question: Diagram description: pentagon ABCDE is inscribed in circle O. CF is tangent to circle O at point C and intersects the extension of AB at point F. Also, AE=DC and ∠E=∠BCD.

source: ZhongKaoGeo


question: Diagram description: pentagon ABCDE is inscribed in circle O. CF is tangent to circle O at point C and intersects the extension of AB at point F. Also, OB=2, AB=BD=DA, and ∠F=45°

source: ZhongKaoGeo


question: Diagram description: in right △ABC, ∠BAC=90°. O is a point on side AB. Circle O, with radius OA, is tangent to side BC at point E. Also, AC=5 and BC=13

source: ZhongKaoGeo


question: Diagram description: in right △ABC, ∠BAC=90°. O is a point on side AB. Circle O, with radius OA, is tangent to side BC at point E. (2) Draw chord EF ⊥ AB at M, and connect AF. Also, ∠F=2∠B.

source: ZhongKaoGeo


question: Diagram description: in rectangle ABCD, AC and BD intersect at point O. BE ⊥ AC and CF ⊥ BD, with foot of the perpendiculars at E and F respectively

source: ZhongKaoGeo


question: Diagram description: in △ABC, AC=CB. O is the midpoint of AB. CA is tangent to circle O at point E, and CO intersects circle O at point D.

source: ZhongKaoGeo


question: Diagram description: in △ABC, AC=CB. O is the midpoint of AB. CA is tangent to circle O at point E, and CO intersects circle O at point D. Also, ∠ACB=80° and point P is a point on circle O (not coinciding with D or E)

source: ZhongKaoGeo


question: Diagram description: line AB // CD. A right-angle triangular plate EFG containing a 60° angle (∠E=60°) has its right-angle vertex F on line AB. Hypotenuse EG intersects AB at point H, and CD intersects FG at point M. Also, ∠AHG=50°.

source: ZhongKaoGeo


question: Diagram description: Point E is on the diagonal AC of square ABCD. Square AFEG and square ABCD have a common point A.  Point G is on AD and F is on AB

source: ZhongKaoGeo


question: Diagram description: Point E is on the diagonal AC of square ABCD. Square AFEG and square ABCD have a common point A.

source: ZhongKaoGeo


question: Diagram description: Point E is on the diagonal AC of square ABCD. Square AFEG and square ABCD have a common point A. Also,  AB = 8 * sqrt(2), AG = (sqrt(2)/2) * AD. Also, points C, G, and E are collinear.

source: ZhongKaoGeo


question: Diagram description: in △ABC, ∠ACB=90°, AC=BC=4. Point D is a point on side BC (not coinciding with B or C). CE is perpendicular to AD and intersects AB at point E, with the foot at point H. Connect BH and extend it to intersect AC at point F. Also, AD is the median on side BC.

source: ZhongKaoGeo


question: Diagram description: in △ABC, ∠ACB=90°, AC=BC=4. Point D is a moving point on side BC. CE ⊥ AD intersects AB at E, foot at H. Connect BH and extend to AC at F. Also, AD bisects ∠CAB.

source: ZhongKaoGeo


question: Diagram description:  in △ABC, ∠ACB=90°, AC=BC=4. Point D is a moving point on side BC. CE ⊥ AD intersects AB at E, foot at H. Connect BH and extend to AC at F. Finally, BD=2CD.

source: ZhongKaoGeo


question: Diagram description: AB is the diameter of circle O. C and D are points on circle O, and AD=CD. Connect BC and extend it to intersect the tangent to circle O passing through D at point E.

source: ZhongKaoGeo


question: Diagram description: AB is the diameter of circle O. C and D are points on circle O, and AD=CD. Connect BC and extend it to point E on the tangent at D. Finnaly,  DE=4 and tanB=4/3.

source: ZhongKaoGeo


question: Diagram description: △ABC is inscribed in circle O. D is a point on the extension of diameter AB. ∠DCB=∠OAC. A line passing through center O parallel to BC intersects the extension of DC at point E.

source: ZhongKaoGeo


question: Diagram description: △ABC is inscribed in circle O. D is on the extension of diameter AB. ∠DCB=∠OAC. OE // BC. Finnaly, CD=4 and CE=6.

source: ZhongKaoGeo


question: Diagram description: in quadrilateral ABCD, ∠B=∠D=90°, ∠A=60°, and AB=4.

source: ZhongKaoGeo


question: Diagram description: in right △ABC, ∠ACB=90°. Bisect ∠ACB, intersecting hypotenuse AB at D. Draw a perpendicular from D to AC at E. Finnaly,  CB=4 and CA=6.

