AM、MR，作辅助线延长AM、BC交于点N，且点S在AN上。H为AM上一点，连接BH、CH，有CN等于AD等于BC等于CH。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4506.jpg" }, { "index": 4507, "ocr_en": "In triangle BCG, point D is the midpoint of side BC. Connect D to G, and BC equals twice DG.Using BD as the base, take a point A outside triangle BCG to form triangle ABD, E lies on segment BG. Connect AE.Choose a point F on CG, and connect AF and EF. Draw auxiliary lines DE, DF, and AC. Point F is such that AF equals DF.", "ocr_zh": "在三角形BCG中，点D为BC边上的中点，连接DG，BC长度是DG两倍。以BD为底边，在三角形BCG外取一点A，作三角形ABD，E是BG边上一点，连接AE。在CG边上取一点F，连接AF、EF，作辅助线连接DE、DF、AC，点F满足AF等于DF。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4507.jpg" }, { "index": 4508, "ocr_en": "In quadrilateral ABEF, connect diagonals AE and BF, which intersect at point P. Select points C and D on AE and BF respectively. Connect CD and BC such that CD is parallel to EF and BC is parallel to AF. Then connect AD.", "ocr_zh": "在四边形ABEF中，连接对角线AE、BF交于点P。分别在AE、BF上取一点C、D，连接CD、BC，满足CD平行于EF，BC平行于AF，连接AD。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4508.jpg" }, { "index": 4509, "ocr_en": "In triangle ABC, let I be the incenter. Point E is the tangent point of the incircle I with side BC. Point D is the tangent point of the A-excircle with BC.Connect AD, which intersects the incircle I at points X and Y. Connect E to Y. Let A′ be the midpoint of segment ED, and connect I to A′. As an auxiliary line, draw EX passing through the center I.", "ocr_zh": "在三角形ABC中有内切圆I，点E为圆I与BC边上的切点，点D为角A内的旁切圆与边BC的切点。连接AD，分别交圆I于点X、Y，连接EY，点A'为ED中点，连接IA'。作辅助线过圆心点I连接EX。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4509.jpg" }, { "index": 4510, "ocr_en": "Two circles intersect, with the larger diameter circle positioned slightly higher and named O1, and the other named O2. Points A, B, C, and D lie on circle O1, where point D is one of the intersection points of O1 and O2. Chords AB and CD intersect at point E, and AB also intersects circle O2 at points E and M. Connect AC and BC.Draw auxiliary lines AD, BD, and DM. Choose a point F on BC. Connect EF and extend it toward E to meet the extension of AC at point G, such that angle CGF equals angle BDM.", "ocr_zh": "图中两个圆相交，令直径稍大、圆心位置偏上的圆为O1，另一圆为O2。点A、B、C、D为圆O1上四点，其中点D为圆O1与O2交点之一，连接AB、CD交于点E，且AB交圆O2于点E、M。连接AC、BC。作辅助线连接AD、BD、DM。在BC上取一点F，连接EF并向点E方向延长交AC延长线于点G，满足角CGF等于角BDM。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4510.jpg" }, { "index": 4511, "ocr_en": "Points D and E lie inside and outside the triangle respectively. Connect AD, CD, and BD. Draw auxiliary lines BE, CE, and DE such that angle CBE equals angle CAD, and angle ACD equals angle BCE.", "ocr_zh": "点D、E分别为三角形内、外一点，连接AD、CD、BD，作辅助线BE、CE、DE，满足角CBE等于角CAD，角ACD等于角BCE。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4511.jpg" }, { "index": 4512, "ocr_en": "In hexagon ABCDEF, connect diagonals AC, AE, BD, DF, and CE. Draw an auxiliary line AD and extend it to point P. Also draw auxiliary lines CP and EP. Point P is chosen such that angle FEA equals angle DEP, and angle EFA equals angle EDP.Triangle EFA is similar to triangle DEP, triangle ABC is similar to triangle PDC, and triangle EFD is similar to triangle AEP.", "ocr_zh": "在六边形ABCDEF中，连接AC、AE、BD、DF、CE，作辅助线连接AD并延长至点P，辅助线连接CP、EP，点P满足角FEA等于角DEP，角EFA等于角EDP。三角形EFA相似于三角形DEP，三角形ABC相似于三角形PDC，三角形EFD相似于三角形AEP。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4512.jpg" }, { "index": 4513, "ocr_en": "In triangle BCF, points A, E, and M are the midpoints of sides BF, BC, and CF, respectively. Connect AC and BM; lines AC, BM, and EF intersect at point D. A semicircle is drawn with AB as its diameter, passing through points A, B, D, and E.", "ocr_zh": "在三角形BCF中，点A、E、M分别为BF、BC、CF边中点，连接AC、BM、EF交于点D。以AB为直径作半圆，A、B、D、E四点都在圆上。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4513.jpg" }, { "index": 4514, "ocr_en": "In triangle ABC, point N lies inside the triangle. Connect AN, CN, and BN. Point K is on the ray BN, outside triangle ABC. Draw auxiliary lines AK and CK such that triangle ACN is congruent to triangle ACK. Points A, N, C, and K lie on the same circle.Point M is another interior point of triangle ABC. Connect AM, BM, and CM. Angle BMA is greater than angle ACB. Triangle ABM is similar to triangle BCK, and triangle ABK is similar to triangle BCM.", "ocr_zh": "在三角形ABC中，点N为内部一点，连接AN、CN、BN，点K是辅助线射线BN上一点，位于三角形ABC外，作辅助线连接AK、CK，满足三角形ACN全等于三角形ACK，A、N、C、K四点共圆。点M为三角形ABC内另一点，连接AM、BM、CM，角BMA大于角ACB，三角形ABM相似于三角形BCK，三角形ABK相似于三角形BCM。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4514.jpg" }, { "index": 4515, "ocr_en": "In triangle ABC, the side lengths satisfy BC is greater than AB, which is greater than AC. Points M, N, and P are the midpoints of sides BC, AC, and AB, respectively. Connect segments PM, PN, and MN. Points M₂, N₂, and P₂ lie on segments PN, PM, and MN respectively. Connect MM₂, NN₂, and PP₂, which intersect at point K. Extend these lines to meet sides AB and BC at points M₁, N₁, and P₁.", "ocr_zh": "在三角形ABC中，边长BC>AB>AC。点M、N、P分别为BC、AC、AB中点，连接PM、PN、MN，点M2、N2、P2分别为PN、PM、MN上一点，连接MM2、NN2、PP2交于点K，并延长分别交AB、BC于点M1、N1、P1。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4515.jpg" }, { "index": 4516, "ocr_en": "In triangle A″AC, points B and A′ lie on segment A″C, such that angle BAA′ equals angle A′AC, and angle A″AA′ equals angle A″A′A. Connect AB and AA′. From point A″, draw a segment that intersects both AB and AA′, with its other endpoint lying on AA′.", "ocr_zh": "在三角形A''AC中，点B、A'在A''C上，满足角BAA'等于角A'AC，角A''AA'等于角A''A'A，连接AB、AA'。由A''作一条线段与AB、AA'都有一个交点，线段另一端在AA'上。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4516.jpg" }, { "index": 4517, "ocr_en": "In isosceles triangle PAC, where PA equals PC, extend lines PA and PC. Construct circle Γ tangent to PA and PC at points A and C, respectively. Point B lies on segment AC. Connect PB, which intersects circle Γ at point Q.Draw the angle bisector of angle AQC, intersecting AC at point R and circle Γ again at point S, with S different from Q. Draw auxiliary lines AS and CS.", "ocr_zh": "在等腰三角形PAC中，PA等于PC，延长PA、PC，作圆Г与PA、PC相切，切点为A、C。B为AC上一点，连接PB交圆Г于点Q。作角AQC的平分线交AC于点R，交圆Г于点S，其中S与Q是不同的两点。作辅助线连接AS、CS。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4517.jpg" }, { "index": 4518, "ocr_en": "In triangle ABC, points D, E, and F are the midpoints of sides BC, AC, and AB, respectively. Connect AD, BE, and CF, which intersect at point G. Triangles AEG, BFG, and CDG form the shaded areas.", "ocr_zh": "在三角形ABC中，点D、E、F分别为BC、AC、AB边上的中点。连接AD、BE、CF交于一点G。三角形AEG、三角形BFG、三角形CDG为阴影覆盖区域。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4518.jpg" }, { "index": 4519, "ocr_en": "Point P lies outside triangle ABC. Connect BP and CP, and extend BP as an auxiliary line to intersect line AC at point F. Triangle ACB is similar to triangle BCF. Extend AC beyond point F to point D. Connect PD, which intersects sides AB and BC at points N and T, respectively.", "ocr_zh": "在三角形ABC外有一点P，连接BP、CP，延长BP作辅助线交直线AC与点F，三角形ACB相似于三角形BCF。于点F的另一端延长AC于点D，连接PD，分别交AB、BC于点N、T。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4519.jpg" }, { "index": 4520, "ocr_en": "On the sides of hexagon A1A2A3A4A5A6, six tangent points B1, B2, B3, B4, B5, and B6 are given. Lines B1B2, B3B4, and B5B6 intersect pairwise at points P, Q, and R respectively. Draw auxiliary lines B1B6, B4B5, and B2B3. Then, draw rays from points P toward A6, from Q toward A2, and from R toward A4.", "ocr_zh": "六边形A1A2A3A4A5A6边上六个切点分别为B1、B2、B3、B4、B5、B6，作B1B2、B3B4、B5B6两两交于点P、Q、R，作辅助线连接B1B6、B4B5、B2B3，作射线PA6、QA2、RA4。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4520.jpg" }, { "index": 4521, "ocr_en": "In triangle ABC, points S and R lie on sides AB and AC respectively, with segment SR parallel to BC. Points P and Q lie on side BC. Connect SP and QR to form square PQRS inscribed in triangle ABC. Let A₁ be the center of square PQRS. Connect A to A₁, and draw auxiliary lines A₁S and A₁R.", "ocr_zh": "在三角形ABC中，点S、R分别为AB、AC上一点，连接SR，有SR平行于BC。点P、Q为点BC上两点，连接SP、QR，得到正方形PQRS内接于三角形ABC。A1为正方形PQRS的中心，连接AA1，作辅助线连接A1S、A1R。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4521.jpg" }, { "index": 4522, "ocr_en": "In hexagon AECDBF, diagonals AD, BE, and CF intersect at a single point. Connect AB, AC, and BC. Let angle BAF equal angle CAE, denoted as alpha; angle ABF equal angle CBD, denoted as beta; and angle BCD equal angle ACE, denoted as gamma.", "ocr_zh": "在六边形AECDBF中，连接对角线AD、BE、CF交于一点。连接AB、AC、BC，记角BAF等于角CAE为α，角ABF等于角CBD为β，角BCD等于角ACE为γ。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4522.jpg" }, { "index": 4523, "ocr_en": "Lines MN, MB, and MC are tangents to circle O at points A, B, and C respectively. Connect chords AB, AC, and BC. Draw AD perpendicular to BC with foot at point D. Connect MD and ND. Let angle MDA be alpha and angle NDA be beta.", "ocr_zh": "MN、MB、MC都是圆O的切线，切点分别为点A、B、C，连接AB、AC、BC，作AD垂直于BC垂足为点D，连接MD、ND，记角MDA为α，角NDA为β。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4523.jpg" }, { "index": 4524, "ocr_en": "In quadrilateral ABCD, connect diagonal AC and draw auxiliary line BD. Points E and G lie on sides CD and BC respectively. Connect BE, AE, AG, and DG such that angle BAG equals angle EAD, both denoted as theta. Let angle CAG be alpha and angle CAE be beta. Lines BE, DG, and AC intersect at point F.", "ocr_zh": "在四边形ABCD中，连接对角线AC，作辅助线连接对角线BD。点E、G分别为CD、BC上一点，连接BE、AE、AG、DG，满足角BAG等于角EAD为θ，记角CAG为α，角CAE为β。BE、DG、AC三线交于一点F。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4524.jpg" }, { "index": 4525, "ocr_en": "Points A, B, C, D, E, and F lie on a circle. Connect chords AB, CD, and EF, which intersect pairwise to form triangle UVW. Point U lies outside the circle, point V is the intersection of AB and EF, and point W is the intersection of AB and CD.Connect BC, intersecting EF at point N; connect DE, intersecting AB at point L; and connect AF, intersecting CD at point M.", "ocr_zh": "点A、B、C、D、E、F在圆上，连接AB、CD、EF，两两相交成三角形UVW，其中点U在圆外，点V为AB和EF交点，点W为AB和CD交点。连接BC交EF于点N，连接DE交AB于点L，连接AF交CD于点M。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4525.jpg" }, { "index": 4526, "ocr_en": "In acute triangle ABC, points E and F lie on sides AC and AB, respectively. Connect BE and CF.", "ocr_zh": "在锐角三角形ABC中，点E、F分别为AC、AB边上一点，连接BE、CF。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4526.jpg" }, { "index": 4527, "ocr_en": "In an isosceles acute triangle ABC, points E and D lie on sides AC and AB, respectively. Connect BE and CD.", "ocr_zh": "在等腰锐角三角形ABC中，点E、D分别为AC、AB边上一点，连接BE、CD。