SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠MBC=20°, O is the center of ⊙O, BM⊥OM. Find the measure of ∠AOB.
问题：如图所示，∠MBC=20°，O是圆O的圆心，BM垂直于OM。求∠AOB的大小。
Construction (CDL):
Shape(OBC,CB)
Shape(BC,CM,MB)
Shape(BM,MO,OB)
Shape(MA,AO,OM)
Shape(OCA,AM,MC)
Shape(OAB,BO,OA)
Collinear(AMB)
Collinear(OMC)
Cocircular(O,CAB)
Conditions (CDL):
Equal(MeasureOfAngle(MBC),20)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(BM,OM)
Goal: Value(MeasureOfAngle(AOB))
Answer: 80
Theorem sequence:
circle_property_chord_perpendicular_bisect_chord(1,O,OM,BA)
arc_property_circumference_angle_external(1,OCA,B)
arc_property_center_angle(1,OCA,O)
perpendicular_bisector_property_bisector(1,OM,BA)
angle_addition(1,AOM,MOB)
Problem: As shown in the diagram, ∠CAB=50°, the center of circle O is O, the diameter of circle O is AB, E is the midpoint of segment CB. Find the measure of ∠ECD.
问题：如图所示，∠CAB=50°，⊙O的圆心为O，⊙O的直径为AB，E平分线段CB。求∠ECD的大小。
Construction (CDL):
Shape(OBE,EB)
Shape(BE,EA,AO,OB)
Shape(OED,DA,AE)
Shape(ODA,AD)
Shape(EC,CD,OED)
Collinear(AOB)
Collinear(ADC)
Collinear(CEB)
Cocircular(O,BEDA)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),50)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsMidpointOfLine(E,CB)
Goal: Value(MeasureOfAngle(ECD))
Answer: 65
Theorem sequence:
diameter_of_circle_property_right_angle(1,BEA,O)
adjacent_complementary_angle(1,BEA,AEC)
mirror_congruent_triangle_judgment_sas(1,EAB,ECA)
mirror_congruent_triangle_property_angle_equal(1,ABE,AEC)
triangle_property_angle_sum(1,BEA)
triangle_property_angle_sum(1,AEC)
triangle_property_angle_sum(1,ABC)
Problem: As shown in the diagram, the length of arc OAC is equal to the length of arc OCB, ∠BAO=37°, O is the center of circle O. Find the measure of ∠COA.
问题：如图所示，弧OAC的长度与⌒OCB的长度相等，∠BAO=37°，⊙O的圆心为O。求∠COA的大小。
Construction (CDL):
Shape(OB,BA,AO)
Shape(OBA,AB)
Shape(OAC,CO,OA)
Shape(OCB,BO,OC)
Cocircular(O,ACB)
Conditions (CDL):
Equal(LengthOfArc(OAC),LengthOfArc(OCB))
Equal(MeasureOfAngle(BAO),37)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COA))
Answer: 127
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
isosceles_triangle_judgment_line_equal(1,OBA)
isosceles_triangle_property_angle_equal(1,OBA)
triangle_property_angle_sum(1,OBA)
angle_addition(1,BOC,COA)
round_angle(1,AOB,BOA)
arc_property_center_angle(1,OAC,O)
arc_property_center_angle(1,OCB,O)
congruent_arc_judgment_length_equal(1,OAC,OCB)
congruent_arc_property_measure_equal(1,OAC,OCB)
Problem: As shown in the diagram, ∠AEB=36°, O is the center of circle O, the diameter of ⊙O is BE, quadrilateral ADCB is a ▱. Find the measure of ∠ADC.