source: ZhongKaoGeo


question: Diagram description: in rhombus ABCD, point P is the midpoint of CD, ∠BCD=60°. Ray AP intersects the extension of BC at E. Ray BP intersects DE at K. Point O is the midpoint of BK.

source: ZhongKaoGeo


question: Diagram description: Given a pentagon ABCDE, AB=AE=6, BC=5, ∠A=∠B=90°, ∠C=135°, ∠E>90°

source: ZhongKaoGeo


question: Diagram description: line AB // CD. A 60° right-angle triangle EFG (∠E=60°) has vertex F on line AB. EG intersects AB at H, CD intersects FG at M. Finnaly, ∠AHG=50°

source: ZhongKaoGeo


question: Diagram description: in isosceles △ABC, AB=AC. Points D and E are on AB and AC respectively, and AD=AE. Connect BE and CD, intersecting at F.

source: ZhongKaoGeo


question: Diagram description: in isosceles △ABC, AB=AC. Points D and E are on AB and AC respectively, AD=AE. BE and CD intersect at F.

source: ZhongKaoGeo


question: Diagram description: in △ABC, ∠A=36°, AB=AC. BD bisects ∠ABC

source: ZhongKaoGeo


question: Diagram description: P is a point on diameter AB. M and N are on circle O, and ∠APM = ∠NPB = 30°. Finnaly, OP = 2cm and AB = 16cm.

source: ZhongKaoGeo


question: Diagram description:  quadrilateral ABCD is a rhombus. Diagonals AC and BD intersect at O. DH ⊥ AB at H. DH intersects AC at E. HO is extended to intersect CD at G.

source: ZhongKaoGeo


question: Diagram description: A, B, and C are points on circle O. ∠ACB=25°

source: ZhongKaoGeo


question: Diagram description:  AB is tangent to circle O at B. OA=2*sqrt(3), ∠BAO=60°. Chord BC // OA.

source: ZhongKaoGeo


question: Diagram description: In right △ABC, AB=AC, ∠BAC=90°. D is the midpoint of BC. Square BDEF is constructed on side BD, with E coinciding with A. Finally, △ABF ∽ △CBE.

source: ZhongKaoGeo


question: Diagram description: in parallelogram ABCD, AB=3cm, BC=5cm, ∠B=60°. G is the midpoint of CD. E is a moving point on AD. The extension of EG intersects the extension of BC at F.

source: ZhongKaoGeo


question: Diagram description: in parallelogram ABCD, AB=3cm, BC=5cm, ∠B=60°. G is the midpoint of CD.

source: ZhongKaoGeo


question: Diagram description: in parallelogram ABCD, AB=3cm, BC=5cm, ∠B=60°. G is the midpoint of CD.

source: ZhongKaoGeo


question: Diagram description: in rectangle ABCD, extend AB to E and CD to F such that BE=DF. Connect EF, intersecting BC and AD at P and Q. Finally, BP=1, PQ=2*sqrt(2), ∠AEF=45°

source: ZhongKaoGeo


question: Diagram description: a large isosceles right triangle plate PCD has vertex P placed at the right-angle vertex of another isosceles right triangle PAB. PCD rotates around P. PC intersects AB at M, PD intersects AB at N. Finnaly,  AB=2, AN=x, BM=y.

source: ZhongKaoGeo


question: Diagram description: in △ABC, arcs with centers A, B and radii AC, BC intersect at C'. Connect BC', AC'. Take M on C'B. Arc with center C' and radius C'M intersects C'C at N. A line through C' bisects MN and intersects BC at P and AB at I. Finnaly, BC = 5/2 and CC' = 4.

source: ZhongKaoGeo


question: Diagram description: in △ABC, AB=CB. Circle O with diameter AB intersects AC at D. CF // AB. E is on CF such that DE=CD.

source: ZhongKaoGeo


question: Diagram description:  in △ABC, D and E are midpoints of AB and AC. BE and CD intersect at O.

source: ZhongKaoGeo


question: Diagram description: In equilateral △ABC, P and Q are on BC such that AP=AQ. Finnaly, ∠BAP=20°

source: ZhongKaoGeo


question: Diagram description: P and Q are points on BC. P is left of Q, AP=AQ. M is the reflection of Q across AC.

source: ZhongKaoGeo


question: Diagram description:  in parallelogram ABCD, E and F are on diagonal AC such that AE=CF.