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4527.jpg" }, { "index": 4528, "ocr_en": "In the quadrilateral CDC′E, points A, B, A′, and B′ lie on sides CE, CD, C′ E, and C′D respectively. Lines A′A, C′C, and B′B are drawn and extended to meet at a point O, which is located above point C. Similarly, lines AB, ED, and A′B′ are drawn and extended to meet at a point F, which lies to the right of point D.", "ocr_zh": "在四边形CDC'E中，点A、B、A'、B'分别在CE、CD、C'E、C'D边上，连接A'A、C'C、B'B并延长三条线交于一点O，O点位于点C上方。连接AB、ED、A'B'并延长三条线交于一点F，点F位于点D右方。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4528.jpg" }, { "index": 4529, "ocr_en": "In circles A and B, they intersect at points C and D. We draw segments AB, CD, AC, and BC. Additionally, circles A and B both intersect with a lower circle at points C, E, and F—point E lies on circle A, point F lies on circle B, and both A and B lie on the lower circle, then we draw segments EF. Auxiliary lines BE, AF, CE, and CF are drawn such that D is the intersection of lines AF and BE.", "ocr_zh": "圆A和圆B交于C、D两点，连接AB、CD、AC、BC。圆A、圆B与下方一圆交于点C、E、F，点E在圆A上，点F在圆B上，点A、B都在下方圆上，连接EF。作辅助线连接BE、AF、CE、CF，且点D为AF、BE交点。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4529.jpg" }, { "index": 4530, "ocr_en": "In triangle ABC, let M₁ and M₂ be the midpoints of sides BC and AC, respectively, and connect M₁M₂ as an auxiliary line. Let H be an interior point of triangle ABC. Draw the line CH. Also draw lines AH and BH and extend them until they intersect the respective sides of the triangle. Reflect triangle CA₁B across M₁ to triangle BHC, then draw the auxiliary line through M₁ connecting to A₁H. Let A₂ be the reflection of A across M₁; draw the auxiliary line connecting A to A₂.", "ocr_zh": "在三角形ABC中，M1、M2分别为BC、AC边上的中点，作辅助线连接M1M2。点H为三角形ABC内一点，作辅助线连接CH，作辅助线连接AH、BH并延长至与三角形边长有交点。作三角形CA1B与三角形BHC关于M1对称，作辅助线过点M1连接A1H。A2为A关于M1的对称点，作辅助线连接AA2。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4530.jpg" }, { "index": 4531, "ocr_en": "In triangle ABC, points D, E, and F lie respectively on sides BC, AC, and AB. Draw segments DE and EF. Let O be the common circumcenter of triangles ABC and DEF. With center O, construct the circumcircles of both triangles. Side BC meets the circumcircle of DEF again at D′; side AC meets it at E′; side AB meets it at F′. Finally, draw the auxiliary line OH perpendicular to BC, with foot H.", "ocr_zh": "在三角形ABC中，点D、E、F分别为BC、AC、AB上一点，连接DE、DE、EF，点O为三角形ABC和三角形DEF的公共外心。以O为圆心分别作三角形ABC和三角形DEF的外接圆，三角形ABC的BC、AC、AB边分别交三角形DEF外接圆于点D和D'、E和E'、F和F'。作辅助线OH垂直于BC，垂足为点H。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4531.jpg" }, { "index": 4532, "ocr_en": "In triangle ABC, point U is the intersection of the angle bisectors of ∠BAC, ∠ACB, and ∠ABC. Connect segments AU, BU, and CU. Point O is the intersection of the perpendicular bisectors OO₂, OO₁, and OO₃ of sides AC, BC, and AB, respectively. Draw auxiliary lines OO₂, OO₁, and OO₃, and connect O₁O₂, O₂O₃, and O₁O₃. Line OO₁ is perpendicular to BC, with foot at point E. Line BU is perpendicular to O₁O₃.", "ocr_zh": "在三角形ABC中，点U为角BAC、角ACB、角ABC角平分线的交点，连接AU、BU、CU。点O为三角形ABC内边AC、BC、AB垂直平分线OO2、OO1、OO3的交点，作辅助线连接OO2、OO1、OO3，连接O1O2、O2O3、O1O3。OO1垂直于BC，垂足为点E，BU垂直于O1O3。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4532.jpg" }, { "index": 4533, "ocr_en": "Construct the circumcircle of triangle ABC centered at O using auxiliary lines.\nLet point I be the intersection of the angle bisectors AD, BK, and CE of angles ∠BAC, ∠ABC, and ∠ACB, respectively—where D lies on BC, K on AB, and E on AC. Connect BK, DE, EK, and DK, and draw auxiliary lines CE and AD. Extend line BK until it meets the circumcircle (O) again at point P; then draw auxiliary lines AP and CP.", "ocr_zh": "用辅助线作出三角形ABC的外接圆圆O。点I为角BAC、角ACB、角ABC角平分线AD、BK、CE的交点，点D、E、K分别在BC、AB、AC边上，连接BK、DE、EK、DK，作辅助线连接CE、AD。作BK辅助线延长线与圆O交于点P，作辅助线连接AP、CP。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4533.jpg" }, { "index": 4534, "ocr_en": "In triangle ABC, point O is the circumcenter. Draw the auxiliary line OA and the auxiliary line ON perpendicular to BC, with foot at point N. Draw altitude AD perpendicular to BC, with foot at point D, and connect OD. Extend ray AB beyond B, and let IA be a point located below triangle ABC and to the right on the extension of AB. Draw the auxiliary line IAF perpendicular to the extension of AB, with foot at point F. Connect A to IA, and extend this line until it intersects OD at point I. Draw auxiliary lines IE perpendicular to AB and IM perpendicular to BC, with feet at points E and M, respectively.", "ocr_zh": "在三角形ABC中，点O为外接圆圆心，作辅助线连接OA，作辅助线ON垂直于BC，垂足为点N。作AD垂直于BC，垂足为点D，连接OD。作射线延长AB边，点IA为三角形ABC下方、AB延长线右方一点，作辅助线IAF垂直于AB延长线，垂足为点F。作辅助线连接AIA，与OD交于点I。作辅助线IE垂直于AB、IM垂直于BC，垂足分别为点E、M。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4534.jpg" }, { "index": 4535, "ocr_en": "Circle O is the circumcircle of isosceles triangle ABC, where the lengths of AB and AC are equal. A smaller circle is tangent to sides AB and AC at points P and Q, respectively. Draw the auxiliary line AM, the angle bisector of ∠BAC, with point M lying on circle O. Points O and the center of the smaller circle both lie on line AM. Draw auxiliary lines connecting PM, QM, and BM. The line PQ intersects AM at point I, which lies between points O and the center of the smaller circle. Finally, draw the auxiliary line BI.", "ocr_zh": "圆O为等腰三角形ABC的外接圆，其中AB长度等于AC长度。小圆与AB、AC相切于点P、Q。作辅助线角BAC平分线AM，点M在圆O上，点O和小圆圆心在AM上。作辅助线连接PM、QM、BM，连接PQ交AM于点I，点I在点O和小圆圆心之间，作辅助线连接BI。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4535.jpg" }, { "index": 4536, "ocr_en": "Points O₁, O₂, and O₃ are three points inside the triangles APS, BQP, and CSQ, respectively. Connect O₁O₂, O₁O₃, and O₂O₃, and use auxiliary lines to construct hexagon O₁P O₂Q O₃S. Point K lies outside triangle ABC. Draw auxiliary lines SK, O₁K, and O₃K such that triangle O₁O₂O₃ is congruent to triangle O₁O₃K.", "ocr_zh": "点O1、O2、O3分别为三角形APS、三角形BQP、三角形CSQ内的三点，连接O1O2、O1O3、O2O3，用辅助线作出六边形O1PO2QO3S。点K为三角形ABC外一点，作辅助线连接SK、O1K、O3K，满足三角形O1O2O3全等于三角形O1O3K。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4536.jpg" }, { "index": 4537, "ocr_en": "Circle O is tangent to equilateral triangle ABC at points B and C. Connect AO, which intersects circle O again at point X and side BC at point M. Extend AO to intersect circle O again at point Y. Draw auxiliary lines OB and OC. Points E and F lie on arcs BY and CX, respectively. Through point M, connect EF, and also connect AE and AF. Draw auxiliary lines OE, OF, EX, FX, and FY.", "ocr_zh": "圆O与等边三角形ABC相切于B、C两点，连接AO交圆O于点X，交BC边于点M，延长AO交圆O于点Y，作辅助线连接OB、OC。点E、F分别为弧BY、弧CX上一点，过点M连接EF，连接AE、AF。作辅助线连接OE、OF、EX、FX、FY。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4537.jpg" }, { "index": 4538, "ocr_en": "In isosceles triangle ABC, auxiliary line AM is the altitude to side BC, and point M is the midpoint of BC. Points D and F are the midpoints of sides AB and AC, respectively. Draw auxiliary line DF, and let point E be a point on DF (not an intersection). Connect C to D, intersecting AM at point G. Draw auxiliary line FM, which intersects CD at point K. Point O lies on line AM above point G and between lines AM and DF. Draw auxiliary line OD and solid line OE.", "ocr_zh": "在等腰三角形ABC中辅助线AM为BC边上的高，点M为BC边上的中点。点D、F分别为AB、AC边上的中点，作辅助线连接DF，点E为DF上一点（非交点）。连接CD，交AM于点G。作辅助线连接FM交CD于点K。点O为AM线上点G上方与DF线之间一点，作辅助线连接OD，作实线连接OE。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4538.jpg" }, { "index": 4539, "ocr_en": "In right triangle ABC, AC is perpendicular to AB with the foot at point A. Point E is the midpoint of AB, and point F lies on side AC. Connect E and F, and construct circle O as the circumcircle of triangle AEF. Extend FE to intersect the extension of CB at point M. Connect AO and extend it to intersect BC at point D. Points I₁ and I₂ are two points on arc EF, with I₂ to the left of I₁. Draw auxiliary lines connecting AI₁, AI₂, DI₁, and DI₂.", "ocr_zh": "在直角三角形ABC中，AC垂直于AB垂足为点A。点E为AB中点，点F为AC边上一点，连接EF，作圆O为三角形AEF的外接圆。延长FE交CB延长线于点M。连接AO并延长，交BC于点D。点I1、I2为圆弧EF上两点，I2在I1左侧，作辅助线连接AI1、AI2、DI1、DI2。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4539.jpg" }, { "index": 4540, "ocr_en": "In triangle ABC, draw AD perpendicular to BC, with foot at point D. Points O₁, O, and O₂ lie below side BC from left to right, with the distances from O₂, O₁, and O to BC increasing in that order. Draw auxiliary lines O₁M and O₂N perpendicular to BC, with feet at points M and N, respectively. Draw OE perpendicular to BC, with foot at point E. Draw auxiliary lines connecting O₁E, O₁D, O₂D, and O₂E.", "ocr_zh": "在三角形ABC中作AD垂直于BC，垂足为点D。点O1、O、O2分别为BC边正下方从左到右三点，O2、O1、O到BC边的距离渐远。作辅助线O1M、O2N垂直于BC边，垂足分别为点M、N。作OE垂直于BC边，垂足为点E。作辅助线连接O1E、O1D、O2D、O2E。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4540.jpg" }, { "index": 4541, "ocr_en": "In isosceles triangle ABC, where AB equals AC, point M is the midpoint of side BC. Connect A to M. Draw a circle passing through points A, B, and M. Point X lies on the circle, and point T lies inside the circle. Connect CT, MX, and MT, with MX perpendicular to MT and foot at point M. Connect TX and BX, where TX equals BX. Connect B to T, and let point N be the midpoint of segment BT. Draw auxiliary lines NX and MN, with NX perpendicular to BT and foot at point N.", "ocr_zh": "在等腰三角形ABC中，AB等于AC，点M为BC边上的中点，连接AM。过点A、B、M作一个圆。点X在圆上，点T在圆内，连接CT、MX、MT，满足MX垂直于MT，垂足为点M。连接TX、BX，有TX等于BX。连接BT，点N为BT中点，作辅助线连接NX、MN，有NX垂直于BT垂足为点N。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4541.jpg" }, { "index": 4542, "ocr_en": "Circles S₁ and S₂ are tangent at point D, and TDT' is their internal common tangent. Points B and C lie on circles S₁ and S₂, respectively. Connect B to D and C to D. Points E and F are the midpoints of segments BD and CD, respectively. Draw auxiliary line BC. Point K lies inside circle S₁ . Connect BK, CK, EK, FK, and DK. Line EK is perpendicular to BD, with foot at point E. Point A lies outside circle S₁. Draw ray AB and connect A to C.", "ocr_zh": "圆S1和圆S2相切，切点为点D，TDT'是圆S1和圆S2的内公切线。点B、C分别为圆S1、S2上一点，连接BD、CD，点E、F分别为BD、CD中点。作辅助线连接BC，点K在圆S1内，连接BK、CK、EK、FK、DK，EK垂直于BD，垂足为点E。点A为圆S1外一点，作射线AB，连接AC。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4542.jpg" }, { "index": 4543, "ocr_en": "Triangles ABC and DEF share the same circumcircle. Line AC intersects DE at point H. Line BC intersects DE and DF at points G and I, respectively. Line AB intersects DF at point J. Points M and N are the midpoints of segments HG and IJ, respectively. Connect MN and AD. Line AD intersects EF and MN at points P and Q, respectively. Draw auxiliary lines NP and MP. Through point N, draw line BE; through point M, draw line CF. Point O lies inside the circle, below BC. Draw auxiliary lines NO and MO.", "ocr_zh": "三角形ABC和三角形DEF外接同一个圆，且AC交DE于点H，BC分别交DE、DF于点G、I，AB交DF于点J。