SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, AD=CD, ∠CAB=70°, O is the center of ⊙O, BC is the diameter of circle O. Find the measure of ∠BOE.
问题：如图所示，AD=CD，∠CAB=70°，⊙O的圆心为O，BC是⊙O的直径。求∠BOE的大小。
Construction (CDL):
Shape(BE,OEB)
Shape(BF,FE,EB)
Shape(EF,FG,GE)
Shape(EG,GD,ODE)
Shape(DG,GC,CD)
Shape(DC,OCD)
Shape(CG,GF,FO,OC)
Shape(FB,BO,OF)
Shape(DA,AE,ODE)
Shape(CB,OBC)
Collinear(BOC)
Collinear(EFO)
Collinear(AEB)
Collinear(DGFB)
Collinear(EGC)
Collinear(ADC)
Cocircular(O,EBCD)
Conditions (CDL):
Equal(LengthOfLine(AD),LengthOfLine(CD))
Equal(MeasureOfAngle(CAB),70)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BC,O)
Goal: Value(MeasureOfAngle(BOE))
Answer: 100
Theorem sequence:
diameter_of_circle_property_right_angle(1,CEB,O)
diameter_of_circle_property_right_angle(1,CDB,O)
perpendicular_bisector_judgment_per_and_mid(1,BD,CA)
perpendicular_bisector_property_distance_equal(1,BD,CA)
isosceles_triangle_judgment_line_equal(1,BCA)
isosceles_triangle_property_angle_equal(1,BCA)
triangle_property_angle_sum(1,EBC)
triangle_property_angle_sum(1,BCA)
arc_property_circumference_angle_external(1,OEB,C)
arc_property_center_angle(1,OEB,O)
Problem: As shown in the diagram, OB=CD, O is the center of ⊙O, the diameter of circle O is BD. Find the measure of ∠BAC.
问题：如图所示，OB=CD，O是⊙O的圆心，⊙O的直径为BD。求∠BAC的大小。
Construction (CDL):
Shape(AB,OBA)
Shape(BC,OCB)
Shape(BA,AE,EB)
Shape(BO,OC,CB)
Shape(OE,EC,CO)
Shape(CE,ED,DC)
Shape(CD,ODC)
Shape(DE,EA,OAD)
Collinear(BOED)
Collinear(AEC)
Cocircular(O,BADC)
Conditions (CDL):
Equal(LengthOfLine(OB),LengthOfLine(CD))
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BD,O)
Goal: Value(MeasureOfAngle(BAC))
Answer: 60
Theorem sequence:
radius_of_circle_property_length_equal(1,OB,O)
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_judgment_line_equal(1,DCO)
isosceles_triangle_property_angle_equal(1,ODC)
isosceles_triangle_property_angle_equal(1,DCO)
triangle_property_angle_sum(1,COD)
arc_property_circumference_angle_external(1,OCB,A)
arc_property_circumference_angle_external(1,OCB,D)
Problem: As shown in the diagram, ∠DAB=100°, the center of ⊙O is O. Find the measure of ∠BOD.
问题：如图所示，∠DAB=100°，圆O的圆心为O。求∠BOD的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(BA,OAB)
Shape(AB,BO,OD,DA)
Shape(DO,OB,BC,CD)
Shape(CB,OBC)
Shape(DC,OCD)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),100)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOD))
Answer: 160
Theorem sequence:
arc_property_circumference_angle_internal(1,ODB,A)
arc_property_center_angle(1,ODB,O)
Problem: As shown in the diagram, ∠ABD=32°, O is the center of circle O. Find the measure of ∠AOC.
问题：如图所示，∠ABD=32°，圆O的圆心为O。求∠AOC的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AB,BD,DA)
Shape(AD,DC,CA)
Shape(AC,OCA)
Shape(CD,DO,OC)
Shape(CO,OD,DB,OBC)
Collinear(ADO)
Collinear(BDC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),32)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AOC))
Answer: 64
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,B)
arc_property_center_angle(1,OCA,O)
Problem: As shown in the diagram, ∠CDB=36°, the center of ⊙O is O, AB is the diameter of circle O. Find the measure of ∠ABC.