问题：如图所示，∠AEB=36°，O是⊙O的圆心，圆O的直径为BE，ADCB是▱。求∠ADC的大小。
Construction (CDL):
Shape(BA,AD,DC,CB)
Shape(AE,EO,OC,CB,BA)
Shape(OBA,AB)
Shape(OAD,DA)
Shape(OAD,ODE,EA)
Shape(OEB,BC,CO,OE)
Collinear(BCOE)
Cocircular(O,BADE)
Conditions (CDL):
Equal(MeasureOfAngle(AEB),36)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BE,O)
Parallelogram(ADCB)
Goal: Value(MeasureOfAngle(ADC))
Answer: 54
Theorem sequence:
diameter_of_circle_property_right_angle(1,BAE,O)
triangle_property_angle_sum(1,AEB)
parallelogram_property_opposite_angle_equal(1,DCBA)
Problem: As shown in the diagram, ∠ACB=130°, the center of ⊙O is O. Find the measure of ∠BOA.
问题：如图所示，∠ACB=130°，⊙O的圆心为O。求∠BOA的大小。
Construction (CDL):
Shape(OA,AC,CB,BO)
Shape(OAC,CA)
Shape(OCB,BC)
Shape(OBA,AO,OB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(ACB),130)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 100
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_internal(1,OAB,C)
Problem: As shown in the diagram, ∠ACD=130°, O is the center of ⊙O, the diameter of ⊙O is AB. Find the measure of ∠BAD.
问题：如图所示，∠ACD=130°，O是⊙O的圆心，圆O的直径为AB。求∠BAD的大小。
Construction (CDL):
Shape(AC,CD,DA)
Shape(BO,OA,AD,DB)
Shape(OAC,CA)
Shape(OCD,DC)
Shape(ODB,BD)
Shape(OBA,AO,OB)
Collinear(AOB)
Cocircular(O,ACDB)
Conditions (CDL):
Equal(MeasureOfAngle(ACD),130)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BAD))
Answer: 40
Theorem sequence:
arc_property_circumference_angle_external(1,ODA,C)
arc_property_circumference_angle_internal(1,ODA,B)
diameter_of_circle_property_right_angle(1,ADB,O)
triangle_property_angle_sum(1,BAD)
Problem: As shown in the diagram, ∠BCA=36°, O is the center of circle O. Find the measure of ∠BOA.
问题：如图所示，∠BCA=36°，圆O的圆心为O。求∠BOA的大小。
Construction (CDL):
Shape(OAB,BO,BA)
Shape(OAB,BC,CA)
Shape(OBC,CB)
Shape(OCA,AC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),36)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 72
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠AOD=30°, O is the center of ⊙O, AB is the diameter of circle O. Find the measure of ∠BCD.
问题：如图所示，∠AOD=30°，O是圆O的圆心，AB是圆O的直径。求∠BCD的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(OAB,BO,BA)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(BC,CD,DO,OB)
Shape(DA,AO,OD)
Collinear(AOB)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(AOD),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BCD))
Answer: 105
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OD,O)
isosceles_triangle_judgment_line_equal(1,ODA)
isosceles_triangle_property_angle_equal(1,ODA)
triangle_property_angle_sum(1,DAO)
arc_property_circumference_angle_external(1,OBD,A)
arc_property_circumference_angle_internal(1,OBD,C)
Problem: As shown in the diagram, AC=8, O is the center of ⊙O, AB is the diameter of circle O, BD⊥OD. Find the length of line OD.
问题：如图所示，AC=8，O是⊙O的圆心，圆O的直径为AB，BD⊥OD。求直线OD的长度。
Construction (CDL):
Shape(OCA,AC)
Shape(OBC,CD,DB)
Shape(OAB,BO,OA)
Shape(CA,AO,OD,DC)
Shape(BD,DO,OB)
Collinear(AOB)
Collinear(CDB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(LengthOfLine(AC),8)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
PerpendicularBetweenLine(BD,OD)
Goal: Value(LengthOfLine(OD))
Answer: 4
Theorem sequence:
line_addition(1,AO,OB)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
diameter_of_circle_property_right_angle(1,BCA,O)
parallel_judgment_corresponding_angle(1,DO,CA,B)
parallel_property_corresponding_angle(2,AC,OD,B)
similar_triangle_judgment_aa(1,BDO,BCA)
similar_triangle_property_line_ratio(1,BDO,BCA)
similar_triangle_property_line_ratio(1,DOB,CAB)
Problem: As shown in the diagram, ∠BOC=50°, the center of ⊙O is O, OB∥AC. Find the measure of ∠OAB.