点M、N分别为HG、IJ中点，连接MN、AD，AD分别交EF、MN于点P、Q。作辅助线连接NP、MP、过点N连接BE、过点M连接CF。点O为圆内BC下方一点，作辅助线连接NO、MO。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4543.jpg" }, { "index": 4544, "ocr_en": "Three circles O₁, O₂, and O₃ share two common intersection points A and B. Line l passes through point A and intersects circles O₁, O₂, and O₃ again at points H, Y, and Z, respectively. From each of the circles O₁, O₂, and O₃, draw perpendicular lines to l, with feet of the perpendiculars at points H₁, H₂, and H₃ on line l, respectively.", "ocr_zh": "三个圆O1、O2、O3有两个共同交点A、B。直线l为过点A一条直线，与圆O1、圆O2、圆O3的另一交点分别为点H、Y、Z。过圆O1、圆O2、圆O3分别作l的垂线交l于垂足点H1、H2、H3。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4544.jpg" }, { "index": 4545, "ocr_en": "Triangle ABC has circumcircle O'. Construct the perpendicular bisector of BC, line MP', which intersects circle O' at points P' and M, with P' and A on the same side of BC. Point P lies on side BC near point C. Below BC, draw circle O passing through point P such that it is tangent to circle O' at point Q. Draw auxiliary lines connecting A to Q, P to P', and O to Q. Connect O to P, A to P, and A to O, and extend AO to intersect circle O' at point M.", "ocr_zh": "三角形ABC外接圆为O'，作辅助线BC的中垂线MP'与圆O'交于点P'、M，其中P'、A在BC边同侧。点P为BC边上靠近点C侧的一点，在BC下方过点P作圆O与圆O'相切于点Q。作辅助线连接AQ、过点P连接P'Q，过点O连接O'Q，连接OP、AP、AO，并作辅助线延长AO交圆O'于点M。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4545.jpg" }, { "index": 4546, "ocr_en": "Below the circle lies a straight line l. Extend the diameter AB perpendicular to line l, with the foot of the perpendicular at point X. Point F is on line l to the right of X. Connect A to F, intersecting the circle at point G. Connect B to F and extend to intersect the circle at point E. Draw auxiliary line AE; AE is perpendicular to EF, with the foot at point E. Point D lies on line l between points X and F. Connect A to D, intersecting the circle at point C, then connect D to E. Draw auxiliary line CF and extend it to intersect the circle at point H. Draw auxiliary line HG.", "ocr_zh": "圆外下方有一直线l，延长直径AB垂直于直线l，垂足为点X。点F为直线l上点X右侧一点，连接AF交圆于点G，连接BF并延长交圆于点E。作辅助线连接AE，有AE垂直于EF，垂足为点E。点D为直线上点X、F中间一点，连接AD交圆于点C，连接DE。作辅助线连接CF并延长交圆于点H，作辅助线连接HG。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4546.jpg" }, { "index": 4547, "ocr_en": "Circles Г₁ and Г₂ intersect at two points, with the upper intersection point labeled C. Points N and A lie on circle Г₁ in the non-overlapping region, with N above A; points M and B lie on circle Г₂ in the non-overlapping region, with M above B. Draw auxiliary lines MN, CN, CM, and BC. Connect A to M, intersecting circles Г₁ and Г₂ at points F and K, respectively. Draw auxiliary lines NK, CK, and BK. Connect B to N, intersecting circles Г₁ and Г₂ at points L and E, respectively. Draw auxiliary lines KL, AL, CL, and LM. Through point E, draw line AC.", "ocr_zh": "圆Г1和Г2相交于两点，上方的交点为点C。点N、A在圆Г1上非重叠区域，点N在点A上方；点M、B在圆Г2上非重叠区域，点M在点B上方。作辅助线连接MN、CN、CM、BC。连接AM分别交圆Г1和Г2于点F、K，作辅助线连接NK、CK、BK；连接BN分别交圆Г1和Г2于点L、E，作辅助线连接KL、AL、CL、LM，过点E连接AC。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4547.jpg" }, { "index": 4548, "ocr_en": "Point D lies inside triangle ABC. Connect AD, BD, and CD, with ∠ABD = 60°, and ∠CAD = ∠ACD = 30°. Point M is the midpoint of side AC. Draw auxiliary line DM, which is perpendicular to AC with foot at M. Point N is the midpoint of CD, and point F lies on segment CM. Draw auxiliary line FN, which is perpendicular to CD with foot at N. Using auxiliary lines through points D, M, F, and N, construct a circle that intersects side BC at two points; the intersection closer to D is labeled point E. Connect DE and EF, and draw auxiliary lines EM and EN.", "ocr_zh": "点D在三角形ABC中，连接AD、BD、CD，有角ABD为60度，角CAD等于角ACD等于30度。点M为AC边中点，作辅助线连接DM，有DM垂直于AC边垂足为点M。点N为CD中点，点F为CM上一点，作辅助线连接FN，有FN垂直于CD垂足为点N。用辅助线过点D、M、F、N作一圆与BC边有两交点，靠近点D的交点为点E。连接DE、EF，作辅助线连接EM、EN。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4548.jpg" }, { "index": 4549, "ocr_en": "Quadrilateral ABCD has a circumcircle, and its incircle has center I. The incircle touches sides AB, BC, CD, and AD at points A₁, B₁, C₁, and D₁, respectively. Draw auxiliary lines connecting AI, A₁I, and D₁I. Connect A₁B₁, B₁C₁, C₁D₁, and A₁D₁. Points E, F, G, and H are the midpoints of sides A₁B₁, B₁C₁, C₁D₁, and A₁D₁, respectively. Connect EF, FG, GH, and EH. Point H lies on auxiliary line AI.", "ocr_zh": "四边形ABCD有外接圆，内切圆圆心为点I，圆I与AB、BC、CD、AD边分别相切于点A1、B1、C1、D1，作辅助线连接AI、A1I、D1I，连接A1B1、B1C1、C1D1、A1D1。点E、F、G、H分别为A1B1、B1C1、C1D1、A1D1边的中点，连接EF、FG、GH、EH，点H在辅助线AI上。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4549.jpg" }, { "index": 4550, "ocr_en": "In triangle ABC, points A₁, B₁, and C₁ lie on sides BC, AC, and AB, respectively. Connect AA₁, BB₁, and CC₁; these three lines intersect at a common point. Construct C₁B₂ perpendicular to AC with foot at B₂, C₁N perpendicular to AA₁ with foot at N, C₁M perpendicular to BB₁ with foot at M, and C₁A₂ perpendicular to BC with foot at A₂. Points B₂, N, M, and A₂ lie on the same straight line.", "ocr_zh": "在三角形ABC中，点A1、B1、C1分别为边BC、AC、AB上一点，连接AA1、BB1、CC1，三条线有同一个交点。作C1B2垂直于AC垂足为点B2，作C1N垂直于AA1垂足为点N，作C1M垂直于BB1垂足为点M，作C1A2垂直于BC垂足为点A2，点B2、N、M、A2在一条直线上。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4550.jpg" }, { "index": 4551, "ocr_en": "Triangles ABC and DEF overlap at a certain angle, sharing the same incircle with center I and the same circumcircle with center O. The vertices are arranged on circle O in the order D, A, F, C, E, B. Connect I to O and extend both ends to intersect circle O at points M and N, with M to the left of N. Draw auxiliary line AI and extend it to intersect circle O at point K. Draw auxiliary line BK. Draw auxiliary line DI and extend it to intersect circle O at point P.", "ocr_zh": "三角形ABC和三角形DEF以一定角度重叠，有同一内切圆圆I，有同一外接圆圆O，各个顶点以DAFCEB的顺序分布在圆O上。连接IO并向两端延长交圆O于M、N两点，点M在点N左侧。作辅助线连接AI并延长交圆O于点K，作辅助线连接BK。作辅助线连接DI并延长交圆O于点P。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4551.jpg" }, { "index": 4552, "ocr_en": "Construct circle O as the circumcircle of triangle ABC. Through point O, draw FM perpendicular to BC, with foot at point M. Extend FM to intersect circle O at point D. Draw auxiliary line AD, intersecting BC at point E. Connect F to A and extend this line to meet the extension of BC at point P. Draw KM perpendicular to AF, with foot at point K.", "ocr_zh": "用辅助线作圆O为三角形ABC的外接圆。过点O作FM垂直于BC垂足为点M，作辅助线延长FM交圆O于点D，作辅助线连接AD交BC于点E。连接FA并延长交BC延长线于点P。作KM垂直于AF，垂足为点K。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4552.jpg" }, { "index": 4553, "ocr_en": "Triangle ABC has a circumcircle. Point D is the midpoint of side BC. Points E and F lie on sides AC and AB, respectively, with AE equal in length to AF. Connect E to F. Point H lies on segment EF. Connect D to H and extend DH beyond H to intersect the circumcircle at point P. Draw auxiliary lines connecting B to P, F to P, C to P, E to P, B to H, and C to H. Extend segment HD toward D to intersect the circumcircle at point M. Draw auxiliary lines connecting B to M and C to M.", "ocr_zh": "三角形ABC有外接圆，点D为BC边上的中点，点E、F分别为AC、AB边上一点，且AE长度等于AE长度，连接EF。点H为EF上一点，连接DH并向点H方向延长交圆于点P，作辅助线连接BP、FP、CP、EP、BH、CH。作辅助线向点D方向延长HD交圆于点M，作辅助线连接BM、CM。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4553.jpg" }, { "index": 4554, "ocr_en": "Circle O is the circumcircle of triangle A₁A₂C, with point Q on arc A₁A₂. Circle S₁, centered at O₁, intersects circle O at points A₁ and Q; circle S₂, centered at O₂, intersects circle O at points A₂ and Q. Circles S₁ and S₂ intersect at points P and Q. Connect segments PQ, OO₁, OO₂, and O₁O₂. Draw auxiliary lines O₁Q, A₁Q, O₂Q, and A₂Q. Circle S₁ intersects segment A₁C at points A₁ and B₁. Connect B₁ to P and extend to intersect circle S₂ at point B₂. Through points O₁ and Q, draw a circle tangent internally to circle O.", "ocr_zh": "圆O为三角形A1A2C的外接圆，点Q在圆弧A1A2上。圆S1圆心为点O1，圆S1与圆O相交于点A1和点Q；圆S2圆心为点O2，圆S2与圆O相交于点A2和点Q；圆S1和圆S2相交于PQ两点。连接PQ、OO1、OO2、O1O2，作辅助线连接O1Q、A1Q、O2Q、A2Q。圆S1交A1C于点A1、B1，连接B1P并延长交圆S2于点B2。过点O1、Q作以圆与圆O内切。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4554.jpg" }, { "index": 4555, "ocr_en": "Circle O is the circumcircle of quadrilateral ABCD. Extend side AB to meet the extension of side CD at point E, and extend side AD to meet the extension of side BC at point F. Connect points E and F. Point G lies on segment EF; draw auxiliary line CG. Points P and Q lie on the circumcircle of triangle AEF. Connect E to P and F to Q.", "ocr_zh": "圆O为四边形ABCD的外接圆，延长AB交CD的延长线于点E，延长AD交BC的延长线于点F，连接EF。点G为EF上一点，作辅助线连接CG。点P、Q为三角形AEF外圆O上两点，连接EP、FQ。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4555.jpg" }, { "index": 4556, "ocr_en": "Two circles intersect at points X and Y, and line XY passes through both points. A horizontal line ABZCD crosses the left circle at points A and C, the right circle at points B and D, and intersects line XY at point Z, from left to right. Points M and N lie on the left and right circles respectively. Connecting C to M and B to N, the lines intersect at point P, which lies on line XY. Connect A to M and extend to intersect line XY at point Q; connect D to N and extend to intersect line XY at point Q'. Points Q and Q' coincide.", "ocr_zh": "两圆相交于X、Y两点，过X、Y作直线XY。直线ABZCD水平，从左至右依次交左圆于A、C两点，交右圆于BD两点，交直线XY于点Z。点M、N分别为左圆和右圆上一点，连接CM、BN交于点P，点P在直线XY上。连接AM并延长交直线XY于点Q，连接DN并延长交直线XY于点Q'，点Q和点Q'重合。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4556.jpg" }, { "index": 4557, "ocr_en": "Points E and F lie on circle T, and line EF passes through the center T. Points M and N lie on the circle, with M above EF and N below EF. Connect M and N; line MN does not pass through center T and intersects EF at point D. Points B and C lie on arcs EM and FM respectively. Draw auxiliary lines BC, BE, BD, BF, CE, CD, CN, and CF. Through point B, construct circle A tangent to circle T and line MN. Through point C, construct circle T₂ tangent to circle T and line MN, with circles A and T₂ tangent to each other.", "ocr_zh": "点E、F为圆T上两点，作辅助线连接EF，EF经过圆心点T。M、N为圆上两点，点M在EF上方，点N在EF下方，连接MN，不经过圆心点T，MN与EF交于点D。点B、C分别为圆弧EM和圆弧FM上一点，作辅助线连接BC、BE、BD、BF、CE、CD、CN、CF。过点B作圆A与圆T和MN相切，过点C作圆T2与圆T和MN相切，且圆A与圆T2相切。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4557.jpg" }, { "index": 4558, "ocr_en": "Circle O is inside arc AB and tangent to it at points C and D. Draw auxiliary lines connecting CD, AC, and BC. From point C, draw tangent CE to the arc, intersecting the extended line of AB at point E; EC is an auxiliary ray.", "ocr_zh": "圆O在弓形AB内，且与弓形AB相切于点C、D，作辅助线连接CD、AC、BC。