问题：如图所示，∠CDB=36°，圆O的圆心为O，圆O的直径为AB。求∠ABC的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AE,EC)
Shape(CE,EB,BC)
Shape(CB,OBC)
Shape(BE,ED,DB)
Shape(BD,ODB)
Shape(DE,EA,OAD)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,CADB)
Conditions (CDL):
Equal(MeasureOfAngle(CDB),36)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ABC))
Answer: 54
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_external(1,OBC,D)
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,CAB)
Problem: As shown in the diagram, ∠COA=100°, the center of ⊙O is O, the diameter of ⊙O is BD, AE is perpendicular to OE. Find the measure of ∠CDB.
问题：如图所示，∠COA=100°，O是⊙O的圆心，BD是⊙O的直径，AE⊥OE。求∠CDB的大小。
Construction (CDL):
Shape(AO,OD,ODA)
Shape(DO,OC,CD)
Shape(DC,OCD)
Shape(OA,AE,EO)
Shape(OE,EC,CO)
Shape(BE,EA,OAB)
Shape(CE,EB,OBC)
Collinear(DOEB)
Collinear(AEC)
Cocircular(O,DABC)
Conditions (CDL):
Equal(MeasureOfAngle(COA),100)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BD,O)
PerpendicularBetweenLine(AE,OE)
Goal: Value(MeasureOfAngle(CDB))
Answer: 25
Theorem sequence:
arc_property_center_angle(1,OAC,O)
circle_property_chord_perpendicular_bisect_arc(1,OAC,OEB)
congruent_arc_judgment_length_equal(1,OAB,OBC)
congruent_arc_property_measure_equal(1,OAB,OBC)
arc_addition_measure(1,OAB,OBC)
arc_property_circumference_angle_external(1,OBC,D)
Problem: As shown in the diagram, ∠BOA=82°, the center of ⊙O is O. Find the measure of ∠BCA.
问题：如图所示，∠BOA=82°，O是⊙O的圆心。求∠BCA的大小。
Construction (CDL):
Shape(BO,OA,OAB)
Shape(AO,OB,BC,CA)
Shape(AC,OCA)
Shape(CB,OBC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),82)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 41
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠CAB=20°, the center of circle O is O, AB is the diameter of ⊙O. Find the measure of ∠CDA.
问题：如图所示，∠CAB=20°，O是⊙O的圆心，⊙O的直径为AB。求∠CDA的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(DC,OCD)
Shape(CB,OBC)
Shape(DA,AC,CD)
Shape(CA,AB,BC)
Shape(BA,OAB)
Collinear(AOB)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),20)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CDA))
Answer: 110
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,CAB)
arc_property_circumference_angle_internal(1,OAC,B)
arc_property_circumference_angle_external(1,OAC,D)
Problem: As shown in the diagram, ∠DAB=56°, the center of circle O is O, the diameter of circle O is AB. Find the measure of ∠ABD.
问题：如图所示，∠DAB=56°，圆O的圆心为O，⊙O的直径为AB。求∠ABD的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(DA,AB,BD)
Shape(OBD,DB)
Collinear(AOB)
Cocircular(O,DAB)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),56)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ABD))
Answer: 34
Theorem sequence:
diameter_of_circle_property_right_angle(1,BDA,O)
triangle_property_angle_sum(1,DAB)
Problem: As shown in the diagram, ∠ACB=67°, ∠AOP=42°, O is the center of circle O. Find the measure of ∠POB.
问题：如图所示，∠ACB=67°，∠AOP=42°，⊙O的圆心为O。求∠POB的大小。
Construction (CDL):
Shape(OB,OBP,PO)
Shape(OP,OPA,AO)
Shape(OAC,CA)
Shape(OCB,BC)
Shape(CB,BO,OA,AC)
Cocircular(O,CBPA)
Conditions (CDL):
Equal(MeasureOfAngle(ACB),67)
Equal(MeasureOfAngle(AOP),42)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(POB))
Answer: 92
Theorem sequence:
arc_property_center_angle(1,OBP,O)
arc_property_center_angle(1,OPA,O)
arc_property_circumference_angle_external(1,OBA,C)
arc_addition_measure(1,OBP,OPA)
Problem: As shown in the diagram, ∠BAO=40°, ∠OHA=90°, O is the center of ⊙O. Find the measure of ∠ABC.