source: ZhongKaoGeo


question: Diagram description: in rectangle ABCD, E is the midpoint of CD. F is on BC such that FC=2BF. Finnaly, AB=2 and AD=3.

source: ZhongKaoGeo


question: Diagram description:  in rectangle ABCD, F is on BC and AF=AD. DE ⊥ AF at E.

source: ZhongKaoGeo


question: Diagram description:  in rectangle ABCD, F is on BC, AF=AD. DE ⊥ AF at E. Arc with center A and radius AB intersects AF at G. Finnaly, BF=FC=1.

source: ZhongKaoGeo


question: Diagram description: EF passes through the intersection O of diagonals of parallelogram ABCD. E is on AD, F is on BC. Finnaly, the perimeter of ABCD is 18 and OE=1.5.

source: ZhongKaoGeo


question: Diagram description: in △ABC, AB=AC. BD and CE are altitudes intersecting at O. Finnaly, ∠ABC=50°.

source: ZhongKaoGeo


question: Diagram description: in rectangle ABCD, AB=20, BC=10. P is a moving point on AB. DP intersects AC at Q.

source: ZhongKaoGeo


question: Diagram description: in rectangle ABCD, AB=20, BC=10. P is a point on AB. DP intersects AC at Q.

source: ZhongKaoGeo


question: Diagram description: in △ABC, EF // BC and AB=3AE. Finnaly, the area of quadrilateral BCFE is 16

source: ZhongKaoGeo


question: Diagram description:  PA is tangent to circle O at A. PO extension intersects circle O at B. AO extension intersects circle O at C. CD ⊥ PB at E, and CD intersects circle O at D.

source: ZhongKaoGeo


question: Diagram description:  △ABC is inscribed in circle O with radius R, and ∠A=60°

source: ZhongKaoGeo


question: Diagram description:  circle O is the circumcircle of △ABC. PC is tangent at C. Extension of BA meets PC at P. Finnaly, AB=PA and PC=12

source: ZhongKaoGeo


question: Diagram description: diagonals AC and BD of rhombus ABCD are 6cm and 8cm respectively.

source: ZhongKaoGeo


question: Diagram description: in △ABC, ∠A=50°, ∠ABC=70°. BD bisects ∠ABC.

source: ZhongKaoGeo


question: Diagram description: line AB passes through C on circle O. AO intersects circle O at E and D. OB intersects circle O at F. Finnaly, OA=OB, CA=CB, DE=10, DF=6.

source: ZhongKaoGeo


question: Diagram description:  in quadrilateral ABCD, AB // DC, AB=AD. Diagonals AC and BD intersect at O. AC bisects ∠BAD. CE ⊥ extension of AB at E.

source: ZhongKaoGeo


question: Diagram description:  in quadrilateral ABCD, AB // DC, AB=AD. AC and BD intersect at O. AC bisects ∠BAD. CE ⊥ AB at E. Finnaly, AB=5 and BD=6

source: ZhongKaoGeo


question: Diagram description: circle O has radius 2. AB is diameter, CD is a chord. AB and CD intersect at M. Arc CD is folded so A coincides with O. AP=OA on extension of OA.

source: ZhongKaoGeo


question: Diagram description: △ABC, ∠ABC=90°. D is on AB such that DB=BC. EF ⊥ AC intersects AC at E and extension of CB at F.

source: ZhongKaoGeo


question: Diagram description:  AB is diameter of semicircle O. P is on extension of BA. PC is tangent at C. BD ⊥ extension of PC at D.

source: ZhongKaoGeo


question: Diagram description: circle O is the circumcircle of △ABC. AB=BC=CD, AB // CD. Connect BD.  Finnaly: AB=10 and cos∠BAC=3/5.

source: ZhongKaoGeo


question: Diagram description:  P is outside circle O. PC is tangent at C. Line PO intersects circle O at A and B. Also, ∠A=30°.

source: ZhongKaoGeo


question: Diagram description:  isosceles △ABC, AC=BC=10, AB=12. BC is the diameter of circle O intersecting AB at D and AC at G. DF ⊥ AC at F. DF intersects extension of CB at E.

source: ZhongKaoGeo


question: Diagram description: in △ABC and △ADE, AB=AD=6, BC=DE, ∠B=∠D=30°. AD intersects BC at P. I is the incenter of △APC.

source: ZhongKaoGeo


question: Diagram description: in △ABC and △ADE, AB=AD=6, BC=DE, ∠B=∠D=30°. Finnaly, AB ⊥ AC.

source: ZhongKaoGeo


question: Diagram description: P is a point on diagonal AC of rhombus ABCD (side=1). M and N are midpoints of AB and BC.