用辅助线过点C作切线CE与弓形相切交AB的辅助延长线于点E，EC为辅助线射线。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4558.jpg" }, { "index": 4559, "ocr_en": "Small circle G1 and large circle G2 intersect at points M and N, with M above N. G1 lies to the left of G2. Draw line MN through points M and N. Draw line CD through point M, where point C lies on G1 and point D on G2. Draw the common tangent line l to G1 and G2, touching them at points A and B respectively, such that line l is parallel to line CD. Let line l intersect line MN at point K. Connect AM and BM. Connect AN and extend it to intersect line CD at point P. Connect BN and extend it to intersect line CD at point Q. Extend lines AC and BD to intersect at point E. Connect EP, EM, and EQ.", "ocr_zh": "小圆Г1和大圆Г2相交于M、N两点，点M在点N上方，小圆在大圆左侧，过点M、N作直线MN。过点M作直线CD，其中点C在圆Г1上，点D在圆Г2上。作圆Г1和圆Г2的公共切线l，在圆Г1和圆Г2上的切点分别为A、B，且直线l与直线CD平行，直线l与直线MN交于点K。连接AM、BM，连接AN交直线CD于点P，连接BN交直线CD于点Q。作直线AC、BD相交于点E，连接EP、EM、EQ。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4559.jpg" }, { "index": 4560, "ocr_en": "Triangle BMM' has a circumcircle, with BB' as its diameter. BB' intersects MM' at point A. Point N' lies on segment BM'. Connect AN', and AN' is perpendicular to BB', with the foot of the perpendicular at point A. Extend ray N'A to intersect the extension of line BM at point N.", "ocr_zh": "三角形BMM'有外接圆，BB'为外接圆直径，BB'与MM'相交于点A。点N'在BM'上，连接AN'，有AN'垂直于BB'，垂足为点A。延长射线N'A与BM延长线交于点N。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4560.jpg" }, { "index": 4561, "ocr_en": "In convex quadrilateral ABCD, connect diagonal AC. Draw a small circle tangent to sides AB and BC at points G and H respectively; the small circle intersects AC at points E and F. Draw a large circle tangent to the extensions of sides DA and DC at points J and K respectively; the large circle also intersects AC at points E and F.", "ocr_zh": "在凸四边形ABCD中，连接对角线AC，作一小圆与AB、BC边分别相切于点G、H，小圆与AC相交于点E、F。作一大圆与DA、DC延长线相切，切点分别为点J、K，大圆与AC交于E、F两点。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4561.jpg" }, { "index": 4562, "ocr_en": "In triangle ABC, draw ray CM perpendicular to AB with foot M, and draw BN perpendicular to AC with foot N. BN and CM intersect at point H. Connect points M and N. Let E and F be the midpoints of sides AB and AC, respectively. Connect EF and let it intersect MN at point P, then connect AP. Let O be the circumcenter of triangle ABC. Connect HO, and draw auxiliary lines AO and AH. Using AO and AH as diameters, construct circles with centers at points O′ and H′, respectively. Connect H′ and O′ with an auxiliary line.", "ocr_zh": "在三角形ABC中，作射线CM垂直于AB垂足为点M，作BN垂直于AC垂足为点N，BN交CM于点H，连接MN。点E、F分别为AB、AC边上的中点，连接EF交MN于点P，连接AP。点O为三角形ABC的外心，连接HO，作辅助线连接AO、AH。分别用辅助线以AO、AH为直径作圆，圆心分别为点O'、H'，作辅助线连接H'O'。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4562.jpg" }, { "index": 4563, "ocr_en": "Between two parallel lines, a large circle O1 and a small circle O2 are tangent to each other at point C. Circle O1 is tangent to the upper line at point A, and circle O2 is tangent to the lower line at point B. Connect AO1, O1O2, and BO2. From point C, draw a line connecting to AB.", "ocr_zh": "在两平行线之间大圆O1与小圆O2相切于点C，圆O1与上方线相切于点A，圆O2与下方线相切于点B。连接AO1、O1O2、BO2，过点C连接AB。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4563.jpg" }, { "index": 4564, "ocr_en": "Between two parallel lines l1 and l2, there are three circles O, O1, and O2, with the diameter of circle O greater than that of O2, which is greater than that of O1. Circle O is tangent to l1 and l2 at points G and F respectively. Circle O1 is tangent to l1 at point A and tangent to circle O at point C. Draw an auxiliary line from point C connecting to AF. Circle O2 is tangent to l2 at point B and tangent to circles O and O1 at points D and E respectively. Connect AD and BC; they intersect at point Q. Draw auxiliary lines CE and EQ, and from point E, draw a line connecting to AB.", "ocr_zh": "在平行线l1、l2之间有三个圆O、O1、O2，且圆O直径大于圆O2大于圆O1。圆O与l1、l2分别相切于点G、F；圆O1与l1相切于点A，与圆O相切于点C，作辅助线过点C连接AF；圆O2与l2相切于点B，与圆O、O1分别相切于点D、E，连接AD、BC相交于点Q。作辅助线连接CE、EQ，过点E连接AB。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4564.jpg" }, { "index": 4565, "ocr_en": "Circles A and B intersect, and line AB is drawn with point M as the midpoint of AB. Circles C and D are two smaller circles inside circle A, each tangent internally to circle A and externally to circle B. Circle C is tangent to circles A and B at points S and T respectively. Circles C and D intersect at points P and Q. Extend line PQ to intersect AB at point M. Draw an auxiliary line connecting A to C and extend it to intersect circle C at point S. From point T, draw a line connecting to B. Connect point C to M.", "ocr_zh": "圆A与圆B相交，连接AB，点M为AB中点。圆C、圆D为圆A内两个小圆，同时与圆A内切且与圆B外切，圆C与圆A、圆B的切点分别为点S、T，圆C和圆D相交于PQ两点。延长PQ与AB交于点M。作辅助线连接AC并延长交圆C于点S，过点T连接BC，连接CM。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4565.jpg" }, { "index": 4566, "ocr_en": "In quadrilateral ABCD, connect diagonals AC and BD, which intersect at point O. Using AB as the diameter, draw auxiliary circle S intersecting AC at point F and BD at point E. Connect AE and BF; they intersect at point H1. Points M and N are the midpoints of sides AD and BC, respectively. Using AD and BC as diameters, draw auxiliary circles M and N. Circles M and N intersect circle S at points E and F, respectively. Point H2 is inside quadrilateral ABCD and is symmetric to point H1 about point O.", "ocr_zh": "在四边形ABCD中，连接对角线AC、BD交于点O。以AB为直径用辅助线作圆S交AC于点F，交BD于点E，作辅助线连接AE、BF交于点H1。点M、N分别为AD、BC中点，分别以AD、BC为直径用辅助线作圆M、圆N。圆M和圆N分别与圆S交于点E、F。点H2为四边形ABCD内关于点O与H1对称的一点。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4566.jpg" }, { "index": 4567, "ocr_en": "In triangle ABC, points Ao and Bo are the midpoints of sides AC and BC, respectively. Connect AoBo. Points O and H are two points inside the triangle ABC. Connect AH and extend it to intersect BC at point A1. Connect BH and extend it to intersect AC at point B1. Line BB1 is perpendicular to AC with foot at B1. Connect A1B1, which intersects AoBo at point C′. Using auxiliary lines, construct three circles passing through points A1, B1, C, H; points Ao, Bo, C, O; and points Ao, A1, Bo, B1, respectively.", "ocr_zh": "在三角形ABC中，点Ao、Bo分别为边BC、AC的中点，连接AoBo。点O和点H为三角形ABC内两点，连接AH并延长交BC边于点A1，连接BH并延长交AC边于点B1，有BB1垂直于AC垂足为点B1，连接A1B1交AoBo于点C'。用辅助线过点A1B1CH、点AoBoCO、点AoA1BoB1分别作一个圆。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4567.jpg" }, { "index": 4568, "ocr_en": "Triangle ABC has a circumcircle. Extend BA and BC beyond the circle to points X and P, respectively. Connect PX, with PX parallel to AC. Draw auxiliary line CM perpendicular to PX, with foot M. Extend MC to intersect the circle at point T. Connect PT. Draw auxiliary line BT and extend it to point N. Draw auxiliary line PN, which is perpendicular to BN with foot at N. Extend AC to intersect PN at point Y. Connect XY.", "ocr_zh": "三角形ABC有外接圆，延长BA、BC至圆外点X、P，连接PX，有PX平行于AC。用辅助线作CM垂直于PX垂足为点M。用辅助线延长MC交圆于点T，连接PT。用辅助线连接BT并延长至点N，辅助线连接PN，有PN垂直于BN垂足为点N。延长AC交PN于点Y，连接XY。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4568.jpg" }, { "index": 4569, "ocr_en": "Extend side BC of triangle ABC in both directions to points H2 and H3, with H2 near point C and H3 near point B, such that CH2 equals BH3. Above side BC outside the triangle are points I2 and I3. Connect I2H2 and I3H3, where I2H2 is perpendicular to CH2 with foot at H2, and I3H3 is perpendicular to BH3 with foot at H3. Connect I3B and I2C, extending them to intersect at point I1. Connect I2I3, with point A lying on line I2I3. Let M be the midpoint of BC. Connect AM. Points I and T lie inside triangles ABM and ACM respectively. Connect AI, IT, and MT. Line IT intersects AM at point G.", "ocr_zh": "用辅助线向两端延长三角形ABC的BC边至点H2、H3，点H2靠近点C，点H3靠近点B，满足CH2等于BH3。在三角形外BC边上方有点I2、I3，连接I2H2、I3H3，有I2H2垂直于CH2垂足为点H2，I3H3垂直于BH3垂足为点H3。用辅助线连接I3B、I2C并延长相交于点I1。用辅助线连接I2I3，点A在I2I3上。点M为BC中点，用辅助线连接AM，点I、点T分别为三角形ABM、三角形ACM中一点，用辅助线连接AI、IT、MT，IT交AM于点G。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4569.jpg" }, { "index": 4570, "ocr_en": "Using auxiliary lines, the circumcircles of triangles APS and BRS intersect at points S and K. Connect AB; point K lies on AB. Draw circle O with center O passing through point P and R and S. Extend lines AP and BR to intersect circle O at point Q. Draw auxiliary line OK.", "ocr_zh": "用辅助线作三角形APS和三角形BRS的外接圆相交于点S、K，连接AB，点K在AB上。过点PRS作圆O圆心为点O，延长AP、BR相交于圆O上一点Q。作辅助线连接OK。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4570.jpg" }, { "index": 4571, "ocr_en": "Rays AB and AC form an acute angle. Circle O is tangent to AB and AC at points B and C respectively. Draw auxiliary lines connecting AO and BC, where AO is perpendicular to BC with foot at point N. Points M and P are inside the circle and symmetric with respect to point O. Connect BM, MO, and PO. Draw ray AM intersecting circle O at points D and E. Draw auxiliary line PR perpendicular to ray AC with foot at point R. Extend BC and PR to intersect at point S.", "ocr_zh": "射线AB、AC呈锐角夹角，圆O与AB、AC相切，切点分别为点B、C。作辅助线连接AO、BC，有AO垂直于BC垂足为点N。点M、P为圆内与点O对称两点，连接BM、MO、PO。作射线AM分别交圆O于点D、E。用辅助线作PR垂直于射线AC垂足为点R，作辅助线延长BC、PR相交于点S。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4571.jpg" }, { "index": 4572, "ocr_en": "In right triangle OQR, draw circle O centered at point O with radius OQ, intersecting line RO and its extension at points A and B. Draw circle Q centered at point Q, tangent to line RO at point H. Circles Q and O intersect at points C and D, where C lies inside triangle OQR and D lies outside. Connect OQ and HQ. Connect line DC and extend it to intersect RO at point P. Lines DP and HQ intersect at point M. Extend line QH in both directions as an auxiliary line, intersecting circles Q and O at points S and T respectively.", "ocr_zh": "在直角三角形OQR以点O为圆心，OQ为半径作圆O交RO及其延长线于点A、B。以点Q为圆心作圆Q与RO相切于点H，圆Q与圆O相交于点C、D，其中点C位于三角形OQR内，点D位于三角形OQR外。连接OQ、HQ，连接DC并延长交RO于点P，DP与HQ相交于点M。作辅助线向两端延长QH与圆Q和圆O分别相交于点S和点T。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4572.jpg" }, { "index": 4573, "ocr_en": "In convex quadrilateral ABCD, diagonals AC and BD intersect at point O. In triangle AOD, draw perpendiculars AA1 to DO and DD1 to AO, intersecting at H2. In triangle BOC, draw perpendiculars BB1 to CO and CC1 to BO, intersecting at H1. Connect H1 and H2. Let E and F be the midpoints of AB and CD.The straight line M1M2 intersects with BO, H1H2 and CO.", "ocr_zh": "在凸四边形ABCD中连接对角线AC、BD相交于点O。在三角形AOD中，用辅助线作AA1垂直于DO垂足为点A1，作DD1垂直于AO垂足为D1，AA1与DD1相交于点H2。在三角形BOC中用，辅助线作BB1垂直于CO垂足为B1，作CC1垂直于BO垂足为C1，BB1与CC1相交于点H1，连接H1H2。