问题：如图所示，∠BOC=50°，⊙O的圆心为O，OB平行于AC。求∠OAB的大小。
Construction (CDL):
Shape(OA,AC,CO)
Shape(OA,AB,BO)
Shape(OAC,CA)
Shape(OCB,BA,AC)
Shape(OBA,AO,OB)
Shape(OCB,BO,OC)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(BOC),50)
IsCentreOfCircle(O,O)
ParallelBetweenLine(OB,AC)
Goal: Value(MeasureOfAngle(OAB))
Answer: 25
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
parallel_property_alternate_interior_angle(2,OB,AC)
isosceles_triangle_judgment_line_equal(1,OAB)
isosceles_triangle_property_angle_equal(1,OAB)
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_external(1,OCB,A)
Problem: As shown in the diagram, ∠CBA=40°, the center of ⊙O is O, AB is the diameter of ⊙O, OE is perpendicular to BE. Find the measure of ∠ECD.
问题：如图所示，∠CBA=40°，圆O的圆心为O，⊙O的直径为AB，OE垂直于BE。求∠ECD的大小。
Construction (CDL):
Shape(OE,EB,BO)
Shape(DE,EC,CD)
Shape(OCD,DC)
Shape(ODB,BE,ED)
Shape(OAC,CE,EO,OA)
Shape(OBA,AO,OB)
Collinear(DEO)
Collinear(CEB)
Collinear(BOA)
Cocircular(O,BACD)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
PerpendicularBetweenLine(OE,BE)
Goal: Value(MeasureOfAngle(ECD))
Answer: 25
Theorem sequence:
triangle_property_angle_sum(1,OEB)
arc_property_center_angle(1,ODB,O)
arc_property_circumference_angle_external(1,ODB,C)
Problem: As shown in the diagram, ∠OCA=30°, O is the center of ⊙O, the diameter of ⊙O is CD, AE is perpendicular to OE. Find the measure of ∠BOE.
问题：如图所示，∠OCA=30°，圆O的圆心为O，CD是⊙O的直径，AE垂直于OE。求∠BOE的大小。
Construction (CDL):
Shape(OAD,DE,EA)
Shape(ODB,BE,ED)
Shape(BO,OE,EB)
Shape(OA,AE,EO)
Shape(OBC,CO,OB)
Shape(OC,CA,AO)
Shape(OCA,AC)
Collinear(AEB)
Collinear(COED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(OCA),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(CD,O)
PerpendicularBetweenLine(AE,OE)
Goal: Value(MeasureOfAngle(BOE))
Answer: 60
Theorem sequence:
circle_property_chord_perpendicular_bisect_arc(1,OAB,OED)
congruent_arc_judgment_length_equal(1,OAD,ODB)
congruent_arc_property_measure_equal(1,OAD,ODB)
arc_property_circumference_angle_external(1,OAD,C)
arc_property_center_angle(1,ODB,O)
Problem: As shown in the diagram, ∠ECB=35°, O is the center of ⊙O, AB is the diameter of circle O. Find the measure of ∠ABD.
问题：如图所示，∠ECB=35°，O是⊙O的圆心，AB是⊙O的直径。求∠ABD的大小。
Construction (CDL):
Shape(ED,DA,AO,OE)
Shape(BD,DE,EB)
Shape(BE,EC,CB)
Shape(OAC,CE,EO,OA)
Shape(OCB,BC)
Shape(OBD,DB)
Shape(ODA,AD)
Collinear(AOEB)
Collinear(DEC)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(ECB),35)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ABD))
Answer: 55
Theorem sequence:
diameter_of_circle_property_right_angle(1,BDA,O)
arc_property_circumference_angle_external(1,OBD,A)
arc_property_circumference_angle_external(1,OBD,C)
triangle_property_angle_sum(1,DAB)
Problem: As shown in the diagram, the diameter of ⊙O is 6, AB=6, BC=3, O is the center of circle O, the diameter of circle O is AB. Find the measure of ∠CDB.