问题：如图所示，∠BAO=40°，∠OHA=90°，圆O的圆心为O。求∠ABC的大小。
Construction (CDL):
Shape(AO,OH,HA)
Shape(AH,HC,OCA)
Shape(CH,HB,BC)
Shape(OBC,CB)
Collinear(OHC)
Collinear(AHB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BAO),40)
Equal(MeasureOfAngle(OHA),90)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ABC))
Answer: 25
Theorem sequence:
triangle_property_angle_sum(1,AOH)
arc_property_center_angle(1,OCA,O)
arc_property_circumference_angle_external(1,OCA,B)
Problem: As shown in the diagram, ∠ABC=20°, O is the center of circle O, the diameter of ⊙O is AB, C bisects arc OBD. Find the measure of ∠DOA.
问题：如图所示，∠ABC=20°，O是圆O的圆心，AB是⊙O的直径，C平分⌒OBD。求∠DOA的大小。
Construction (CDL):
Shape(OCA,AF,FC)
Shape(OC,CF,FO)
Shape(CO,OB,BC)
Shape(OBC,CB)
Shape(FA,OAD,DF)
Shape(OF,FD,DO)
Shape(BO,OD,ODB)
Collinear(AFOB)
Collinear(CFD)
Cocircular(O,CADB)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),20)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsMidpointOfArc(C,OBD)
Goal: Value(MeasureOfAngle(DOA))
Answer: 100
Theorem sequence:
arc_property_center_angle(1,OCA,O)
arc_property_center_angle(1,OCD,O)
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_external(1,OCA,B)
arc_addition_measure(1,OBC,OCA)
flat_angle(1,AOB)
angle_addition(1,DOF,FOC)
congruent_arc_judgment_length_equal(1,OCD,OBC)
congruent_arc_property_measure_equal(1,OCD,OBC)
Problem: As shown in the diagram, ∠CAB=50°, ∠COE=∠EOB, the center of ⊙O is O, E bisects segment BC. Find the measure of ∠BDO.
问题：如图所示，∠CAB=50°，∠COE=∠EOB，O是圆O的圆心，E平分线段BC。求∠BDO的大小。
Construction (CDL):
Shape(AB,BO,OC,CA)
Shape(OB,BE,EO)
Shape(OE,EC,CO)
Shape(EB,BD,DE)
Shape(OBD,DB)
Shape(CE,ED,ODC)
Collinear(BEC)
Collinear(OED)
Cocircular(O,ABDC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),50)
Equal(MeasureOfAngle(COE),MeasureOfAngle(EOB))
IsCentreOfCircle(O,O)
IsMidpointOfLine(E,BC)
Goal: Value(MeasureOfAngle(BDO))
Answer: 65
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
angle_addition(1,COD,DOB)
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OD,O)
isosceles_triangle_judgment_line_equal(1,OBD)
isosceles_triangle_property_angle_equal(1,OBD)
triangle_property_angle_sum(1,OBD)
Problem: As shown in the diagram, ∠CAB=50°, the center of circle O is O, AB is the diameter of ⊙O, ∠ABC and ∠BCA are the base angles of the isosceles triangle ABC. Find the measure of ⌒ODE.
问题：如图所示，∠CAB=50°，圆O的圆心为O，圆O的直径为AB，AB和AC是等腰三角形ABC的腰。求⌒ODE的角度。
Construction (CDL):
Shape(OEA,AE)
Shape(EA,AD,ODE)
Shape(AO,OB,BD,DA)
Shape(ODE,DC,CE)
Shape(OBD,DB)
Collinear(BDC)
Collinear(AOB)
Collinear(AEC)
Cocircular(O,ABDE)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),50)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsoscelesTriangle(ABC)
Goal: Value(MeasureOfArc(ODE))
Answer: 50
Theorem sequence:
diameter_of_circle_property_right_angle(1,BDA,O)
altitude_of_triangle_judgment(1,AD,ABC)
isosceles_triangle_property_line_coincidence(1,ABC,D)
angle_addition(1,CAD,DAB)
arc_property_circumference_angle_external(1,ODE,A)
Problem: As shown in the diagram, CB=CD, ∠ACD=45°, ∠DAC=30°, O is the center of circle O. Find the measure of ∠OAB.
