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The dataset generation failed
Error code: DatasetGenerationError
Exception: ArrowInvalid
Message: JSON parse error: Invalid value. in row 0
Traceback: Traceback (most recent call last):
File "/usr/local/lib/python3.12/site-packages/datasets/packaged_modules/json/json.py", line 174, in _generate_tables
df = pandas_read_json(f)
^^^^^^^^^^^^^^^^^^^
File "/usr/local/lib/python3.12/site-packages/datasets/packaged_modules/json/json.py", line 38, in pandas_read_json
return pd.read_json(path_or_buf, **kwargs)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
File "/usr/local/lib/python3.12/site-packages/pandas/io/json/_json.py", line 791, in read_json
json_reader = JsonReader(
^^^^^^^^^^^
File "/usr/local/lib/python3.12/site-packages/pandas/io/json/_json.py", line 905, in __init__
self.data = self._preprocess_data(data)
^^^^^^^^^^^^^^^^^^^^^^^^^^^
File "/usr/local/lib/python3.12/site-packages/pandas/io/json/_json.py", line 917, in _preprocess_data
data = data.read()
^^^^^^^^^^^
File "/usr/local/lib/python3.12/site-packages/datasets/utils/file_utils.py", line 813, in read_with_retries
out = read(*args, **kwargs)
^^^^^^^^^^^^^^^^^^^^^
File "<frozen codecs>", line 322, in decode
UnicodeDecodeError: 'utf-8' codec can't decode byte 0x89 in position 0: invalid start byte
During handling of the above exception, another exception occurred:
Traceback (most recent call last):
File "/usr/local/lib/python3.12/site-packages/datasets/builder.py", line 1815, in _prepare_split_single
for _, table in generator:
^^^^^^^^^
File "/usr/local/lib/python3.12/site-packages/datasets/packaged_modules/json/json.py", line 177, in _generate_tables
raise e
File "/usr/local/lib/python3.12/site-packages/datasets/packaged_modules/json/json.py", line 151, in _generate_tables
pa_table = paj.read_json(
^^^^^^^^^^^^^^
File "pyarrow/_json.pyx", line 342, in pyarrow._json.read_json
File "pyarrow/error.pxi", line 155, in pyarrow.lib.pyarrow_internal_check_status
File "pyarrow/error.pxi", line 92, in pyarrow.lib.check_status
pyarrow.lib.ArrowInvalid: JSON parse error: Invalid value. in row 0
The above exception was the direct cause of the following exception:
Traceback (most recent call last):
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1339, in compute_config_parquet_and_info_response
parquet_operations = convert_to_parquet(builder)
^^^^^^^^^^^^^^^^^^^^^^^^^^^
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 972, in convert_to_parquet
builder.download_and_prepare(
File "/usr/local/lib/python3.12/site-packages/datasets/builder.py", line 894, in download_and_prepare
self._download_and_prepare(
File "/usr/local/lib/python3.12/site-packages/datasets/builder.py", line 970, in _download_and_prepare
self._prepare_split(split_generator, **prepare_split_kwargs)
File "/usr/local/lib/python3.12/site-packages/datasets/builder.py", line 1702, in _prepare_split
for job_id, done, content in self._prepare_split_single(
^^^^^^^^^^^^^^^^^^^^^^^^^^^
File "/usr/local/lib/python3.12/site-packages/datasets/builder.py", line 1858, in _prepare_split_single
raise DatasetGenerationError("An error occurred while generating the dataset") from e
datasets.exceptions.DatasetGenerationError: An error occurred while generating the datasetNeed help to make the dataset viewer work? Make sure to review how to configure the dataset viewer, and open a discussion for direct support.