问题：如图所示，⊙O的直径为6，AB=6，BC=3，O是圆O的圆心，AB是圆O的直径。求∠CDB的大小。
Construction (CDL):
Shape(OAD,DE,EO,OA)
Shape(ODB,BD)
Shape(OBC,CB)
Shape(OCA,AO,OC)
Shape(CO,OE,EC)
Shape(CE,EB,BC)
Shape(BE,ED,DB)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(DiameterOfCircle(O),6)
Equal(LengthOfLine(AB),6)
Equal(LengthOfLine(BC),3)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CDB))
Answer: 30
Theorem sequence:
circle_property_length_of_radius_and_diameter(1,O)
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OC,O)
isosceles_triangle_judgment_line_equal(1,COB)
isosceles_triangle_judgment_line_equal(1,OBC)
equilateral_triangle_judgment_isosceles_and_isosceles(1,COB)
equilateral_triangle_property_angle(1,OBC)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,D)
Problem: As shown in the diagram, ∠CAB=40°, O is the center of circle O. Find the measure of ∠COB.
问题：如图所示，∠CAB=40°，O是⊙O的圆心。求∠COB的大小。
Construction (CDL):
Shape(OBC,CO,OB)
Shape(AB,BO,OC,CA)
Shape(OCA,AC)
Shape(OAB,BA)
Cocircular(O,BCA)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COB))
Answer: 80
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, the diameter of circle O is 8, AB=8, ∠EBO=30°, O is the center of circle O, AB is the diameter of ⊙O, AB is parallel to CD. Find the length of line AD.
问题：如图所示，⊙O的直径为8，AB=8，∠EBO=30°，⊙O的圆心为O，⊙O的直径为AB，AB∥CD。求直线AD的长度。
Construction (CDL):
Shape(OAC,CE,EA)
Shape(OCD,DC)
Shape(ODB,BD)
Shape(BO,OA,AE,EB)
Shape(BE,ED,DB)
Shape(DE,EC,CD)
Shape(OBA,AO,OB)
Collinear(AOB)
Collinear(AED)
Collinear(BEC)
Cocircular(O,ACDB)
Conditions (CDL):
Equal(DiameterOfCircle(O),8)
Equal(LengthOfLine(AB),8)
Equal(MeasureOfAngle(EBO),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
ParallelBetweenLine(AB,CD)
Goal: Value(LengthOfLine(AD))
Answer: 4*sqrt(3)
Theorem sequence:
parallel_property_alternate_interior_angle(2,AB,CD)
arc_property_circumference_angle_external(1,ODB,A)
arc_property_circumference_angle_external(1,ODB,C)
diameter_of_circle_property_right_angle(1,ADB,O)
triangle_property_angle_sum(1,BAD)
sine_theorem(1,ADB)
Problem: As shown in the diagram, ∠CEA=20°, O is the center of ⊙O, AB is the diameter of circle O. Find the measure of ∠CDB.
问题：如图所示，∠CEA=20°，O是圆O的圆心，AB是⊙O的直径。求∠CDB的大小。
Construction (CDL):
Shape(EA,AF,FE)
Shape(EF,FO,OB,BE)
Shape(FC,CD,DB,BO,OF)
Shape(OAC,CF,FA)
Shape(OCD,DC)
Shape(ODB,BD)
Shape(OBE,EB)
Shape(OEA,AE)
Collinear(AFOB)
Collinear(EFC)
Cocircular(O,ACDBE)
Conditions (CDL):
Equal(MeasureOfAngle(CEA),20)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CDB))
Answer: 110
Theorem sequence:
diameter_of_circle_property_right_angle(1,BEA,O)
angle_addition(1,BEF,FEA)
arc_property_circumference_angle_internal(1,OBC,E)
arc_property_circumference_angle_external(1,OBC,D)
Problem: As shown in the diagram, ∠APD=80°, ∠DAP=55°, O is the center of circle O. Find the measure of ∠CBP.