问题：如图所示，CB=CD，∠ACD=45°，∠DAC=30°，O是⊙O的圆心。求∠OAB的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(OCD,DC)
Shape(OBC,CB)
Shape(CD,DE,EC)
Shape(ED,DA,AE)
Shape(EA,AO,OB,BE)
Shape(BC,CE,EB)
Shape(OA,AB,BO)
Collinear(BED)
Collinear(AEC)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(LengthOfLine(CB),LengthOfLine(CD))
Equal(MeasureOfAngle(ACD),45)
Equal(MeasureOfAngle(DAC),30)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(OAB))
Answer: 15
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,D)
arc_property_circumference_angle_external(1,OBD,A)
arc_property_circumference_angle_external(1,ODA,B)
arc_property_circumference_angle_external(1,OCD,A)
arc_property_circumference_angle_external(1,ODA,C)
arc_property_circumference_angle_external(1,OBC,D)
arc_property_circumference_angle_external(1,OCD,B)
arc_property_circumference_angle_external(1,ODB,C)
congruent_arc_judgment_chord_equal(1,OBC,OCD)
congruent_arc_property_measure_equal(1,OBC,OCD)
round_arc(1,OBD,ODB)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
isosceles_triangle_judgment_line_equal(1,OAB)
isosceles_triangle_property_angle_equal(1,OAB)
triangle_property_angle_sum(1,OAB)
triangle_property_angle_sum(1,CDB)
triangle_property_angle_sum(1,DAB)
Problem: As shown in the diagram, ∠AOC=Mul(2,MeasureOfAngle(ABC))°, ∠COD=Sub(147,Mul(2,MeasureOfAngle(ABC)))°, ∠ODC=90°, Sub(MeasureOfAngle(BCA)=MeasureOfAngle(ABC)), O is the center of ⊙O. Find the measure of ∠AOD.
问题：如图所示，∠AOC=Mul(2,MeasureOfAngle(ABC))°，∠COD=Sub(147,Mul(2,MeasureOfAngle(ABC)))°，∠ODC=90°，Sub(MeasureOfAngle(BCA)=MeasureOfAngle(ABC))，O是圆O的圆心。求∠AOD的大小。
Construction (CDL):
Shape(AB,BO,OA)
Shape(OB,BD,DO)
Shape(OD,DC,CO)
Shape(AO,OC,CA)
Collinear(BDC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(AOC),Mul(2,MeasureOfAngle(ABC)))
Equal(MeasureOfAngle(COD),Sub(147,Mul(2,MeasureOfAngle(ABC))))
Equal(MeasureOfAngle(ODC),90)
Equal(Sub(MeasureOfAngle(BCA),MeasureOfAngle(ABC)),33)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AOD))
Answer: 147
Theorem sequence:
angle_addition(1,AOC,COD)
Problem: As shown in the diagram, ∠CEO=90°, ∠DOC=80°, ∠OED=90°, O is the center of ⊙O, the diameter of ⊙O is AB. Find the measure of ∠OAC.
问题：如图所示，∠CEO=90°，∠DOC=80°，∠OED=90°，O是圆O的圆心，AB是⊙O的直径。求∠OAC的大小。
Construction (CDL):
Shape(OA,AC,CO)
Shape(EO,OC,CE)
Shape(DO,OE,ED)
Shape(OBD,DE,EB)
Shape(OCB,BE,EC)
Collinear(AOEB)
Collinear(DEC)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(CEO),90)
Equal(MeasureOfAngle(DOC),80)
Equal(MeasureOfAngle(OED),90)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(OAC))
Answer: 20
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OC,O)
radius_of_circle_property_length_equal(1,OD,O)
isosceles_triangle_judgment_line_equal(1,OAC)
isosceles_triangle_judgment_line_equal(1,OCD)
isosceles_triangle_property_angle_equal(1,OAC)
isosceles_triangle_property_angle_equal(1,OCD)
triangle_property_angle_sum(1,OAC)
triangle_property_angle_sum(1,EOC)
triangle_property_angle_sum(1,OCD)
flat_angle(1,EOA)
angle_addition(1,EOC,COA)
Problem: As shown in the diagram, ∠CAD=25°, O is the center of ⊙O, AB is the diameter of circle O, AD∥OC, DA∥CO. Find the measure of ∠BOC.
问题：如图所示，∠CAD=25°，O是圆O的圆心，圆O的直径为AB，AD∥OC，DA∥CO。求∠BOC的大小。
Construction (CDL):
Shape(CA,AD,ODC)
Shape(OA,AC,CO)
Shape(BO,OC,OCB)
Shape(OAD,DA)
Collinear(AOB)
Cocircular(O,ADCB)
Conditions (CDL):
Equal(MeasureOfAngle(CAD),25)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
ParallelBetweenLine(AD,OC)
ParallelBetweenLine(DA,CO)
Goal: Value(MeasureOfAngle(BOC))
Answer: 50
Theorem sequence:
angle_addition(1,OAC,CAD)
parallel_property_corresponding_angle(1,OC,AD,B)
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_external(1,OCB,A)
Problem: As shown in the diagram, ∠CAB=24°, O is the center of ⊙O, AB is the diameter of circle O. Find the measure of ∠CDA.