source: ZhongKaoGeo


question: Diagram description:  B, A, D, E are collinear. AB=AC=AD, ∠B=55°, BC=1.8m, DE=2m

source: ZhongKaoGeo


question: Diagram description:  BD ⊥ AC at D, CE ⊥ AB at E, AD=AE.

source: ZhongKaoGeo


question: Diagram description:  ABCD, E is on AD. Connect BE. Assume BE=BC and CF ⊥ BE at F.

source: ZhongKaoGeo


question: Diagram description: there is a rectangle ABCD, E is on AD. Connect BE. Assume Area(ABCD)=20.

source: ZhongKaoGeo


question: Diagram description:  in rhombus ABCD, cosA=1/3. CE ⊥ extension of AB at E. EF ⊥ AD at F. Assume Area(ABCD)=24

source: ZhongKaoGeo


question: Diagram description: parallelogram ABCD, ∠A=60°, AB=6, AD=5. E is on CD and CE=2. F is on BC. EG ⊥ EF intersects a side at G. Finnaly, EF·EG=7*sqrt(3).

source: ZhongKaoGeo


question: Diagram description:  in right △ABC, ∠ABC=90°. AB is the diameter of semicircle O intersecting AC at D. E is the midpoint of BC.

source: ZhongKaoGeo


question: Diagram description: right △ABC, ∠ABC=90°. AB is diameter of semicircle O. E is midpoint of BC. If ∠BAC=30° and DE=2.

source: ZhongKaoGeo


question: Diagram description: △ABC is inscribed in circle O. AB is diameter. D is outside circle O, DG // BC. DG intersects AC at G, AB at E, circle O at F. Connect DB, CF. ∠A=∠D.

source: ZhongKaoGeo


question: Diagram description: △ABC is inscribed in circle O. AB is diameter. DG // BC. If AE=OE, CF bisects ∠ACB, and BD=12

source: ZhongKaoGeo


question: Diagram description: rhombus ABCD, DE ⊥ CD intersects AC at E. Connect BE. P is a moving point on BE. P' is the reflection of P across DE. Q is a moving point on AC. Let AE=14 and CE=18.

source: ZhongKaoGeo


question: Diagram description:  triangle ABC congruent triangle ADE, Point A is their common vertex. if angle B = 70, angle C = 30, angle DAC = 35.

source: MathVerse


question: Diagram description:  in the rhombus ABCD, M and N are respectively AB and CD, and AM = CN, MN and AC intersect at point O. Connect BO. angle DAC = 28.

source: MathVerse


question: Diagram description: Circle I is the inscribed circle of triangle ABC, D, E, F are 3 tangent points, angle DEF = 52.

source: MathVerse


question: Diagram description:  the points B, E, C, and F are on the same straight line, triangle ABC congruent triangle DEF, angle B = 45, angle F = 65.

source: MathVerse


question: Diagram description:  AB parallel CD, point E is on the extended line of CA. angle BAE = 40.

source: MathVerse


question: Diagram description:  the perimeter of parallelogram ABCD is 32, AC, BD intersect at point O, and OE perpendicular AC and it intersects AD at point E.

source: MathVerse


question: Diagram description:  the circle O is the circumscribed circle of triangle ABC, and the bisector of angle BAC and angle ABC intersects at point I. Extend AI and it intersects circle O at point D. Connect BD and DC. the radius of circle O is 8, angle BAC = 120.

source: MathVerse


question: Diagram description:  the diameter AB of circle O is perpendicular to the chord CD, the foot of perpendicular is the point E, angle CAO = 22.5, OC = 6.

source: MathVerse


question: Diagram description:  in circle O, AB is the diameter, CD is the chord, AB perpendicular CD, the foot of perpendicular is the point E. angle BOC = 30.

source: MathVerse


question: Diagram description:  it is known that circle O is the circumscribed circle of triangle ABC, and AB is the diameter of circle O, if OC = 5, AC = 6.

source: MathVerse


question: Diagram description: In regular hexagon UVWXYZ, WY is perpendicular to XU, and the side is 12 centimeters long.

source: MathVerse


question: Diagram description: Two triangles MQN and MPO are similar with a shared vertex M. QN and PO are parallel. MQ = 5, MN = 6

source: MathVerse


question: Diagram description: In triangle ABC above, angle A is 60° and triangle ABC is equilateral. The length of AB is 3, and D is the midpoint of AC.

source: MathVerse


question: Diagram description: In triangle ABC above, AB = AC,point E lies on line AB, point D lies on line AC, E is the midpoint of line AB, and D is the midpoint of line AC.