问题：如图所示，∠APD=80°，∠DAP=55°，⊙O的圆心为O。求∠CBP的大小。
Construction (CDL):
Shape(PC,CB,BP)
Shape(AP,PD,DA)
Shape(OAC,CP,PA)
Shape(OCB,BC)
Shape(ODA,AD)
Shape(OBD,DP,PB)
Collinear(CPD)
Collinear(APB)
Cocircular(O,CBDA)
Conditions (CDL):
Equal(MeasureOfAngle(APD),80)
Equal(MeasureOfAngle(DAP),55)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CBP))
Answer: 45
Theorem sequence:
triangle_property_angle_sum(1,APD)
arc_property_circumference_angle_external(1,OAC,B)
arc_property_circumference_angle_external(1,OAC,D)
Problem: As shown in the diagram, ∠ACB=45°, the radius of circle O is 6, O is the center of circle O, BD is perpendicular to OD. Find the length of line OD.
问题：如图所示，∠ACB=45°，⊙O的半径为6，O是圆O的圆心，BD⊥OD。求直线OD的长度。
Construction (CDL):
Shape(OBA,AD,DB)
Shape(OAC,CA)
Shape(OCB,BC)
Shape(CB,BO,OA,AC)
Shape(OB,BD,DO)
Shape(OD,DA,AO)
Collinear(BDA)
Cocircular(O,BAC)
Conditions (CDL):
Equal(MeasureOfAngle(ACB),45)
Equal(RadiusOfCircle(O),6)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(BD,OD)
Goal: Value(LengthOfLine(OD))
Answer: 3*sqrt(2)
Theorem sequence:
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OA,O)
arc_property_circumference_angle_external(1,OBA,C)
arc_property_center_angle(1,OBA,O)
right_triangle_judgment_angle(1,BDO)
right_triangle_judgment_angle(1,AOB)
right_triangle_property_pythagorean(1,BDO)
right_triangle_property_pythagorean(1,AOB)
line_addition(1,BD,DA)
circle_property_chord_perpendicular_bisect_chord(1,O,OD,BA)
Problem: As shown in the diagram, ∠BCD=110°, the center of ⊙O is O, AB is the diameter of circle O. Find the measure of ∠AED.
问题：如图所示，∠BCD=110°，圆O的圆心为O，⊙O的直径为AB。求∠AED的大小。
Construction (CDL):
Shape(ODA,AE,ED)
Shape(OAE,EA)
Shape(BC,CA,AO,OB)
Shape(ODA,AC,CD)
Shape(OCD,DC)
Shape(OBC,CB)
Shape(OEB,BO,OA,AE)
Collinear(AOB)
Cocircular(O,EBCDA)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),110)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(AED))
Answer: 20
Theorem sequence:
angle_addition(1,BCA,ACD)
diameter_of_circle_property_right_angle(1,BCA,O)
arc_property_circumference_angle_external(1,ODA,E)
arc_property_circumference_angle_external(1,ODA,C)
Problem: As shown in the diagram, ∠BAO=35°, ∠BOD=76°, AB is parallel to CD. Find the measure of ∠OCD.
问题：如图所示，∠BAO=35°，∠BOD=76°，AB∥CD。求∠OCD的大小。
Construction (CDL):
Shape(AO,OB,BA)
Shape(OC,CD,DO)
Shape(BO,OD)
Collinear(AOD)
Collinear(BOC)
Conditions (CDL):
Equal(MeasureOfAngle(BAO),35)
Equal(MeasureOfAngle(BOD),76)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(OCD))
Answer: 41
Theorem sequence:
parallel_property_alternate_interior_angle(1,AB,CD)
adjacent_complementary_angle(1,BOD,DOC)
triangle_property_angle_sum(1,OCD)
Problem: As shown in the diagram, ∠ABF=39°, ∠FCE=90°, EF∥AB. Find the measure of ∠AED.