问题：如图所示，∠CAB=24°，O是⊙O的圆心，AB是圆O的直径。求∠CDA的大小。
Construction (CDL):
Shape(DA,AE,ED)
Shape(CD,DE,EC)
Shape(EA,AO,OB,BE)
Shape(CE,EB,OBC)
Collinear(AOB)
Collinear(DEB)
Collinear(AEC)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),24)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CDA))
Answer: 114
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_external(1,OAC,D)
arc_addition_measure(1,OAB,OBC)
flat_angle(1,BOA)
Problem: As shown in the diagram, ∠DCE=65°, O is the center of circle O. Find the measure of ∠DAB.
问题：如图所示，∠DCE=65°，O是圆O的圆心。求∠DAB的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(ODA,AD)
Shape(DC,CE)
Shape(AB,BC,CD,DA)
Collinear(BCE)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(DCE),65)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DAB))
Answer: 65
Theorem sequence:
arc_property_circumference_angle_external(1,ODB,C)
flat_angle(1,BCE)
angle_addition(1,BCD,DCE)
arc_property_circumference_angle_internal(1,ODB,A)
Problem: As shown in the diagram, the length of ⌒ODA is equal to the length of ⌒OCB, ∠ACE=50°, the center of circle O is O. Find the measure of ∠BEC.
问题：如图所示，⌒ODA的长度与⌒OCB的长度相等，∠ACE=50°，O是⊙O的圆心。求∠BEC的大小。
Construction (CDL):
Shape(EA,AC,CE)
Shape(DE,EB,OBD)
Shape(EC,OCB,BE)
Shape(AE,ED,ODA)
Collinear(AEB)
Collinear(CED)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(LengthOfArc(ODA),LengthOfArc(OCB))
Equal(MeasureOfAngle(ACE),50)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BEC))
Answer: 100
Theorem sequence:
congruent_arc_judgment_length_equal(1,ODA,OCB)
congruent_arc_property_measure_equal(1,ODA,OCB)
arc_property_circumference_angle_external(1,OCB,A)
arc_property_circumference_angle_external(1,ODA,C)
triangle_property_angle_sum(1,EAC)
flat_angle(1,BEA)
angle_addition(1,BEC,CEA)
Problem: As shown in the diagram, AB=AC, ∠CAB=50°, O is the center of circle O, the diameter of circle O is BE. Find the measure of ∠CDB.
问题：如图所示，AB=AC，∠CAB=50°，⊙O的圆心为O，⊙O的直径为BE。求∠CDB的大小。
Construction (CDL):
Shape(AB,BO,OD,DA)
Shape(DO,OB,BC,CD)
Shape(ED,DC,CE)
Collinear(ADC)
Collinear(EDOB)
Cocircular(O,ABCE)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(MeasureOfAngle(CAB),50)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BE,O)
Goal: Value(MeasureOfAngle(CDB))
Answer: 75
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,ABC)
isosceles_triangle_property_angle_equal(1,ABC)
triangle_property_angle_sum(1,DBC)
triangle_property_angle_sum(1,ABC)
triangle_property_angle_sum(1,BCE)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_external(1,OBC,E)
diameter_of_circle_property_right_angle(1,BCE,O)
Problem: As shown in the diagram, ∠COB=20°, ∠OEA=90°, the center of circle O is O, the diameter of ⊙O is AB. Find the measure of ∠ACD.
问题：如图所示，∠COB=20°，∠OEA=90°，O是圆O的圆心，AB是圆O的直径。求∠ACD的大小。
Construction (CDL):
Shape(AE,ED,ODA)
Shape(EA,AO,OE)
Shape(CD,DE,EC)
Shape(CE,EO,OC)
Shape(CO,OB,OBC)
Shape(OCD,DC)
Collinear(AOB)
Collinear(DEO)
Collinear(AEC)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(COB),20)
Equal(MeasureOfAngle(OEA),90)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ACD))
Answer: 40
Theorem sequence:
arc_property_center_angle(1,ODA,O)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_external(1,ODA,C)
triangle_property_angle_sum(1,EAO)
Problem: As shown in the diagram, AB=OA, the center of ⊙O is O. Find the measure of ∠BDA.