点E、F分别为AB、CD中点。直线M1M2与BO、H1H2、CO相交。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4573.jpg" }, { "index": 4574, "ocr_en": "Point C lies between rays OA and OB. Connect OC. Draw CD perpendicular to OA with foot D, and CE perpendicular to OB with foot E. Draw EM perpendicular to OA with foot M, and DN perpendicular to OB with foot N. Connect MN; MN is perpendicular to OC with foot H.", "ocr_zh": "点C为射线OA、OB夹角之间一点，连接OC，作CD垂直于OA垂足为点D，作CE垂直于OB垂足为点E。作EM垂直于OA垂足为点M，作DN垂直于OB垂足为点N，连接MN，MN垂直于OC垂足为点H。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4574.jpg" }, { "index": 4575, "ocr_en": "Right triangles BCD and ABE intersect at point B. Angle ABC is obtuse, and angle DBE is acute. Connect DE. In the right triangles, BC is perpendicular to CD with foot C, and AB is perpendicular to AE with foot A. Draw CF perpendicular to BD with foot F, and AG perpendicular to BE with foot G. Extend CF and AG to intersect at point P. Connect BP, then extend BP to intersect DE at point H. Point O lies on line BP.", "ocr_zh": "直角三角形BCD和直角三角形ABE相交于点B，角ABC为钝角，角DBE为锐角，连接DE。直角三角形中，BC垂直于CD垂足为点C，AB垂直于AE垂足为点A。作CF垂直于BD垂足为点F，作AG垂直于BE垂足为点G，延长CF、AG相交于点P。连接BP，并作辅助线延长BP交DE于点H，点O为BP上一点。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4575.jpg" }, { "index": 4576, "ocr_en": "Point O is the circumcenter of triangle ABC. Connect BO and CO. Point T is inside the triangle BOC. Draw auxiliary line OT, intersecting BC at point M. Points D and E lie on sides AB and AC respectively, with DE parallel to BC. Line DE intersects OB and OC each at one point. Connect AT, intersecting DE at point H. Connect DM and EM. Draw an auxiliary line to extend DM to meet the extension of AC at point G, and draw another auxiliary line to extend EM to meet the extension of AB at point F. Draw auxiliary lines FT, GT, and BT.", "ocr_zh": "点O为三角形ABC内一点，连接BO、CO。点T为三角形BOC内一点，作辅助线连接OT交BC于点M。点D、E分别为AB、AC上一点，连接DE，有DE平行于BC，且DE与OB、OC分别有一个交点，连接AT交DE于点H，连接DM、EM。作辅助线延长DM交AC延长线于点G，作辅助线延长EM交AB延长线于点F，作辅助线连接FT、GT、BT。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4576.jpg" }, { "index": 4577, "ocr_en": "In pentagon ABCQD, connect AQ, AC, BD, BQ, and CD. Diagonals AC and BD intersect at point P. Draw an auxiliary arc passing through points A, D, and Q. Connect QP and extend it to intersect the arc at point Q′.", "ocr_zh": "在五边形ABCQD中，连接AQ、AC、BD、BQ、CD，AC交BD于点P。用辅助线过点A、D、Q作圆弧，连接QP并延长交圆弧于点Q'。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4577.jpg" }, { "index": 4578, "ocr_en": "Circle O1 lies inside circle O1′ and is tangent to it internally at point A. Point T1 is on circle O1. Connect AO1 and O1T1. Draw auxiliary lines O1O1′ and AT1, extending AT1 to intersect circle O1′ at point T1′. Circle O1 has a tangent line l1 passing through point A. Circle O2 intersects circle O1 at points A and B, and intersects circle O1′ at points A and B1. Point T2 lies on circle O2. Connect AO2 and O2T2. Draw the tangent to circle O2 at point T2, intersecting line l1 at point M2. Line l2 passes through point A and is tangent to circle O2. Tangents to circles O1 and O1′ at points T1 and T1′ intersect line l2 at points M1 and M1′ respectively.", "ocr_zh": "圆O1在圆O1'内，并与圆O1'内切，切点为点A，点T1为圆O1上一点，连接AO1、O1T1。作辅助线连接O1O1'、AT1，并延长AT1交圆O1'于点T1'。圆O1有一过点A的切线l1。圆O2与圆O1交于点A、B，与圆O1'交于点A、B1。T2为圆O2上一点，连接AO2、O2T2。过点T2作圆O2的切线与直线l1相交于点M2。直线l2过点A与圆O2相切。过点T1、T1'分别作圆O1、O1'的切线与直线l2分别相交于点M1、M1'。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4578.jpg" }, { "index": 4579, "ocr_en": "In parallelogram ABCD, there is a point O. Connect AO, BO, CO, and DO, with angle ODC equal to angle OBC. Draw a circle passing through points A, B, and O. Draw auxiliary diameter OO′ of this circle, and connect AO′ and BO′ with auxiliary lines.", "ocr_zh": "平行四边形ABCD中有一点O，连接AO、BO、CO、DO，满足角ODC等于角OBC。过点A、B、O作圆，用辅助线作直径OO'，作辅助线连接AO'、BO'。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4579.jpg" }, { "index": 4580, "ocr_en": "Triangle ABC has circumcircle O. A similar transformation maps triangle ABC to triangle A'B'C', with points A', B', and C' lying on sides BC, AC, and AB respectively. Circle N is the circumcircle of triangle A'B'C'. Draw auxiliary lines connecting A'B', A'C', B'C', and A'N. Point Pa lies outside circle O. Connect PaA' and AO, extending them to intersect at point Qa. From point Pa, draw the tangent PaA'' to circle O, touching at point A''. ", "ocr_zh": "三角形ABC的外接圆为圆O，将三角形ABC进行位似变化得到三角形A'B'C'，点A'、B'、C'分别位于边BC、AC、AB上，作辅助线连接A'B'、A'C'、B'C'，圆N为三角形A'B'C'的外接圆，作辅助线连接A'N。点Pa为圆O外一点，连接PaA'、AO并延长交于点Qa，过点Pa作切线PaA''与圆O相切，切点为点A''。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4580.jpg" }, { "index": 4581, "ocr_en": "In square ABCD, connect diagonal BD. Points E and F lie on sides BC and CD respectively, with BE equal to DF, then connect EF. Connect AE and AF, intersecting BD at points P and Q respectively. Draw auxiliary lines EQ and FP. Extend side CD to point E′ such that DE′ equals DF. Connect AE′ with an auxiliary line.", "ocr_zh": "在正方形ABCD中，连接对角线BD。点E、F分别为边BC、CD上一点，且有BE等于DF，连接EF，连接AE、AF分别与BD交于点P、Q。作辅助线连接EQ、FP。作辅助线延长CD至点E'，DE'等于DF，作辅助线连接AE'。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4581.jpg" }, { "index": 4582, "ocr_en": "Quadrilateral ABCD is cyclic. Connect diagonal BD, with BD equal to AD. Point Q lies inside triangle ABD. Connect DQ and extend it to intersect AB at point P. Connect AQ and extend it to intersect BD at point R. Connect BQ and extend it to intersect AD at point H. Draw auxiliary line HR. Extend DA to point D′ such that H is the midpoint of DD′. Draw auxiliary line BD′.", "ocr_zh": "四边形ABCD为圆内接四边形，连接对角线BD，有BD等于AD。点Q为三角形ABD内一点，连接DQ并延长交AB于点P，连接AQ并延长交BD于点R，连接BQ并用辅助线延长交AD于点H，用辅助线连接HR。用辅助线延长DA至点D'，有点H为DD'中点，作辅助线连接BD'。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4582.jpg" }, { "index": 4583, "ocr_en": "In quadrilateral ABCD, connect diagonals AC and BD. Point K lies on BD near point B. Connect AK. Triangles ABK and ADC are similar. Points I′ and I lie inside triangles ABK and ADC respectively. Connect II′, intersecting BD at point X. Draw auxiliary lines AI′ and AI. Using points A, X, I, and D, draw an auxiliary arc.", "ocr_zh": "四边形ABCD中，连接对角线AC、BD。点K为BD上靠近点B的一点，连接AK，有三角形ABK与三角形ADC相似。在三角形ABK和三角形ADC中分别有点I'、I，连接II'与BD相交于点X，作辅助线连接AI'、AI。过点A、X、I、D用辅助线作圆弧。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4583.jpg" }, { "index": 4584, "ocr_en": "Square ABCD is folded along line l passing through sides AB and CD, line l is a dashed line. . After folding, point D maps to D′ on side BC, and point A maps to A′ outside the square. Segment A′D′ intersects AB at point E. Using auxiliary lines, complete the folded squares ABCD and A′B′C′D′. Point D lies on side B′C′. Triangle BD′E has its largest inscribed circle.", "ocr_zh": "将正方形ABCD沿过AB、CD边的直线l折叠，直线l为虚线。折叠后点D对应的D'落在BC边上，点A对应的A'落在正方形外，A'D'与AB相交于点E。用辅助线分别补全折叠后的正方形ABCD和正方形A'B'C'D'。点D在边B'C'上。三角形BD'E中有一最大内切圆。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4584.jpg" }, { "index": 4585, "ocr_en": "In an isosceles acute triangle ABC with AB equal to AC and angle A equal to 20 degrees, point D lies on AB. Connect CD, where angle BCD is 50 degrees. Point E lies on AC. Connect DE and BE, where angle CBE is 60 degrees. Point F lies on CE. Draw auxiliary lines connecting DF and BF. Through point B, draw an auxiliary line intersecting segment EF.", "ocr_zh": "在等腰锐角三角形ABC中，AB等于AC，角A等于20度。点D为AB上一点，连接CD，有角BCD等于50度。点E为AC上一点，连接DE、BE，有角CBE等于60度。点F为CE上一点，作辅助线连接DF、BF。用辅助线过点B作与EF相交的线段。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4585.jpg" }, { "index": 4586, "ocr_en": "In triangle ABC, extend line BA to point C1 and connect CC1. Point E lies on CC1; connect AE. Take point C2 and connect CC2. Draw auxiliary line BC2 such that triangle ACC1 is congruent to triangle BCC2. Point D lies on CC2; connect BD, with AE equal to BD. Point F lies on AB; draw auxiliary lines EF, DF, and DE. Point O lies outside triangle ABC above side AB. Draw auxiliary lines OA, OB, OC, OC1,OC2 and OD, satisfying OC equals OC1 equals OC2 and OA equals OB.", "ocr_zh": "在三角形ABC中，作辅助线延长BA至点C1，连接CC1。点E为CC1上一点，连接AE。取点C2，连接CC2，作辅助线连接BC2，满足三角形ACC1全等于三角形BCC2，点D为CC2上一点，连接BD，有AE等于BD。点F在AB上，作辅助线连接EF、DF、DE。点O在三角形ABC外AB边上方，作辅助线连接OA、OB、OC、OC1、OC2、OD，满足OC等于OC1等于OC2，OA等于OB。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4586.jpg" }, { "index": 4587, "ocr_en": "In triangle APD, point M is the midpoint of AD. Draw auxiliary line PM and extend it to point P′, with PM equal to P′M. Connect AP′ with an auxiliary line. Points B and C lie on AM and DM respectively. Connect PC and PB and draw auxiliary line P′B, with PC equal to P′B.", "ocr_zh": "在三角形APD中，点M为AD中点。作辅助线连接PM并延长至点P'，有PM等于P'M，作辅助线连接AP'。点B、C分别为AM、DM上一点，连接PB、PC，作辅助线连接P'B，满足PC等于P'B。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4587.jpg" }, { "index": 4588, "ocr_en": "Circle O is the circumcircle of quadrilateral ABTC. Connect diagonals BC and AT. Through point T, draw the incircle O1 of circle O, touching at point T. Circle O1 intersects AT, BT, and CT at points A1, B1, and C1 respectively. Connect A and B1.", "ocr_zh": "圆O为四边形ABTC的外接圆，连接对角线BC、AT。过点T作圆O的内切圆O1，切点为点T，圆O1分别交AT、BT、CT于点A1、B1、C1，连接AB1。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4588.jpg" }, { "index": 4589, "ocr_en": "Triangle ABC has a circumcircle, and point D lies on the arc BC of the circle. Lines BD and CD are drawn and extended to points E and F, respectively, and line EF is connected. Let point M be the midpoint of EF, and connect BM and CM. Reflect point B over point M to obtain point B', and reflect point C over point M to obtain point C'. Then draw the auxiliary lines B'M, C'M, B'F, C'E, BC', B'C', and B'C.", "ocr_zh": "三角形ABC有外接圆，点D为圆弧BC上一点，连接BD、CD并分别延长至点E、F，连接EF。点M为EF中点，连接BM、CM。将点B、C分别关于点M对称得到点B'、C'，作辅助线连接B'M、C'M、B'F、C'E、BC'、B'C'、B'C。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4589.jpg" }, { "index": 4590, "ocr_en": "In hexagon ABCDEF, diagonal CF is drawn and auxiliary line BE is connected. Point G is located within quadrilateral ABCF, and point H is the reflection of point G across CF and lies within quadrilateral CDEF. Lines AG, BG, DH, and EH are drawn. Triangle ABG is rotated counterclockwise by 60 degrees around point A to form the auxiliary line triangle AMG; triangle DEH is rotated clockwise by 60 degrees around point E to form the auxiliary line triangle ENH'. Auxiliary lines BM, DN, MN, GM, and HN are drawn.", "ocr_zh": "在六边形ABCDEF中，连接对角线CF，作辅助线连接BE。点G在四边形ABCF内，点H与点G关于CF对称位于四边形CDEF内，连接AG、BG、DH、EH。将三角形ABG绕点A逆时针旋转60度，得到用辅助线作的三角形AMG；将三角形DEH绕点E顺时针旋转60度，得到用辅助线作的三角形ENH'。用辅助线连接BM、DN、MN、GM、HN。