问题：如图所示，∠ABF=39°，∠FCE=90°，EF∥AB。求∠AED的大小。
Construction (CDL):
Shape(CE,EF,FC)
Shape(EA,AB,BF,FE)
Shape(AE,ED)
Collinear(DEF)
Collinear(CEA)
Collinear(CFB)
Conditions (CDL):
Equal(MeasureOfAngle(ABF),39)
Equal(MeasureOfAngle(FCE),90)
ParallelBetweenLine(EF,AB)
Goal: Value(MeasureOfAngle(AED))
Answer: 51
Theorem sequence:
parallel_property_collinear_extend(1,EF,AB,D)
parallel_property_alternate_interior_angle(2,DE,AB)
triangle_property_angle_sum(1,CAB)
Problem: As shown in the diagram, ∠COA=52°, O is the center of circle O, AC is perpendicular to OC. Find the measure of ∠BED.
问题：如图所示，∠COA=52°，O是圆O的圆心，AC⊥OC。求∠BED的大小。
Construction (CDL):
Shape(OAD,DC,CA)
Shape(ODB,BC,CD)
Shape(ODB,BE,ED)
Shape(OA,AC,CO)
Shape(OBE,EB)
Shape(OEA,AO,OC,CD,CE)
Collinear(ACB)
Collinear(OCD)
Cocircular(O,ADBE)
Conditions (CDL):
Equal(MeasureOfAngle(COA),52)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AC,OC)
Goal: Value(MeasureOfAngle(BED))
Answer: 26
Theorem sequence:
circle_property_chord_perpendicular_bisect_arc(1,OAB,OCD)
congruent_arc_judgment_length_equal(1,OAD,ODB)
congruent_arc_property_measure_equal(1,OAD,ODB)
arc_property_center_angle(1,OAD,O)
arc_property_circumference_angle_external(1,ODB,E)
Problem: As shown in the diagram, ∠BDE=15°, ∠ECA=25°, the center of ⊙O is O. Find the measure of ∠BOA.
问题：如图所示，∠BDE=15°，∠ECA=25°，O是⊙O的圆心。求∠BOA的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(OBD,DB)
Shape(ODC,CE,ED)
Shape(OAE,EC,CA)
Shape(OEB,BD,DE)
Shape(OAE,OEB,BO,OA)
Cocircular(O,AEBDC)
Conditions (CDL):
Equal(MeasureOfAngle(BDE),15)
Equal(MeasureOfAngle(ECA),25)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 80
Theorem sequence:
arc_addition_measure(1,OAE,OEB)
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAE,C)
arc_property_circumference_angle_external(1,OEB,D)
Problem: As shown in the diagram, ∠EDB=35°, the center of ⊙O is O, the diameter of circle O is AB. Find the measure of ∠BCO.
问题：如图所示，∠EDB=35°，圆O的圆心为O，AB是圆O的直径。求∠BCO的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(AO,OC,CA)
Shape(OE,EC,CO)
Shape(CE,EB,BC)
Shape(CB,OBC)
Shape(DE,EA,OAD)
Shape(BE,ED,DB)
Shape(BD,ODB)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,CADB)
Conditions (CDL):
Equal(MeasureOfAngle(EDB),35)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BCO))
Answer: 55
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,D)
arc_property_center_angle(1,OBC,O)
radius_of_circle_property_length_equal(1,OC,O)
radius_of_circle_property_length_equal(1,OB,O)
isosceles_triangle_judgment_line_equal(1,OBC)
isosceles_triangle_property_angle_equal(1,OBC)
triangle_property_angle_sum(1,COB)
Problem: As shown in the diagram, ∠AFO=90°, ∠BOC=70°, O is the center of circle O. Find the measure of ∠CDA.