source: MathVerse


question: Diagram description:  the two chords AB and CD in the circle intersect at E, ∠D = 35, ∠AEC = 105.

source: MathVista


question: Diagram description:  points A, B, C, and D are on circle O, and point E is on the extended line of AD. ∠ABC = 60.

source: MathVista


question: Diagram description:  AB is the diameter of circle O, DB and DC are respectively tangent to circle O at points B and C. If ∠ACE = 25.

source: MathVista


question: Diagram description:  ∠BAC = 110, if A and B are symmetrical with respect to the line MP, A and C are symmetrical with respect to the line NQ.

source: MathVista


question: Diagram description:  points A and B are three points on circle O and AB = AC. Connect BO and CO,  ∠ABC = 65.

source: MathVista


question: Diagram description:  in right triangle ABC, ∠BAC = 90, AD ⊥ BC at D, DE ⊥ AB at E, AD = 3, DE = 2.

source: MathVista


question: Diagram description:  AB is the diameter of circle O, point C is on circle O, AE is the tangent of circle O, A is the tangent point, connect BC and extend to intersect AE at point D. ∠AOC = 80.

source: MathVista


question: Diagram description:  in the two concentric circles, the chord AB of the great circle is tangent to the small circle at point C.  AB = 6.

source: MathVista


question: Diagram description:  line segment AB = 10, M is the midpoint of line segment AB, C is the midpoint of line segment MB, N is a point of line segment AM, and MN = 1.

source: MathVista


question: Diagram description:  in ▱ABCD, the diagonal AC and BD intersect at point O, if AC = 12, BD = 8, AB = 7.

source: MathVista


question: Diagram description:  PA and PB are tangents of circle O, the tangent point of point A and B, AC is the diameter of circle O, given that ∠P = 50.

source: MathVista


question: Diagram description:  AB is the diameter of circle O, point D is on the extended line of AB, passing point D is the tangent of circle O, and the tangent point is C,  ∠A = 25.

source: MathVista


question: Diagram description:  in right triangle ABC, ∠C = 90, ∠A = 30, BC = 2, the radius of circle C is 1, point P is the point on the hypotenuse AB, passing point P is a tangent PQ of circle C (Point Q is the tangent point).

source: MathVista


question: Diagram description:  triangle ABC is the inscribed triangle of circle O, angle C = 30, the radius of circle O is 5, point P is a point on circle O, in triangle ABP, PB = AB.

source: MathVerse


question: Diagram description:  in the right triangle ABC, angle CAB = 90, angle ABC = 72, AD is the angle bisector of angle CAB and intersects BC at point D, and CE is the altitude from point C onto side AD in triangle ACD.

source: MathVerse


question: Diagram description:  JKLM is a rectangle, MLPR is a rhombus, connect MK and connect MP. angle JMK = angle RMP, angle JMK = 55, and angle MRP = 70.

source: MathVerse


question: Diagram description:  in triangle ABC, angle C = 90, AC = BC, AD bisects angle CAB and it intersects BC at D, DE perpendicular AB at E, AB = 6.

source: MathVerse


question: Diagram description:  AB is the diameter of circle O, C is a point on circle O, the tangent of circle O passing through point C intersects the extended line of AB at point E, OD perpendicular AC at point D, angle E = 30, CE = 6.

source: MathVerse


question: Diagram description:  BD is the angular bisector of triangle ABC, AE perpendicular BD, and the foot of perpendicular is F, angle ABC = 35, angle C = 50.

source: MathVerse


question: Diagram description:  O is a point in the quadrilateral ABCD, OA = OB = OC, angle ABC = angle ADC = 65.

source: MathVerse


question: Diagram description:  the perimeter of parallelogram ABCD is 16, AC and BD intersect at point O, and OE perpendicular AC and it intersects AD at point E.

source: MathVerse


question: Diagram description:  in triangle PQR, ST parallel RQ, triangle PST is similar to triangle PRQ with a shared vertex P, PS = 12.5, SR = 5, PT = 15.

source: MathVerse


question: Diagram description:  AB is the diameter of circle O, CD is the chord of circle O, and the extended lines of AB and CD intersect at point E, AB = 2 DE, angle E = 16.

source: MathVerse


question: Diagram description:  in triangle ABC, AB = AC, angle A = 36, the perpendicular bisector of AB intersects AC at D, and intersects AB at E.

source: MathVerse


question: Diagram description:  in triangle ABC, angle BAC = 90, AD perpendicular BC at point D, AE bisects angle DAC, angle B = 50.