index int64 | ocr_en string | ocr_zh string | class string | difficulty int64 | image_path string |
|---|---|---|---|---|---|
1 | 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 ... | 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。 | Plane Geometry | 1 | fig_1.jpg |
2 | 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... | 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。
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。
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。 | Plane Geometry | 1 | fig_2.jpg |
3 | 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 perpendicula... | 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。
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。 | Plane Geometry | 1 | fig_3.jpg |
4 | 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. | 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。 | Plane Geometry | 1 | fig_4.jpg |
5 | 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. | 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。 | Plane Geometry | 1 | fig_5.jpg |
6 | 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 AD... | 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。 | Plane Geometry | 1 | fig_6.jpg |
7 | 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 fo... | 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。 | Plane Geometry | 1 | fig_7.jpg |
8 | 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... | 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。 | Plane Geometry | 1 | fig_8.jpg |
9 | 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. | 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。 | Plane Geometry | 2 | fig_9.jpg |
10 | 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 D... | 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。 | Plane Geometry | 2 | fig_10.jpg |
11 | 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. | 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。 | Analytic Geometry | 2 | fig_11.jpg |
12 | 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... | 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。 | Plane Geometry | 2 | fig_12.jpg |
13 | 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 poin... | 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。 | Analytic Geometry | 2 | fig_13.jpg |
14 | **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 h... | 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... | Analytic Geometry | 3 | fig_14.jpg |
15 | 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 l... | 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。 | Analytic Geometry | 2 | fig_15.jpg |
16 | Points A and B are two points on opposite sides of line l. Connect AB, intersecting line l at point P. | 点A、B为直线l异侧两点，连接AB，交直线l于点P。 | Plane Geometry | 1 | fig_16.jpg |
17 | 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. | 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。 | Plane Geometry | 2 | fig_17.jpg |
18 | 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. | 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。 | Plane Geometry | 1 | fig_18.jpg |
19 | 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... | 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。 | Plane Geometry | 2 | fig_19.jpg |
20 | 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. | 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。 | Plane Geometry | 1 | fig_20.jpg |
21 | 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. | 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。 | Plane Geometry | 1 | fig_21.jpg |
22 | 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. | 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。 | Plane Geometry | 2 | fig_22.jpg |
23 | 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'... | 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''。 | Plane Geometry | 3 | fig_23.jpg |
24 | 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 \( l \). | l为一条直线，点A、B为直线l外两点且A，B异侧。 | Plane Geometry | 1 | fig_24.jpg |
25 | 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. | 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。 | Plane Geometry | 1 | fig_25.jpg |
26 | 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 ... | 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。 | Plane Geometry | 1 | fig_26.jpg |
27 | 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. | 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。 | Plane Geometry | 1 | fig_27.jpg |
28 | 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. | 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。 | Plane Geometry | 1 | fig_28.jpg |
29 | 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. | 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。 | Plane Geometry | 2 | fig_29.jpg |
30 | 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. | 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。 | Plane Geometry | 1 | fig_30.jpg |
31 | 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. | 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。 | Plane Geometry | 1 | fig_31.jpg |
32 | In the plane rectangular coordinate system xOy, the coordinates of point A are (0, 3), and the coordinates of point B are (2, 0). | 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）。 | Analytic Geometry | 1 | fig_32.jpg |
33 | 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. | 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。 | Plane Geometry | 1 | fig_33.jpg |
34 | There is an angle AOB, and point P is inside angle AOB. Connect OP. | 有一角AOB，点P在角AOB内，连接OP。 | Plane Geometry | 1 | fig_34.jpg |
35 | 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. | 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。 | Plane Geometry | 1 | fig_35.jpg |
36 | There is an angle MON, and point P is inside angle MON. | 有一角MON，点P在角MON内。 | Plane Geometry | 1 | fig_36.jpg |
37 | There is an angle MON, and points A and B are on rays OM and ON, respectively. | 有一角MON，点A，B分别在射线OM，ON上。 | Plane Geometry | 1 | fig_37.jpg |
38 | 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. | 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。 | Analytic Geometry | 2 | fig_38.jpg |
39 | **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 \( d \).
**Upper right figure:** Line \( l \) is a straight line, and segment \( d \) is a line segment par... | 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... | Plane Geometry | 2 | fig_39.jpg |
40 | 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. | 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。 | Analytic Geometry | 1 | fig_40.jpg |
41 | There are two parallel lines \( l_1 \) and \( l_2 \), and point A is a point above line \( l_1 \). | 有两条平行直线l1，l2，点A为直线l1上方一点。 | Plane Geometry | 1 | fig_41.jpg |
42 | 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. | 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。 | Solid Geometry | 1 | fig_42.jpg |
43 | 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. | 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。 | Solid Geometry | 1 | fig_43.jpg |
44 | 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. | 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。 | Solid Geometry | 1 | fig_44.jpg |
45 | 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. | 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。 | Solid Geometry | 1 | fig_45.jpg |
46 | There is a cylinder, and points A and B are on the same cross-section passing through the axis of the cylinder. | 有一圆柱，点A,B在过圆柱轴的同一横截面上。 | Solid Geometry | 1 | fig_46.jpg |
47 | **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.
**Figure B:** The sector OA(A) is the lateral surface development of ... | 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），... | Solid Geometry | 2 | fig_47.jpg |
48 | 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. | 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。 | Solid Geometry | 1 | fig_48.jpg |
49 | 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. | 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。 | Solid Geometry | 1 | fig_49.jpg |
50 | In triangle ABC, point D is the midpoint of side BC, and point F is a point on side AB. Connect DF. | 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。 | Plane Geometry | 1 | fig_50.jpg |
51 | In triangle ABD, point D is the midpoint of side BC. Connect AD. | 在三角形ABD中，点D为BC边中点。连接AD。 | Plane Geometry | 1 | fig_51.jpg |
52 | 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 F... | 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。 | Plane Geometry | 1 | fig_52.jpg |
53 | 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. | 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。 | Plane Geometry | 1 | fig_53.jpg |
54 | In triangle ABC, point D is the midpoint of side BC. Connect AD. | 在三角形ABC中，点D为BC的中点，连接AD。 | Plane Geometry | 1 | fig_54.jpg |
55 | 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. | 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。 | Plane Geometry | 2 | fig_55.jpg |
56 | Left diagram: In triangle ABC, AB equals AC, and point D is the midpoint of BC.