问题：如图所示，∠AFO=90°，∠BOC=70°，⊙O的圆心为O。求∠CDA的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(DA,AE,ED)
Shape(DE,EF,FO,OD)
Shape(DO,OB,OBD)
Shape(OF,FB,BO)
Shape(CE,EA,OAC)
Shape(FE,EC,CF)
Shape(BF,FC,OCB)
Collinear(AEFB)
Collinear(OFC)
Collinear(DEC)
Cocircular(O,DACB)
Conditions (CDL):
Equal(MeasureOfAngle(AFO),90)
Equal(MeasureOfAngle(BOC),70)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CDA))
Answer: 35
Theorem sequence:
arc_property_center_angle(1,OCB,O)
circle_property_chord_perpendicular_bisect_arc(1,OAB,OFC)
congruent_arc_judgment_length_equal(1,OAC,OCB)
congruent_arc_property_measure_equal(1,OAC,OCB)
arc_property_circumference_angle_external(1,OAC,D)
Problem: As shown in the diagram, ∠ABO=30°, ∠OCA=40°, O is the center of ⊙O. Find the measure of ∠COB.
问题：如图所示，∠ABO=30°，∠OCA=40°，O是圆O的圆心。求∠COB的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AC,OCA)
Shape(CO,OB,OBC)
Shape(AB,BO,OA)
Shape(AO,OC,CA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(ABO),30)
Equal(MeasureOfAngle(OCA),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COB))
Answer: 140
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OC,O)
isosceles_triangle_judgment_line_equal(1,OAB)
isosceles_triangle_judgment_line_equal(1,OCA)
isosceles_triangle_property_angle_equal(1,OAB)
isosceles_triangle_property_angle_equal(1,OCA)
angle_addition(1,CAO,OAB)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,OBC,O)
Problem: As shown in the diagram, the length of arc OAB is equal to the length of ⌒OBC, ∠CDB=20°, the center of ⊙O is O. Find the measure of ∠BOA.
问题：如图所示，⌒OAB的长度与⌒OBC的长度相等，∠CDB=20°，⊙O的圆心为O。求∠BOA的大小。
Construction (CDL):
Shape(AO,OB,BD,ODA)
Shape(CD,DB,OBC)
Shape(DC,OCD)
Shape(BO,OA,OAB)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(LengthOfArc(OAB),LengthOfArc(OBC))
Equal(MeasureOfAngle(CDB),20)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 40
Theorem sequence:
congruent_arc_judgment_length_equal(1,OAB,OBC)
arc_property_circumference_angle_external(1,OBC,D)
congruent_arc_property_measure_equal(1,OAB,OBC)
arc_property_center_angle(1,OAB,O)
Problem: As shown in the diagram, ∠CAB=50°, ∠BCA is the right angle of the right △ BCA. Find the measure of ∠CBD.
问题：如图所示，∠CAB=50°，△BCA为直角△。求∠CBD的大小。
Construction (CDL):
Shape(CA,AB,BC)
Shape(CB,BD,DC)
Collinear(ABD)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),50)
RightTriangle(BCA)
Goal: Value(MeasureOfAngle(CBD))
Answer: 140
Theorem sequence:
triangle_property_angle_sum(1,CAB)
adjacent_complementary_angle(1,ABC,CBD)
Problem: As shown in the diagram, ∠ABD=28°, the center of ⊙O is O, the diameter of ⊙O is AB. Find the measure of ∠DAB.
问题：如图所示，∠ABD=28°，O是⊙O的圆心，AB是⊙O的直径。求∠DAB的大小。
Construction (CDL):
Shape(DB,OBD)
Shape(AD,ODA)
Shape(AE,ED,DA)
Shape(DE,EB,BD)
Shape(CE,EA,OAC)
Shape(EC,CB,BE)
Shape(BC,OCB)
Collinear(DEC)
Collinear(AEOB)
Cocircular(O,DACB)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),28)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(DAB))
Answer: 62
Theorem sequence:
diameter_of_circle_property_right_angle(1,BDA,O)
triangle_property_angle_sum(1,DAB)
Problem: As shown in the diagram, ∠APB=70°, ∠CDO=90°, ∠OEC=90°, O is the center of circle O. Find the measure of ∠ECD.