问题：如图所示，AB=OA，⊙O的圆心为O。求∠BDA的大小。
Construction (CDL):
Shape(BO,OE,EB)
Shape(BE,EA,AB)
Shape(OAB,BA)
Shape(ED,DA,AE)
Shape(ODA,AD)
Collinear(OEA)
Collinear(BED)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(OA))
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BDA))
Answer: 30
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,BOA)
isosceles_triangle_judgment_line_equal(1,OAB)
equilateral_triangle_judgment_isosceles_and_isosceles(1,BOA)
equilateral_triangle_property_angle(1,OAB)
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,D)
Problem: As shown in the diagram, ∠BOC=130°, the center of ⊙O is O, the diameter of ⊙O is AB. Find the measure of ∠CDA.
问题：如图所示，∠BOC=130°，圆O的圆心为O，AB是⊙O的直径。求∠CDA的大小。
Construction (CDL):
Shape(OC,OCB,BO)
Shape(CO,OE,EC)
Shape(ED,DA,AE)
Collinear(AEOB)
Collinear(CED)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(BOC),130)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CDA))
Answer: 25
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_external(1,OAC,D)
arc_addition_measure(1,OAC,OCB)
flat_angle(1,BOA)
Problem: As shown in the diagram, ∠AOB=122°, O is the center of circle O. Find the measure of ∠BCA.
问题：如图所示，∠AOB=122°，O是圆O的圆心。求∠BCA的大小。
Construction (CDL):
Shape(CA,AO,OB,BC)
Shape(OA,AD,DB,BO)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(AOB),122)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 119
Theorem sequence:
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_internal(1,OBA,C)
Problem: As shown in the diagram, Add(MeasureOfAngle(ADC)=MeasureOfAngle(CBA)), ∠BQA=110°, the center of ⊙O is O, QA is the tangent to ⊙O, QB is the tangent to circle O. Find the sum of the measure of ∠ADC and the measure of ∠CBQ.
问题：如图所示，Add(MeasureOfAngle(ADC)=MeasureOfAngle(CBA))，∠BQA=110°，⊙O的圆心为O，圆O的切线为QA，⊙O的切线为QB。求∠ADC的大小与∠CBQ的大小之和。
Construction (CDL):
Shape(AD,DC,CB,BA)
Shape(OAD,DA)
Shape(ODC,CD)
Shape(OCB,BC)
Shape(OBA,AB)
Shape(QA,OBA,BQ)
Cocircular(O,DCBA)
Conditions (CDL):
Equal(Add(MeasureOfAngle(ADC),MeasureOfAngle(CBA)),180)
Equal(MeasureOfAngle(BQA),110)
IsCentreOfCircle(O,O)
IsTangentOfCircle(QA,O)
IsTangentOfCircle(QB,O)
Goal: Value(Add(MeasureOfAngle(ADC),MeasureOfAngle(CBQ)))
Answer: 215
Theorem sequence:
tangent_of_circle_property_length_equal(1,QA,QB,O)
isosceles_triangle_judgment_line_equal(1,QAB)
isosceles_triangle_property_angle_equal(1,QAB)
triangle_property_angle_sum(1,ABQ)
angle_addition(1,CBA,ABQ)
Problem: As shown in the diagram, CD=AD, DA=DC, ∠BCD=90°, ∠DCO=90°, ∠OAB=90°, CF is the angle bisector of ∠BCD, the center of ⊙O is O, AC is the median of △ ABO. Find the measure of ∠EOC.