Right 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. | 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。 | Plane Geometry | 1 | fig_56.jpg |
57 | 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. | 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。 | Plane Geometry | 1 | fig_57.jpg |
58 | 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... | 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。 | Plane Geometry | 2 | fig_58.jpg |
59 | 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 w... | 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。 | Plane Geometry | 2 | fig_59.jpg |
60 | 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 perpendicul... | 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。 | Plane Geometry | 2 | fig_60.jpg |
61 | 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 ... | 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。 | Plane Geometry | 2 | fig_61.jpg |
62 | Left diagram: In triangle ABC, point D is the midpoint of AB.
Right 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. | 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。 | Plane Geometry | 1 | fig_62.jpg |
63 | 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. | 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。 | Plane Geometry | 2 | fig_63.jpg |
64 | 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 perpendi... | 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。 | Plane Geometry | 2 | fig_64.jpg |
65 | 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 bein... | 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。 | Plane Geometry | 2 | fig_65.jpg |
66 | 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. | 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。 | Plane Geometry | 2 | fig_66.jpg |
67 | 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. | 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。 | Plane Geometry | 2 | fig_67.jpg |
68 | 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. | 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。 | Plane Geometry | 2 | fig_68.jpg |
69 | 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.
Right 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. | 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。 | Plane Geometry | 1 | fig_69.jpg |
70 | 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. | 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。 | Plane Geometry | 2 | fig_70.jpg |
71 | 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. | 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。 | Plane Geometry | 1 | fig_71.jpg |
72 | 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 a... | 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。 | Plane Geometry | 2 | fig_72.jpg |
73 | 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 midpo... | 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。 | Plane Geometry | 2 | fig_73.jpg |
74 | 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. | 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。 | Plane Geometry | 2 | fig_74.jpg |
75 | 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... | 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。 | Plane Geometry | 2 | fig_75.jpg |
76 | 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. | 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。 | Plane Geometry | 2 | fig_76.jpg |
77 | In square ABCD, point E is on BC, and point F is on DC. Connect AE, AF, and EF. | 在正方形ABCD中，点E在BC上，点F在DC上。连接AE、AF、EF。 | Plane Geometry | 1 | fig_77.jpg |
78 | 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.
Lower diagram: In angle AOB, OA equals OB. Points E and F are points inside angle AOB with point ... | 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。 | Plane Geometry | 1 | fig_78.jpg |
79 | 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. | 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。 | Plane Geometry | 2 | fig_79.jpg |
80 | 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. | 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。 | Plane Geometry | 1 | fig_80.jpg |
81 | 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 ... | 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。 | Plane Geometry | 2 | fig_81.jpg |
82 | 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 symmet... | 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。 | Plane Geometry | 2 | fig_82.jpg |
83 | 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, wit... | 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。 | Plane Geometry | 2 | fig_83.jpg |
84 | In square ABCD, point E is the midpoint of BC, and point F is the midpoint of CD. Connect AE, AF, and EF. | 在正方形ABCD中，点E为BC中点，点F为CD中点，连接AE、AF、EF。 | Plane Geometry | 1 | fig_84.jpg |
85 | 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. | 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。 | Plane Geometry | 2 | fig_85.jpg |
86 | 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. | 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。 | Plane Geometry | 2 | fig_86.jpg |
87 | 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. | 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。 | Plane Geometry | 2 | fig_87.jpg |
88 | 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. | 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。 | Plane Geometry | 1 | fig_88.jpg |
89 | 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. | 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。 | Plane Geometry | 2 | fig_89.jpg |
90 | 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. | 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。 | Plane Geometry | 1 | fig_90.jpg |
91 | 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. | 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。 | Plane Geometry | 1 | fig_91.jpg |
92 | 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. | 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。 | Plane Geometry | 1 | fig_92.jpg |
93 | 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. | 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。 | Plane Geometry | 2 | fig_93.jpg |
94 | 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. | 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。 | Plane Geometry | 2 | fig_94.jpg |
95 | 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. | 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。 | Plane Geometry | 2 | fig_95.jpg |
96 | 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. | 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。 | Plane Geometry | 2 | fig_96.jpg |
97 | 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. | 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。 | Plane Geometry | 2 | fig_97.jpg |
98 | 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. | 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。 | Plane Geometry | 2 | fig_98.jpg |
99 | 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. | 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。 | Plane Geometry | 2 | fig_99.jpg |
100 | In pentagon ABCDE, AB equals AE. Connect AD. | 在五边形ABCDE中，AB等于AE。连接AD。 | Plane Geometry | 1 | fig_100.jpg |
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