问题：如图所示，∠APB=70°，∠CDO=90°，∠OEC=90°，⊙O的圆心为O。求∠ECD的大小。
Construction (CDL):
Shape(AD,DC,OCA)
Shape(CD,DO,OE,EC)
Shape(CE,EB,OBC)
Shape(BP,OPB)
Shape(BO,OA,AP,PB)
Shape(PA,OAP)
Collinear(OEB)
Collinear(ADO)
Cocircular(O,CAPB)
Conditions (CDL):
Equal(MeasureOfAngle(APB),70)
Equal(MeasureOfAngle(CDO),90)
Equal(MeasureOfAngle(OEC),90)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ECD))
Answer: 40
Theorem sequence:
arc_property_circumference_angle_external(1,OBA,P)
arc_property_center_angle(1,OBA,O)
quadrilateral_property_angle_sum(1,CDOE)
Problem: As shown in the diagram, ∠DOA=50°, O is the center of ⊙O, AO is parallel to DC. Find the measure of ∠CBA.
问题：如图所示，∠DOA=50°，圆O的圆心为O，AO∥DC。求∠CBA的大小。
Construction (CDL):
Shape(AB,OBA)
Shape(BA,AO,OC,CB)
Shape(OD,DC,CO)
Shape(DO,OA,OAD)
Shape(CD,ODC)
Shape(BC,OCB)
Cocircular(O,BADC)
Conditions (CDL):
Equal(MeasureOfAngle(DOA),50)
IsCentreOfCircle(O,O)
ParallelBetweenLine(AO,DC)
Goal: Value(MeasureOfAngle(CBA))
Answer: 65
Theorem sequence:
parallel_property_alternate_interior_angle(2,AO,DC)
radius_of_circle_property_length_equal(1,OD,O)
radius_of_circle_property_length_equal(1,OC,O)
isosceles_triangle_judgment_line_equal(1,ODC)
isosceles_triangle_property_angle_equal(1,ODC)
triangle_property_angle_sum(1,ODC)
angle_addition(1,COD,DOA)
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,B)
Problem: As shown in the diagram, ∠COB=80°, the center of circle O is O. Find the measure of ∠CAB.
问题：如图所示，∠COB=80°，圆O的圆心为O。求∠CAB的大小。
Construction (CDL):
Shape(BO,OA,OAB)
Shape(OB,BD,DO)
Shape(OD,DA,AO)
Shape(CD,DB,BC)
Shape(AD,DC,CA)
Shape(AC,OCA)
Shape(CB,OBC)
Collinear(ODC)
Collinear(BDA)
Cocircular(O,BCA)
Conditions (CDL):
Equal(MeasureOfAngle(COB),80)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 40
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠DBC=70°, ∠OAD=20°, O is the center of ⊙O. Find the measure of ∠BCA.
问题：如图所示，∠DBC=70°，∠OAD=20°，⊙O的圆心为O。求∠BCA的大小。
Construction (CDL):
Shape(AO,OC,OCA)
Shape(OA,AD,DO)
Shape(OD,DC,CO)
Shape(BD,DA,OAB)
Shape(BC,CD,DB)
Shape(CB,OBC)
Collinear(ADC)
Collinear(ODB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(DBC),70)
Equal(MeasureOfAngle(OAD),20)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 50
Theorem sequence:
vertical_angle(1,ADO,CDB)
arc_property_circumference_angle_external(1,OAB,C)
arc_property_center_angle(1,OAB,O)
triangle_property_angle_sum(1,OAD)
triangle_property_angle_sum(1,CDB)