source: MathVerse


question: Diagram description:  in circle O, chord BC and radius OA intersect at point D. Connect AB and OC. angle A = 60, angle ADC = 90.

source: MathVerse


question: Diagram description:  the radius of circle O is 5, AB and CD are chords with central angles angle AOB and angle COD respectively, angle AOB is complementary to angle COD, and chord CD = 8.

source: MathVerse


question: Diagram description:  in circle O, point M is the midpoint of arc AB. Connect MO and extend it to intersect circle O at point N, connect BN, angle AOB = 140.

source: MathVerse


question: Diagram description:  JKLM is a rectangle, MLPR is a rhombus, ML = MR, angle JMK congruent angle RMP, m angle JMK = 55, and m angle MRP = 70.

source: MathVerse


question: Diagram description:  in circle O, chord AC and BD intersect at point E, arc AB = arc BC = arc CD, angle BEC = 110.

source: MathVerse


question: Diagram description:  in circle A, the chord BC = 8, DE = 6, angle BAC + angle EAD = 180.

source: MathVerse


question: Diagram description:  the quadrilateral ABCD has diagonals that intersect and are perpendicular to each other, AB = AD, BC = DC, the length from D to the diagonal intersection point is 5, and the length from C to the diagonal intersection point is 12.

source: MathVerse


question: Diagram description:  AB is the diameter of circle O, points C and D are on circle O, and point C is the midpoint of arc BD. Through point C draw the perpendicular line EF of AD, and it intersects straight line AD at point E. The radius of circle O is 2.5, and the length of AC is 4.

source: MathVerse


question: Diagram description:  in the inscribed pentagon ABCDE of circle O, angle CAD = 35, angle AED = 115.

source: MathVerse


question: Diagram description:  PA and PB are tangent to circle O at points A and B respectively, the tangent EF of circle O intersects PA and PB at points E and F respectively, and the tangent point C is on the arc AB, PA = 2.

source: MathVerse


question: Diagram description:  in triangle ABC, R, S, T are midpoints of AC, AB, and CB. AT, BR, and CS intersect at M. MC = 7, RM = 4, and AT = 16.

source: MathVerse


question: Diagram description:  in the circle O with a radius of 2, C is a point on the extended line of the diameter AB, CD is tangent to the circle at point D. Connect AD, angle DAC = 30.

source: MathVista


question: Diagram description:  the diameter CD of ⊙O crosses the midpoint G of chord EF, ∠DCF = 20.

source: MathVista


question: Diagram description:  in triangle ABC, D is the intersection point of the angle bisectors BD and CD of triangle ABC, angle A = 50.

source: MathVerse


question: Diagram description:  the angle between the diameter AB of circle O and the chord AC is 30, the tangent PC passing through point C and the extended line of AB intersect at point P, the radius of circle O is 2.

source: MathVerse


question: Diagram description:  point P is a point on the extended line of the diameter AB of circle O, passing through point P to draw the tangent PC of circle O, and the tangent point is C, AO = OB = PB = 1.

source: MathVerse


question: Diagram description:  a square DEFG is cut from a piece of triangular paper ABC. Among them, G and F are on BC, D and E are on AB and AC respectively, AH perpendicular BC and it intersects DE at M, BC = 12, AH = 8.

source: MathVerse


question: Diagram description:  PA and PB are tangents of circle O, with A and B the tangent points, AC is the diameter of circle O, angle P = 50.

source: MathVerse


question: Diagram description:  points A, B, and C are on circle O, and the tangent line of circle O passing through point A intersects the extended line of OC at point P, angle B = 30, OP = 3.

source: MathVerse


question: Diagram description:  polygon ABCD ~ polygon AEFG, angle AGF = 108, GF = 14, AD = 12, DG = 4.5, EF = 8, and AB = 26.

source: MathVerse


question: Diagram description:  in triangle ABC, the bisector of angle BAC is perpendicular to BC at point D, AB is equal to AC, AB = 6, BD = 3.

source: MathVerse


question: Diagram description:  the straight lines AB and CD intersect at point O, ray OM bisects angle AOC, ON perpendicular OM, angle AOC = 70.

source: MathVerse


question: Diagram description:  the diameter CD of circle O crosses the midpoint G of chord EF, angle DCF = 20.

source: MathVerse


question: Diagram description:  AB is the diameter of circle O, C and D are two points on circle O, angle BAC = 30, arc AD = arc CD.