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4590.jpg" }, { "index": 4591, "ocr_en": "Circle W and circle W1 are externally tangent and their centers are O and O1 respectively. Draw an auxiliary line connecting OO1 passing through the point of tangency. Circle O is the circumcircle of triangle ABC. Extend line CA beyond point O₁ as an auxiliary line, and draw auxiliary lines connecting OA, OB, and OC. From points A, B, and C, draw tangents AA₁, BB₁, and CC₁ to circle W₁, with points A₁, B₁, and C₁ being the respective points of tangency. Draw the auxiliary line O₁A₁.", "ocr_zh": "圆W和圆W1外切且圆心分别为O、O1，用辅助线连接OO1过切点。圆O为三角形ABC的外接圆，作辅助线延长CA过点O1，作辅助线连接OA、OB、OC。分别过点A、B、C作圆W1的切线AA1、BB1、CC1，切点分别为点A1、B1、C1，作辅助线连接O1A1。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4591.jpg" }, { "index": 4592, "ocr_en": "Below vertex F of rectangle EFGH, there is a line segment BC, with the same length as side EF. Point B is near vertex E, and point C is near vertex G. Connect lines CH and BG; they intersect at point A. Line BG intersects EF at point K. Draw auxiliary lines AF and CF. Draw perpendiculars EE′ and FF′ from points E and F to line BC, with feet of the perpendiculars at points E′ and F′ respectively. Point E′ lies to the right of point B, and point F′ lies to the left of point C. Point D is a point between E′ and F′. Connect line AD, and let point O lie on segment AD.", "ocr_zh": "在矩形EFGH的顶点F正下方有一线段BC，且BC边长与EF边长相等，点B靠近正方形顶点E，点C靠近顶点G。连接CH、BG相交于点A，BG交EF于点K，作辅助线连接AF、CF。用辅助线作EE'、FF'垂直于BC垂足分别为点E'、F'，点E'在点B右侧，点F'在点C左侧。点D为E'F'之间一点，连接AD，点O在AD上。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4592.jpg" }, { "index": 4593, "ocr_en": "Circle O is the circumcircle of triangle ABC, with center at point O. Connect lines OA and OB. Point P lies on the arc AB of the circle. Draw the auxiliary line OP. Points R and T lie on sides AC and BC, respectively. Connect PR and PT, which intersect AB at points Q and S, respectively. PR and PT are perpendicular to OA and OB, with feet of the perpendiculars at points G and H, respectively.", "ocr_zh": "圆O为三角形ABC外接圆圆心为点O，连接OA、OB。点P为圆弧AB上一点，用辅助线连接OP。点R、T分别在AC、BC上，连接PR、PT分别交AB于点Q、S，且PR、PT分别垂直于OA、OB垂足为点G、H。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4593.jpg" }, { "index": 4594, "ocr_en": "The circumcenter of triangle ABC is point O. Draw a perpendicular from point A to line BC, with foot of the perpendicular at point P. Connect lines OC and OP. Extend lines CO and AP; they intersect the circumcircle (circle O) at points D and F, respectively. Draw the auxiliary line BD. Also, connect AO and extend it to intersect the circle at point E. Then draw the auxiliary line EF.", "ocr_zh": "三角形ABC外接圆圆心为点O。作AP垂直于BC垂足为点P，连接OC、OP。作辅助线延长CO、AP分别交圆O于点D、F，作辅助线连接BD。作辅助线连接AO并延长交圆O于点E，作辅助线连接EF。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4594.jpg" }, { "index": 4595, "ocr_en": "Circle O is internally tangent to circle O₁ at point T. Triangle ABC is inscribed in circle O₁. Circle O is tangent to sides AB and AC at points P and Q, respectively. Connect PQ. Connect AO and extend it to intersect circle O₁ at point E. Line segment AE intersects PQ at point I. Draw auxiliary lines BI and BE. Connect centers O and O₁ and extend the line to intersect circle O at points M and T, where point M lies on arc AB, and point T lies on arc CE.", "ocr_zh": "圆O相切于圆O1内，切点为点T。三角形ABC内接于圆O1，圆O与AB、AC边分别相切于点P、Q，连接PQ。连接AO并延长交圆O1于点E，AE交PQ于点I，作辅助线连接BI、BE。连接圆心OO1并延长交圆于点M、T，点M在圆弧AB上，点T在圆弧CE上。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4595.jpg" }, { "index": 4596, "ocr_en": "Circle I is the incircle of quadrilateral ABCD. Connect the center to the points of tangency on each side. Connect the diagonals AC and BD. Angle BAC is denoted as alpha (α), angle BDC as theta (θ), angle ACD as gamma (γ), and angle ABD as beta (β).", "ocr_zh": "圆I为四边形ABCD的内切圆，连接圆心与各边上的切点，连接对角线AC、BD。角BAC为α，角BDC为θ，角ACD为γ，角ABD为角β。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4596.jpg" }, { "index": 4597, "ocr_en": "Triangle ABC has circumcircle O and incircle I. Connect CI. Connect BO and extend it to intersect circle O at point K. Connect IK, with angle IKC equal to beta (β). Extend lines AB and CB to points N and M, respectively, and connect MN, with angle BMN equal to alpha (α).", "ocr_zh": "三角形ABC外接圆为圆O，内切圆为圆I，连接CI。连接BO并延长交圆O于点K，连接IK，角IKC为β。延长AB、CB至点N、M，连接MN，角BMN为α。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4597.jpg" }, { "index": 4598, "ocr_en": "In equilateral triangle ACE, points M, N, and G are the midpoints of sides AC, AE, and CE, respectively. Connect AG, and draw the auxiliary line MN intersecting AG at point D. Connect ED and extend it to intersect AC at point B. Connect CD and extend it to intersect AE at point F.", "ocr_zh": "在等边三角形ACE中，点M、N、G分别为AC、AE、CE中点，连接AG，作辅助线连接MN交AG于点D。连接ED并延长交AC于点B，连接CD并延长交AE于点F。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4598.jpg" }, { "index": 4599, "ocr_en": "Quadrilateral ABCD has a circumcircle. The diagonals AC and BD intersect at point R, and BD is the diameter of the circumcircle. Point B′ is the reflection of point B about line AC. Connect DB′ and extend it to intersect the extension of CA at point Q. Point A′ is the reflection of point A about line BD. Connect CA′ and extend it to intersect the extension of DB at point P. Draw auxiliary lines AA′, BB′, A′B, AP, BQ, and PQ.", "ocr_zh": "四边形ABCD有外接圆，连接对角线AC、BD交于点R，BD为外接圆的直径。点B'为点B关于AC对称的一点，连接DB'并延长与CA延长线交于点Q。点A'为点A关于BD对称的一点，连接CA'并延长与DB延长线交于点P。作辅助线连接AA'、BB'、A'B、AP、BQ、PQ。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4599.jpg" }, { "index": 4600, "ocr_en": "In triangle ACE, there is a point D. Connect AD. Extend lines DE and CD to intersect AC and AE at points B and F, respectively. Connect BF. Take points Q, R, and S as the midpoints of CD, BD, and BC, respectively. Draw auxiliary lines SR, QR, and SQ, and extend them to intersect BF, AD, and CE at points N, M, and P, respectively. Then draw the auxiliary line connecting points N and MP through point N.", "ocr_zh": "在三角形ACE中有一点D，连接AD，连接DE、CD并延长分别交AC、AE于点B、F，连接BF。分别取CD、BD、BC中点Q、R、S。作辅助线连接SR、QR、SQ并延长分别交BF、AD、CE于点N、M、P，用辅助线过点N连接MP。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4600.jpg" }, { "index": 4601, "ocr_en": "In quadrilateral ABCD, connect the diagonals AC and BD, which intersect at point Q. Point L is the midpoint of BC. Extend lines BA and CD to intersect at point M. Connect LM, which intersects AD, AC, and BD at points H, H′, and L′, respectively. Draw auxiliary line MQ and extend it to intersect BC at point R. Line MR intersects AD at point T. Extend lines AD and BC to intersect at point P. Connect PQ and extend it to intersect LM at point K. Line PK intersects CD at point N. Finally, draw the auxiliary line MP.", "ocr_zh": "在四边形ABCD中连接对角线AC、BD交于点Q，点L为BC中点。延长BA、CD交于点M，连接LM交AD、AC、BD于点H、H'、L'，作辅助线连接MQ并延长交BC于点R，MR交AD于点T。作辅助线延长AD、BC交于点P，作辅助线连接PQ并延长交LM于点K，PK交CD于点N，作辅助线连接MP。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4601.jpg" }, { "index": 4602, "ocr_en": "In quadrilateral ABCD, connect the diagonal AC. Point G is the midpoint of side BC. Connect DG, which intersects line AC at point F. Connect BF and extend it to intersect CD at point E. Connect AG and AE. Draw auxiliary lines connecting EG and DB, extending them to intersect at point N. Line AC intersects DB and EG at points H and M, respectively. Finally, draw the auxiliary line AN.", "ocr_zh": "在四边形ABCD中，连接对角线AC。点G为BC边上的中点，连接DG交AC于点F。连接BF并延长交CD于点E，连接AG、AE。作辅助线连接EG、DB并延长交于点N，AC交DB、EG于点H、M。作辅助线连接AN。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4602.jpg" }, { "index": 4603, "ocr_en": "In triangle ABC, there is a point O. Connect AO and extend AO to point P. Draw auxiliary lines BP and CP. Extend AO further to point R such that OR equals AO. Draw the auxiliary line RC. Lines l and h pass through points B and C, respectively, and both intersect AO at the same point Q (i.e., points Q and Q₁ coincide).", "ocr_zh": "在三角形ABC中有一点O，连接AO。延长OA至点P，作辅助线连接BP、CP。作辅助线延长AO至点R，使得OR等于AO，作辅助线连接RC。直线l、h分别为过点B、C的直线，分别交AO于点Q、Q1，点Q、Q1重合。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4603.jpg" }, { "index": 4604, "ocr_en": "In triangle ABC, point F is the midpoint of AB, and point E lies on AC. Connect FE and extend it to intersect the extension of BC at point G. Draw AO perpendicular to EF, with the foot of the perpendicular at point O. Extend AO to intersect BC at point D. Connect OB, OC, DF, and DE. Draw auxiliary lines BE and CF.", "ocr_zh": "在三角形ABC中点F为AB中点，点E为AC上一点，连接FE并延长交BC延长线于点G。作AO垂直于EF垂足为点O，延长AO交BC于点D。连接OB、OC、DF、DE。作辅助线连接BE、CF。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4604.jpg" }, { "index": 4605, "ocr_en": "In triangle ABC, the largest incircle touches side BC at point E. Connect AE, which intersects the circle again at point D. Draw auxiliary line CD. Connect BD, which intersects the circle again at point G. Connect CG, which intersects AE at point F. Through point D, draw the tangent line DH to the circle, which intersects the extension of BC at point H. The extension of BC is drawn as an auxiliary line.", "ocr_zh": "三角形ABC内有一最大内切圆与BC边相切于点E。连接AE交圆于点D，作辅助线连接CD，连接BD与圆交于点G，连接CG与AE交于点F。用辅助线过点D作圆的切线DH交BC延长线于点H，BC延长线为辅助线。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4605.jpg" }, { "index": 4606, "ocr_en": "In triangle ABC, AD is the angle bisector of angle BAC, with point D on side BC. Point M lies on AD. Connect BM and CM, and extend them to intersect AC and AB at points E and F, respectively. Connect DF and EF. Extend FE and BC as auxiliary lines to intersect at point G. On side AB, take a point S such that AS equals AC. Draw auxiliary lines DS and MS.", "ocr_zh": "三角形ABC中，AD为角BAC的平分线，点D为BC上一点。点M为AD上一点，连接BM、CM并延长分别交AC、AB于点E、F，连接DF、EF。用辅助线延长FE、BC交于点G。在AB上取一点S，有AS等于AC，作辅助线连接DS、MS。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4606.jpg" }, { "index": 4607, "ocr_en": "Let the point of intersection between the circle and the straight line be point A, and let another point A on the circle be point A1.In obtuse triangle ABC, there is a circumcircle with tangent line l touching the circle at point A. Using point A as the center and AC as the radius, construct a semicircle intersecting line l at points F and E, where F is closer to point B and E is closer to point C. The semicircle intersects AB at point D. Connect DE. Draw the angle bisector of angle BAC as an auxiliary line, intersecting DE and BC at points I and M, respectively, and intersecting the circumcircle again at point A₁. Connect FD and extend it to intersect the angle bisector of angle BAC at point IA. Draw auxiliary line AD and extend it to intersect the auxiliary extension of CB at point P. Finally, draw auxiliary lines connecting P and IA, C and IA, and A₁ and B.", "ocr_zh": "设图中圆与直线的切点为点A，圆上另一点A设为点A1。