问题：如图所示，CD=AD，DA=DC，∠BCD=90°，∠DCO=90°，∠OAB=90°，CF平分∠BCD，O是⊙O的圆心，AC是三角形ABO的中线。求∠EOC的大小。
Construction (CDL):
Shape(OC,CE,EO)
Shape(OA,AC,CO)
Shape(OAC,CA)
Shape(OAC,AD,DC)
Shape(CD,DF,FC)
Shape(CF,FB,BC)
Collinear(ECF)
Collinear(OCB)
Collinear(ADFB)
Cocircular(O,ACE)
Conditions (CDL):
Equal(LengthOfLine(CD),LengthOfLine(AD))
Equal(LengthOfLine(DA),LengthOfLine(DC))
Equal(MeasureOfAngle(BCD),90)
Equal(MeasureOfAngle(DCO),90)
Equal(MeasureOfAngle(OAB),90)
IsBisectorOfAngle(CF,BCD)
IsCentreOfCircle(O,O)
IsMedianOfTriangle(AC,ABO)
Goal: Value(MeasureOfAngle(EOC))
Answer: 90
Theorem sequence:
angle_addition(1,BCF,FCD)
vertical_angle(1,OCE,BCF)
radius_of_circle_property_length_equal(1,OC,O)
radius_of_circle_property_length_equal(1,OE,O)
isosceles_triangle_judgment_line_equal(1,OCE)
isosceles_triangle_property_angle_equal(1,OCE)
triangle_property_angle_sum(1,OCE)
Problem: As shown in the diagram, ∠CAB=35°, O is the center of circle O, AB is the diameter of circle O. Find the measure of ∠ADC.
问题：如图所示，∠CAB=35°，O是圆O的圆心，⊙O的直径为AB。求∠ADC的大小。
Construction (CDL):
Shape(CA,AO,OE,EC)
Shape(CE,EB,BC)
Shape(OBC,CB)
Shape(EO,OA,AD,DE)
Shape(BE,ED,ODB)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),35)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ADC))
Answer: 55
Theorem sequence:
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_external(1,OCA,D)
arc_property_circumference_angle_external(1,OBC,A)
arc_addition_measure(1,OBC,OCA)
flat_angle(1,AOE)
Problem: As shown in the diagram, the length of ⌒OAC is equal to the length of arc OCB, ∠BOA=90°, the measure of arc OAC is equal to the measure of ⌒OCB, O is the center of ⊙O. Find the measure of ∠CPA.
问题：如图所示，⌒OAC的长度与弧OCB的长度相等，∠BOA=90°，⌒OAC的角度与⌒OCB的角度相等，O是圆O的圆心。求∠CPA的大小。
Construction (CDL):
Shape(OA,AD,DO)
Shape(PD,DE,EP)
Shape(ED,DA,OAC,CE)
Shape(EC,OCB,BE)
Shape(PE,EB,OBP)
Collinear(ODEB)
Collinear(PDA)
Collinear(PEC)
Cocircular(O,ACBP)
Conditions (CDL):
Equal(LengthOfArc(OAC),LengthOfArc(OCB))
Equal(MeasureOfAngle(BOA),90)
Equal(MeasureOfArc(OAC),MeasureOfArc(OCB))
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CPA))
Answer: 45/2
Theorem sequence:
arc_addition_measure(1,OAC,OCB)
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAC,P)
Problem: As shown in the diagram, ∠CEA=64°, ∠COA=90°, O is the center of circle O, AB is the diameter of circle O. Find the measure of ∠DAB.
问题：如图所示，∠CEA=64°，∠COA=90°，O是圆O的圆心，圆O的直径为AB。求∠DAB的大小。
Construction (CDL):
Shape(DA,AO,OE,ED)
Shape(OA,OAC,CO)
Shape(EO,OC,CE)
Shape(DE,EB,OBD)
Collinear(AOEB)
Collinear(DEC)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(CEA),64)
Equal(MeasureOfAngle(COA),90)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(DAB))
Answer: 19
Theorem sequence:
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,D)
triangle_property_angle_sum(1,DAE)
flat_angle(1,CED)
angle_addition(1,CEA,AED)
Problem: As shown in the diagram, ∠COB=100°, the center of ⊙O is O, AD is parallel to CB, DA is parallel to BC. Find the measure of ∠ADC.
问题：如图所示，∠COB=100°，⊙O的圆心为O，AD平行于CB，DA平行于BC。求∠ADC的大小。
Construction (CDL):
Shape(AD,DO,OC,CA)
Shape(OB,BC,CO)
Collinear(DOC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(COB),100)
IsCentreOfCircle(O,O)
ParallelBetweenLine(AD,CB)
ParallelBetweenLine(DA,BC)
Goal: Value(MeasureOfAngle(ADC))
Answer: 40
Theorem sequence:
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OC,O)
isosceles_triangle_judgment_line_equal(1,OBC)
isosceles_triangle_property_angle_equal(1,OBC)
triangle_property_angle_sum(1,OBC)
parallel_property_alternate_interior_angle(1,DA,BC)