source: MathVerse


question: Diagram description:  BA is the tangent of circle O, and OB intersects circle O at point C. angle B = 45 and the length of AB is 2.

source: MathVerse


question: Diagram description:  AB perpendicular DC and GH perpendicular FE, AC = 4.4, GH = 3.15, GE = 6.3, triangle ACD similar to triangle GEF.

source: MathVerse


question: Diagram description:  in parallelogram ABCD, BD is connected, BD perpendicular to AB, and BD perpendicular to DC, BD = 6m, AD = 10m.

source: MathVerse


question: Diagram description:  triangle ABC is inscribed in the circle with center O and diameter AC, and AB = AO.

source: MathVerse


question: Diagram description:  AB parallel CD, CP intersects AB at O, AO = PO, angle C = 50.

source: MathVerse


question: Diagram description:  AB parallel CD, AD bisects angle BAC, and angle C = 80.

source: MathVerse


question: Diagram description:  the straight line AB parallel CD, AE bisects angle CAB, angle ACD = 40.

source: MathVerse


question: Diagram description:  AB parallel CD, AE bisects angle CAB and intersects CD at point E, angle C = 70.

source: MathVerse


question: Diagram description:  points A, B, and C are on circle O, angle ABO = 22, angle ACO = 42.

source: MathVerse


question: Diagram description:  point A is a point outside circle O, AB, AC, and BD are the tangents of circle O, and the tangent points are P, C, and D respectively. AB = 5, AC = 3.

source: MathVerse


question: Diagram description:  circle O has a radius of 13 inches, with radii OC and OB. Radius OB is perpendicular to chord CD, which is 24 inches long. The arc CD = 134.

source: MathVerse


question: Diagram description:  two triangles GJI and GKH are similar with a shared vertex G, KH parallel JI, GJ = 8, GH = 12, and HI = 4.

source: MathVerse


question: Diagram description:  AB and CD are the two diameters of circle O, chord DE parallel AB, arc DE is the arc of 50.

source: MathVerse


question: Diagram description:  circle O has a radius of 13 inches, which is the length of OC. Radius OB is perpendicular at point X to chord CD, and CD is 24 inches long.

source: MathVerse


question: Diagram description:  a rectangle is inscribed in a circle, with two sides 8 inches and 6 inches.

source: MathVerse


question: Diagram description:  in circle G, CE and AB are diameters, angle AGD is a right angle, and angle AGC = 60.

source: MathVerse


question: Diagram description:  in triangle ABD, BC is perpendicular to AD, with AB length 32 and BD length 35, angle A = 50, angle D = 45.

source: MathVerse


question: Diagram description:  in triangle ABD, BC is perpendicular to AD, AB = 58, BD = 72, angle A = 57, angle D = 31.

source: MathVerse


question: Diagram description:  in triangle ABC, angle A = 60, angle B = 35, CD is the perpendicular from C to AB, and CD = 9.

source: MathVerse


question: Diagram description:  the quadrilateral ABCD is composed of two right triangles ABC and ACD, angle B is 35, angle BCD is a right angle, the length of BC is 13.

source: MathVerse


question: Diagram description:  the figure consists of a quadrilateral and a triangle, CDE is an equilateral triangle and ABCE is a square with an area of 1, forming polygon ABCDE.

source: MathVerse


question: Diagram description:  in the sector OAB with a radius of 1 and a central angle of 90, OA and OB are respectively the diameters of two semicircles, and there is a shaded part in the figure.

source: MathVerse


question: Diagram description:  AB is the diameter of circle O, point C is on circle O, passing point C to draw the tangent of circle O and it intersects the extended line of AB at point D. Connect AC. angle D = 50.

source: MathVerse


question: Diagram description:  the circle with center O has radius 7, AB is a diameter, and AC = BC.

source: MathVerse


question: Diagram description:  in right triangle GEF, the hypotenuse GF is 16 and HF is 12. EH is the altitude of triangle GEF drawn to the hypotenuse.

source: MathVerse


question: Diagram description:  circle O is the circumscribed circle of triangle ABC. Connect OB and OC, the radius of circle O is 2, angle BAC = 60.

source: MathVerse


question: Diagram description:  AB is tangent to circle O at point B, and the extended line of AO intersects circle O at point C. Connect BC, angle A = 36.

source: MathVerse


question: Diagram description:  in parallelogram ABCD, the diagonals AC and BD intersect at point O, angle DAC = 42, angle CBD = 23.

source: MathVerse


question: Diagram description:  in the circle O with a radius of 2, C is a point on the extended line of the diameter AB, CD is tangent to the circle at point D. Connect AD, angle DAC = 30.