钝角三角形ABC有外接圆，直线l为外接圆的切线，切点为点A。以点A为圆心、AC为半径作一半圆交直线l于点F、E，F靠近点B，E靠近点C。半圆交AB于点D，连接DE。用辅助线作角BAC的平分线交DE、BC于点I、M，交外接圆于点A1。连接FD并延长交角BAC平分线于点IA。作辅助线连接AD并延长交CB的辅助延长线于点P。作辅助线连接PIA、CIA、A1B。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4607.jpg" }, { "index": 4608, "ocr_en": "In triangle ABC, the circumcircle has center O. Draw AD perpendicular to BC, with foot of the perpendicular at point D. Connect OD. Point IA is the excenter of triangle ABC opposite side BC. Draw auxiliary line AIA, which intersects OD at point I, BC at point E, and the circumcircle O at point M. Connect OM with an auxiliary line. Draw IAF perpendicular to BC, with foot of the perpendicular at point F.", "ocr_zh": "三角形ABC的外接圆为圆心为点O。作AD垂直于BC垂足为点D，连接OD。点IA为三角形ABC的BC边下方的旁心，作辅助线连接AIA交OD于点I、交BC于点E、交圆O于点M。作辅助线连接OM。用辅助线作IAF垂直于BC垂足为点F。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4608.jpg" }, { "index": 4609, "ocr_en": "In triangle ABC, the largest incircle touches sides AB, AC, and BC at points F, E, and D, respectively. Connect AD. DD′ is the diameter of the circle. Draw the tangent D′X to the circle at point D′, where point X lies on line AD. Draw auxiliary line FE and extend it to intersect the extension of BC at point K. Draw auxiliary line D′K, which intersects the circle again at point N. Connect NX. Connect CN and BN, which intersect the circle again at points Q and P, respectively. Draw auxiliary lines DN and PQ.", "ocr_zh": "三角形ABC中有最大内切圆，与AB、AC、BC分别切于点F、E、D，连接AD。DD'为圆直径，过点D'作圆的切线D'X，点X在AD上。作辅助线连接FE并延长交BC延长线于点K，作辅助线连接D'K交圆于点N。连接NX，连接CN、BN分别交圆于点Q、P，作辅助线连接DN、PQ。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4609.jpg" }, { "index": 4610, "ocr_en": "Circle O1 is the circumcircle of triangle ADE. Draw AF perpendicular to DE with foot at point F. Circles O1 and O2 intersect at points A and B. Draw auxiliary line AB. Through point A, draw the tangent to circle O1, which intersects circle O2 at point C. Connect center O1 to point C and extend to D such that CD is perpendicular to AE with foot at point H, and CD is tangent to circle O2 at point C. Draw auxiliary line CB and extend it to intersect circle O1 at point G. Draw auxiliary lines BD, BH, BE, and DG.", "ocr_zh": "圆O1为三角形ADE的外接圆，作AF垂直于DE垂足为点F。圆O1和圆O2相交于A、B两点，作辅助线连接AB。过点A作圆O1切线交圆O2于点C，过圆心O1连接CD，有CD垂直于AE垂足为点H，且CD与圆O2相切切点为点C。作辅助线连接CB并延长交圆O1于点G，作辅助线连接BD、BH、BE、DG。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4610.jpg" }, { "index": 4611, "ocr_en": "The left circle and the right circle intersect at points A and P, with point A above point P. Connect AP. Points B and C lie on the left circle and right circle respectively. Connect AB, AC, and BC. Points M and N are the midpoints of sides AC and AB, respectively. Connect MN, which intersects AP at point T. Connect BM and CN, which intersect at point G. Construct the circumcircle of triangle AMN.", "ocr_zh": "左圆与右圆相交于点A、P，点A在点P上方，连接AP。点B、C分别在左圆、右圆上，连接AB、AC、BC，点M、N分别为AC、AB边的中点，连接MN交AP于点T。连接BM、CN交于点G。作三角形AMN的外接圆。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4611.jpg" }, { "index": 4612, "ocr_en": "Point P is an interior point of triangle ABC. Using P as the center of inversion with inversion power equal to one unit, points A, B, and C are inverted to points A′, B′, and C′, respectively. Connect lines BP, CP, A′P, A′B′, A′C′, and B′C′. Points A, B′, and C′ lie on lines A′P, BP, and CP, respectively.", "ocr_zh": "点P为三角形ABC内一点，以P为反演中心，单位长度为反演幂，点A、B、C的反点分别为A'、B'、C'。连接BP、CP、A'P、A'B'、A'C'、B'C'，点A、B'、C'分别在A'P、BP、CP上。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4612.jpg" }, { "index": 4613, "ocr_en": "Circle O is the largest incircle of quadrilateral ABCD, tangent to sides AB, BC, CD, and AD at points E, F, G, and H, respectively. Connect HE and GF, and extend them to intersect at point P. Connect OP and the diagonal AC.", "ocr_zh": "圆O为四边形ABCD的最大内切圆，与AB、BC、CD、AD边分别相切于点E、F、G、H。连接HE、GF并延长交于点P，连接OP和对角线AC。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4613.jpg" }, { "index": 4614, "ocr_en": "In triangle AN'M', points B′, C′, and T′ are the midpoints of segments AN′, AM′, and M′N′, respectively. Connect AT′, N′C′, and M′B′, which intersect at point P′.", "ocr_zh": "在三角形AN'M'中，点B'、C'、T'分别为AN'、AM'、M'N'的中点，连接AT'、N'C'、M'B交于点P'。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4614.jpg" }, { "index": 4615, "ocr_en": "Quadrilateral ABCD has a circumcircle. Inside the circumcircle, there is a smaller circle tangent internally to the circumcircle at point N, and also tangent to side CD at point M. Connect diagonals AC and BD, which intersect at point F. Extend lines AD and BC to intersect at point E. As auxiliary lines, extend BA and CD to intersect at point P. Through point N, draw the tangent NR to the circumcircle, with point R located above point P. Connect DR. Point S is the midpoint of side AB.", "ocr_zh": "四边形ABCD有外接圆，圆内有一小圆与外接圆内切于点N，小圆还与CD边相切，切点为点M。连接AC、BD交于点F。延长AD、BC交于点E，作辅助线延长BA、CD交于点P。过点N作外接圆的切线NR，点R在点P上方，连接DR。点S为AB中点。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4615.jpg" }, { "index": 4616, "ocr_en": "Between two vertical parallel lines, there is circle O′ and another circle tangent to each other. Circle O′ is above, and both circles are tangent to the two parallel lines. Draw a straight line passing through the centers of the two circles. The lower circle is tangent to the left and right parallel lines at points A′ and B′, respectively. Connect A′B′. Point C lies on segment A′B′. Points O and E are located above points A′ and B′ on the left and right parallel lines, respectively, with A′O equal to B′E, both equal to the radius of the circles.", "ocr_zh": "在两竖直平行线之间有圆O'与另一圆相切，圆O'在上方，且两圆都与两条平行线相切。过两圆圆心作一直线。下方圆与左、右平行线的切点分别为A'、B'，连接A'B'，点C在A'B'上。点O、E分别在左右平行线上点A'、B'上方，且A'O等于B'E等于圆的半径", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4616.jpg" }, { "index": 4617, "ocr_en": "Between two vertical parallel lines l₁ and l₂, there are two arcs A′B′ arranged one above the other, intersecting each other. Both arcs are tangent to lines l₁ and l₂. The line K′L′ intersects the two arcs at exactly two points, with point K′ above point L′, and segment K′L′ is closer to line l₂.", "ocr_zh": "在左右两竖直平行线l1、l2之间有上下两圆弧A'B'相交，且两圆弧都与l1、l2相切。直线K'L'与两圆弧只有两个交点，点K'在点L'上方，且K'L'离l2较近。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4617.jpg" }, { "index": 4618, "ocr_en": "Circle O is the incircle of triangle ABC, and it is tangent to sides AB, AC, and BC at points F, G, and E respectively. Draw AD perpendicular to BC, with point D being the foot of the perpendicular. Using an auxiliary line from point E, draw diameter EM of circle O, and connect AE. Draw an auxiliary line connecting GF and extend it to intersect the extension of CB's auxiliary line at point K. GF intersects AD at point H. Draw an auxiliary line connecting KM and extend it to intersect circle O at point I, and IM intersects AE at point L. Draw auxiliary lines connecting BI, EI, and CI.", "ocr_zh": "圆O为三角形ABC的内切圆，与边AB、AC、BC分别切于点F、G、E。作AD垂直于BC，垂足为点D。过点E用辅助线作圆O的直径EM，作辅助线连接AE。作辅助线连接GF并延长交CB的辅助线延长线于点K，GF交AD于点H。作辅助线连接KM交圆O于点I，IM交AE于点L。作辅助线连接BI、EI、CI。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4618.jpg" }, { "index": 4619, "ocr_en": "Between the two horizontal parallel lines l1 and l2, there is circle O1 which is tangent to line l1 at point Y; circle O2 is externally tangent to circle O1 and has point X as the point of tangency, and is also tangent to line l2 at point Z. Connect O1O2 and draw line XZ through point X.", "ocr_zh": "在两条水平平行线l1、l2之间，有圆O1与直线l1相切，切点为点Y；圆O2与圆O1外切，切点为点X，与直线l2相切，切点为点Z。连接O1O2，过点X连接YZ。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4619.jpg" }, { "index": 4620, "ocr_en": "The incenter of triangle ABC is point I. Points D, E, and H are the points of tangency of circle I with sides AB, AC, and BC respectively. Draw a line through point A parallel to BC, and call this line AF. Draw a line connecting ED and extend it to intersect AF at point F. Point M is the midpoint of BC. Connect AM and intersect circle I at points K and L. Point K is closer to point A, and AM intersects ED at point G. Draw a line parallel to BC through point L and intersect circle I at point Y. Connect AY and extend it to intersect BC at point Q. Draw a line connecting GH, draw a line connecting YG and extend it to intersect circle I at point X. Connect AX and extend it to intersect BC at point P.", "ocr_zh": "三角形ABC内切圆圆心为点I，圆I与AB、AC、BC的切点分别为点D、E、H。用辅助线过点A作BC的平行线AF，作辅助线连接ED并延长交AF于点F。点M为BC中点，连接AM交圆I于点K、L，点K靠近点A，AM交ED于点G。过点L作BC的平行线交圆I于点Y，连接AY并延长交BC于点Q。作辅助线连接GH，作辅助线连接YG并延长交圆I于点X。连接AX并延长交BC于点P。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4620.jpg" }, { "index": 4621, "ocr_en": "Quadrilateral ABCD has an inscribed circle, which is tangent to sides AB, BC, CD, and AD at points E, F, G, and H respectively. Draw auxiliary lines connecting EF and HG and extend them to intersect at point M. Draw auxiliary lines connecting EH and FG and extend them to intersect at point N. Draw auxiliary lines extending BA and CD to intersect at point O, and draw auxiliary lines extending BC and AD to intersect at point P.", "ocr_zh": "四边形ABCD有内切圆，与AB、BC、CD、AD边分别相切于点E、F、G、H。作辅助线连接EF、HG并延长交于点M，作辅助线连接EH、FG并延长交于点N。作辅助线延长BA、CD交于点O，作辅助线延长BC、AD交于点P。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4621.jpg" }, { "index": 4622, "ocr_en": "In triangle ABC, AD and CE are the angle bisectors of angles BAC and ACB respectively. They intersect at point I. Points D and E are on sides BC and AB respectively. Points D' and E' are on side BC, with CD equal to CD' and AE equal to AE'. Auxiliary lines are drawn connecting IE' and ID'.", "ocr_zh": "在三角形ABC中AD、CE分别为角BAC、角ACB的平分线，AD交CE于点I，点D、E分别在BC、AB边上。点D'、E'在BC边上，满足CD等于CD'、AE等于AE'，作辅助线连接IE'、ID'。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4622.jpg" }, { "index": 4623, "ocr_en": "Circle A is externally tangent to circle B at point M. Connect AB. CD is tangent to circle A and circle B, with points of tangency being D and C respectively. Connect AD and BC. Use auxiliary lines to make AE perpendicular to BC and the foot of the perpendicular is point E. Point P is inside the quadrilateral ADCB and outside circles A and B. Connect AP and BP, and make PQ perpendicular to CD with the foot of the perpendicular being point Q.", "ocr_zh": "圆A与圆B外切于点M，连接AB。CD与圆A、圆B相切，切点分别为点D、C。连接AD、BC，用辅助线作AE垂直于BC垂足为点E。点P在四边形ADCB内且在圆A、圆B外，连接AP、BP，作PQ垂直于CD垂足为点Q。