source: MathVerse


question: Diagram description:  point P is outside circle O, PA and PB are connected, the straight lines PA and PB are the two tangents of circle O, angle APB = 120, the radius of circle O is 10.

source: MathVerse


question: Diagram description:  in triangle ABC, angle A = 80, angle B = 60, point D is on AB and point E is on AC, DE parallel BC.

source: MathVerse


question: Diagram description:  AB is the diameter of circle O, C and D are two points on circle O. Connect AC, BC, CD, and OD respectively. angle DOB = 140.

source: MathVerse


question: Diagram description:  two triangles XNM and XZY are similar with a shared vertex X, and MN parallel YZ. XN = 6, XM = 2, XY = 10.

source: MathVerse


question: Diagram description:  triangle ABE and triangle ACD are similar with a shared vertex A, and BE parallel CD. AB = 12, AC = 16, ED = 5.

source: MathVerse


question: Diagram description:  triangle ABE and triangle ACD are similar with a shared vertex A, BE parallel CD, AB = 12, AC = 16, ED = 5.

source: MathVerse


question: Diagram description:  in trapezoid JKLM, JK is parallel to ML, JM = 6, KL = 6, and angle JML is 80 degrees.

source: MathVerse


question: Diagram description:  PA and PB are tangent to circle O at A and B respectively, and angle C = 65.

source: MathVerse


question: Diagram description:  AB parallel CD, point E is on BC, and CD = CE, angle D = 74.

source: MathVerse


question: Diagram description:  AB is a diameter of the circle, C is a point on the circle, AC = 8 inches, and BC = 15 inches.

source: MathVista


question: Diagram description:  triangle ABC is the inscribed triangle of circle O, angle ACB = 30, AB = 6.

source: MathVerse


question: Diagram description:  BC and BA intersect at point B, BD bisects angle ABC, CD parallel AB, angle BCD = 70.

source: MathVerse


question: Diagram description:  triangle ABC congruent triangle DEF, points A and D, B and E are the corresponding vertices, BC = 5, BF = 7.

source: MathVerse


question: Diagram description:  the straight lines AB and CD intersect at point O, EO perpendicular AB, and the foot of perpendicular is point O, angle BOD = 50.

source: MathVerse


question: Diagram description:  AB is the diameter of circle O, points C and D are two points on the circle, and angle AOC = 126.

source: MathVerse


question: Diagram description:  in circle O, the length of chord AB is 10, and the circumference angle ACB = 45.

source: MathVerse


question: Diagram description:  AB is the diameter of circle O, C and D are two points on the circle, angle D = 34.

source: MathVerse


question: Diagram description:  AB is the diameter of circle O, and points C and D are on circle O, angle BOD = 130.

source: MathVerse


question: Diagram description:  in parallelogram ABCD, CE bisects angle BCD and intersects side AD at point E, and DE = 3.

source: MathVista


question: Diagram description:  ABCE is a square with an area of 1, and CDE is an equilateral triangle constructed on side CE with point D outside the square, so that ABCDE forms a pentagon.

source: MathVista


question: Diagram description:  in triangle ABC, the medians AD, BE, and CF intersect at point Q, where D, E, F are the midpoints of BC, CA, AB respectively, and BE = 9.

source: MathVerse


question: Diagram description:  in rectangle ABDF, point C is the midpoint of side BD and point E is the midpoint of side DF.

source: MathVerse


question: Diagram description:  triangles WSR and TVR are similar and share the vertex R, with WS parallel to VT, WS = 8, and VT = 10.

source: MathVerse


question: Diagram description:  triangle ABC is inscribed in circle O, AB is the diameter of circle O, point D is on circle O, and angle ACD = 40.

source: MathVista


question: Diagram description:  line AB is tangent to circle O at point A, the radius of circle O is 1, and angle OBA = 30.

source: MathVerse


question: Diagram description:  OA = OB = OC, and angle ACB = 30.

source: MathVerse


question: Diagram description:  AC is a diameter of circle J and BD is a diameter of circle K, and these diameters lie on the same straight line with the points in the order A, B, C, D. Circle J has radius 10, circle K has radius 8, and BC = 5.4.

source: MathVerse


question: Diagram description:  AC is the diameter of circle O, point B is on circle O, and angle OBC = 40.

source: MathVerse


question: Diagram description:  AB and AD are chords of circle O, and angle BOD = 50.

source: MathVerse


question: Diagram description:  triangle ABC is inscribed in circle O, and angle OAB = 35.

source: MathVerse