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4623.jpg" }, { "index": 4624, "ocr_en": "In isosceles trapezoid ABCD, point M is the midpoint of AB, and it satisfies that AM is equal to AD. Using points A and B as centers and AD as the radius, draw arcs DM and CM within the quadrilateral. Circle P is a circle located inside the quadrilateral, tangent to side CD, arc DM, and arc CM. Connect AP and BP.", "ocr_zh": "在等腰梯形ABCD中，点M为AB中点，满足AM等于AD。分别以A、B为圆心AD为半径在四边形内作圆弧DM、CM。圆P为四边形内一圆，与CD、圆弧DM和圆弧CM相切。连接AP、BP。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4624.jpg" }, { "index": 4625, "ocr_en": "The circumcircle of triangle ABC is circle O. Connect AO, BO, and CO, and extend them to intersect sides BC, AC, and AB at points D, E, and F, respectively.Angle OBC is equal to angle OCB. Extend AD beyond the circle to a point A′ outside the circle. Connect A′B.", "ocr_zh": "三角形外接圆为圆O，连接AO、BO、CO并延长分别交BC、AC、AB于点D、E、F，角OBC等于角OCB。延长AD至圆外一点A'，连接A'B。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4625.jpg" }, { "index": 4626, "ocr_en": "Triangle ABC has a circumcircle. The angle bisectors of angles ∠ABC, ∠ACB, and ∠BAC intersect the circumcircle at points B1, C1, and A1, respectively. Lines AA1, BB1, and CC1 intersect at a single point inside triangle ABC.", "ocr_zh": "三角形ABC有外接圆，角ABC、角ACB、角BAC的平分线分别交圆于点B1、C1、A1。AA1、BB1、CC1在三角形ABC内交于一点。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4626.jpg" }, { "index": 4627, "ocr_en": "Point P is a point inside triangle A′B′C′. Draw PA perpendicular to B′C′, with foot of the perpendicular at point A. Draw PB perpendicular to A′C′, with foot at point B. Draw PC perpendicular to A′B′, with foot at point C. Connect AB, AC, and BC.", "ocr_zh": "点P为三角形A'B'C'内一点，作PA垂直于B'C'垂足为点A，作PB垂直于A'C'垂足为点B，作PC垂直于A'B'垂足为点C。连接AB、AC、BC。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4627.jpg" }, { "index": 4628, "ocr_en": "Triangle ABC has a circumcircle. In triangle ABC, points Ao, Bo, and Co are the midpoints of sides BC, AC, and AB, respectively. Connect the medians AAo, BBo, and CCo, and extend them to intersect the circumcircle at points A1, B1, and C1, respectively. The medians AAo, BBo, and CCo intersect at point G.", "ocr_zh": "三角形ABC有外接圆。在三角形ABC中，点Ao、Bo、Co分别为BC、AC、AB边上的中点，连接AAo、BBo、CCo并延长分别交圆于点A1、B1、C1。中线AAo、BBo、CCo交于点G。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4628.jpg" }, { "index": 4629, "ocr_en": "Quadrilateral ABCD has a circumcircle, and there is a point O inside the quadrilateral. Connect AO, BO, CO, and DO. From point O, draw PO perpendicular to AB with foot at point P, QO perpendicular to BC with foot at point Q, RO perpendicular to CD with foot at point R, and SO perpendicular to AD with foot at point S. Connect PQ, QR, RS, and PS to form quadrilateral PQRS, and construct the incircle of PQRS as an auxiliary figure. Angle OAS is angle 1, angle OPS is angle 2, angle OPQ is angle 3, angle OBQ is angle 4, angle ORS is angle 5, and angle ODS is angle 6.", "ocr_zh": "四边形ABCD有外接圆，四边形内有一点O，连接AO、BO、CO、DO。过点O作PO垂直于AB垂足为点P、QO垂直于BC垂足为点Q、RO垂直于CD垂足为点R、SO垂直于AD垂足为点S。连接PQ、QR、RS、PS，并用辅助线作四边形PQRS的内切圆。角OAS为角1，角OPS为角2，角OPQ为角3，角OBQ为角4，角ORS为角5，角ODS为角6。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4629.jpg" }, { "index": 4630, "ocr_en": "In triangle ABC, let sides BC, AC, and AB be denoted as a, b, and c, respectively. Point H is a point inside triangle ABC. Connect AH, BH, and CH. Extend AH to a point H′, then connect BH′ and CH′ such that angle BCH′ is equal to angle BCH.", "ocr_zh": "在三角形ABC中，令BC、AC、AB边分别为a、b、c。点H为三角形ABC内一点，连接AH、BH、CH。作辅助线延长AH至点H'，作辅助线连接BH'、CH'，满足角BCH'等于角BCH。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4630.jpg" }, { "index": 4631, "ocr_en": "In triangle ABC, points A₁, B₁, and C₁ lie on sides BC, AC, and AB respectively. Connect AA₁, BB₁, and CC₁, which intersect at point P. Connect A₁B₁, B₁C₁, and A₁C₁.", "ocr_zh": "在三角形ABC中，点A1、B1、C1分别在边BC、AC、AB上，连接AA1、BB1、CC1交于点P，连接A1B1、B1C1、A1C1。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4631.jpg" }, { "index": 4632, "ocr_en": "In the Cartesian coordinate plane xOy, line l lies on the x-axis. Point A is on the positive half of the y-axis, point B is in the third quadrant, and point C is in the fourth quadrant. Connect points A and B, A and C, and B and C. Point D is the intersection of segment BC and the negative half of the y-axis. Point M lies inside triangle ABC on the negative half of the x-axis. Connect BM and CM. Through point M, draw line d' parallel to the y-axis. Point P is a point on segment AB located in the second quadrant. Through point P, draw a line parallel to the y-axis intersecting BM at point E. Point Q is a point on segment AC located in the first quadrant. Through point Q, draw a line parallel to the y-axis intersecting CM at point F. Connect points P and Q.", "ocr_zh": "在平面直角坐标系xOy中，直线l在x轴上。点A在y轴的正半轴上，点B在第三象限，点C在第四象限，连接AB、AC、BC。点D为BC与y轴负半轴的交点。点M在三角形ABC内，x轴的负半轴上，连接BM、CM。过点M作直线d'平行于y轴。点P为AB上第二象限内一点，过点P作y轴的平行线交BM于点E。点Q为AC上第一象限内一点，过点Q作平行于y轴的直线交CM于点F，连接PQ。", "class": "Analytic Geometry", "difficulty": 4, "image_path": "fig_4632.jpg" }, { "index": 4633, "ocr_en": "In the Cartesian coordinate plane xOy, circle O is centered at the origin O and is the incircle of triangle APQ. In triangle ABC, point A lies on the positive half of the y-axis, point P is in the third quadrant, and point Q is in the fourth quadrant. Circle O is tangent to segments AP and AQ at points B and C, respectively. Point S is the intersection of segment AP with the x-axis. Connect points C and B, and extend this line to point R such that CR is parallel to the x-axis. Draw a line passing through points P and R. As an auxiliary line, connect points S and Q. Let segment PQ be denoted as line l.", "ocr_zh": "在平面直角坐标系xOy中，圆O以原点O为圆心，是三角形APQ的内切圆。三角形ABC中，点A在y轴的正半轴上，点P在第三象限中，点Q在第四象限中。圆O与AP、AQ的切点分别为B、C，AP与x轴的交点为点S。连接CB并延长至点R，CR平行于x轴，过点P、R作一条直线。作辅助线过点S连接QR。令PQ为l。", "class": "Analytic Geometry", "difficulty": 4, "image_path": "fig_4633.jpg" }, { "index": 4634, "ocr_en": "In quadrilateral ABCD, points Q₁, R₁, S₁, and P₁ are the midpoints of sides AB, BC, CD, and AD, respectively. Connect P₁Q₁, Q₁R₁, R₁S₁, and P₁S₁. Points U and V are the midpoints of segments P₁S₁ and Q₁R₁, respectively. Draw an auxiliary line connecting U and V. Points P, Q, R, and S are points directly above P₁, Q₁, R₁, and S₁, respectively. Connect PQ, QR, RS, PS, PA, PD, QA, QB, RB, RC, SC, and SD. Points E, F, G, and H are the midpoints of segments PS, PQ, QR, and RS, respectively. Connect E and G. Points M and N are the midpoints of segments PD and SD, respectively. Draw auxiliary lines connecting E to M, E to P₁, E to U, E to S₁, E to N, M to P₁, and N to S₁. Also draw auxiliary lines connecting G to Q₁, G to V, and G to R₁.", "ocr_zh": "四边形ABCD中，点Q1、R1、S1、P1分别为AB、BC、CD、AD的中点，连接P1Q1、Q1R1、R1S1、P1S1，点U、V分别为P1S1、Q1R1的中点，作辅助线连接UV。点P、Q、R、S分别为点P1、Q1、R1、S1正上方一点，连接PQ、QR、RS、PS、PA、PD、QA、QB、RB、RC、SC、SD。点E、F、G、H分别为PS、PQ、QR、RS的中点，连接EG，点M、N分别为PD、SD中点。作辅助线连接EM、EP1、EU、ES1、EN、MP1、NS1。作辅助线连接GQ1、GV、GR1。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4634.jpg" }, { "index": 4635, "ocr_en": "In the isosceles trapezoid DABE, side DE is longer than side AB, and DE is parallel to AB. Point M is the midpoint of DE. Two squares are constructed on sides AD and BE, respectively, intersecting at point C. Angle ACB is an acute angle.", "ocr_zh": "在等腰梯形DABE中，DE边比AB边长，且DE平行于AB，点M为DE中点。分别以AD、BE为边作两个正方形相交于点C，角ACB为锐角。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4635.jpg" }, { "index": 4636, "ocr_en": "Quadrilateral abcd has points h, k, f, and g as the midpoints of sides ab, bc, cd, and ad, respectively. Draw auxiliary lines connecting h to f and g to k, which intersect at point o.", "ocr_zh": "abcd为四边形，点h、k、f、g分别为ab、bc、cd、ad中点，作辅助线连接hf、gk交于点o。", "class": "Plane Geometry", "difficulty": 1, "image_path": "fig_4636.jpg" }, { "index": 4637, "ocr_en": "In quadrilateral ABCD, connect diagonals AC and BD, which intersect at point O. Through point O, draw auxiliary line MN parallel to AB, intersecting AD and BC at points M and N, respectively. Draw OQ perpendicular to BC with foot Q. Draw auxiliary line NP perpendicular to AD with foot P. Connect points O to P, B to P, and C to P.", "ocr_zh": "在四边形ABCD中，连接对角线AC、BD交于点O。用辅助线过点O作MN平行于AB，分别交AD、BC于点M、N。作OQ垂直于BC，垂足为点Q。用辅助线作NP垂直于AD，垂足为点P，连接OP、BP、CP。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4637.jpg" }, { "index": 4638, "ocr_en": "In quadrilateral ABCD, points M and N lie on side AB, with point M closer to point A; points P and Q lie on side CD, with point Q closer to point D. Connect points A to P, N to P, M to Q, and C to M, such that MQ is parallel to NP and AP is parallel to CM. Draw auxiliary lines connecting AC and MP.", "ocr_zh": "在四边形ABCD中，M、N在为AB边上两点，点M更靠近点A；P、Q在CD边上，点Q更靠近点D，连接AP、NP、MQ、CM，满足MQ平行于NP，AP平行于CM。作辅助线连接AC、MP。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4638.jpg" }, { "index": 4639, "ocr_en": "In triangle ABC, there is a point P. Draw auxiliary lines connecting A to P, B to P, and C to P. Through point P, draw A'P perpendicular to AB with foot A'; draw B'P perpendicular to BC with foot B'; and draw C'P perpendicular to AC with foot C'.", "ocr_zh": "在三角形ABC中有一点P，用辅助线连接AP、BP、CP。过点P作A'P垂直于AB垂足为点A'，作B'P垂直于BC垂足为点B'，作C'P垂直于AC垂足为点C'。", "class": "Plane Geometry", "difficulty": 2, "image_path": "fig_4639.jpg" }, { "index": 4640, "ocr_en": "In triangle ABC, a line DE intersects sides AB and AC at points D and E, respectively, such that AD equals AE. Point I is the incenter of triangle ABC, and point Z is the midpoint of BC. Connect B to I and C to I, extending these lines to intersect line DE at points Y and X, respectively. Connect B to X, X to Z, E to I, and Y to Z. Connect point D to the intersection points of XZ and BI.", "ocr_zh": "在三角形ABC中，有一直线DE与AB、AC边分别相交于点D、E，满足AD等于AE。点I为三角形ABC的内心，点Z为BC中点，连接BI、CI并延长分别交直线DE于点Y、X。连接BX、XZ、EI、YZ。连接点D与XZ和BI的交点。", "class": "Plane Geometry", "difficulty": 4, "image_path": "fig_4640.jpg" }, { "index": 4641, "ocr_en": "Point P lies inside triangle ABC. Draw auxiliary lines connecting A to P, B to P, and C to P. Draw MP perpendicular to AB with foot M, PQ perpendicular to BC with foot Q, and NP perpendicular to AC with foot N. Through point P, draw line DE intersecting sides AB and AC at points D and E, respectively, such that point D is below point M and point E is below point N. Connect points C to D and B to E.", "ocr_zh": "点P在三角形ABC中，作辅助线连接AP、BP、CP。作MP垂直于AB垂足为点M，作PQ垂直于BC垂足为点Q，作NP垂直于AC垂足为点N。过点P作DE分别与AB、AC边交于点D、E，且点D在点M下方，点E在点N下方。连接CD、BE。", "class": "Plane Geometry", "difficulty": 3, "image_path": "fig_4641.jpg" } ]