SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠ACB=90°, ∠ACE=60°, the center of circle O is O, the diameter of ⊙O is AB. Find the measure of ∠AOE.
问题：如图所示，∠ACB=90°，∠ACE=60°，⊙O的圆心为O，AB是圆O的直径。求∠AOE的大小。
Construction (CDL):
Shape(OBE,EO,OB)
Shape(OE,EF,FO)
Shape(OEA,AF,FE)
Shape(FA,AC,CF)
Shape(OCB,BC)
Shape(BO,OF,FC,CB)
Shape(OAC,CA)
Collinear(BOFA)
Collinear(EFC)
Cocircular(O,BEAC)
Conditions (CDL):
Equal(MeasureOfAngle(ACB),90)
Equal(MeasureOfAngle(ACE),60)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(AOE))
Answer: 120
Theorem sequence:
arc_property_center_angle(1,OEA,O)
arc_property_circumference_angle_external(1,OEA,C)
Problem: As shown in the diagram, ∠AOB=45°, O is the center of ⊙O. Find the measure of ∠ACB.
问题：如图所示，∠AOB=45°，圆O的圆心为O。求∠ACB的大小。
Construction (CDL):
Shape(OAC,CA)
Shape(OCB,BC)
Shape(AC,CB,BO,OA)
Shape(OBA,AO,OB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(AOB),45)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ACB))
Answer: 45/2
Theorem sequence:
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_external(1,OBA,C)
Problem: As shown in the diagram, the length of ⌒ODB is equal to the length of arc OBC, ∠CAB=20°, the center of circle O is O, CF is perpendicular to BF. Find the measure of ∠DOA.
问题：如图所示，弧ODB的长度与⌒OBC的长度相等，∠CAB=20°，⊙O的圆心为O，CF⊥BF。求∠DOA的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(CA,AO,OF,FC)
Shape(OBC,CF,FB)
Shape(ODB,BF,FD)
Shape(OD,DF,FO)
Shape(OAD,DO,OA)
Collinear(AOFB)
Collinear(CFD)
Cocircular(O,CADB)
Conditions (CDL):
Equal(LengthOfArc(ODB),LengthOfArc(OBC))
Equal(MeasureOfAngle(CAB),20)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(CF,BF)
Goal: Value(MeasureOfAngle(DOA))
Answer: 140
Theorem sequence:
congruent_arc_judgment_length_equal(1,OBC,ODB)
congruent_arc_property_measure_equal(1,OBC,ODB)
arc_property_center_angle(1,ODB,O)
arc_property_circumference_angle_external(1,OBC,A)
adjacent_complementary_angle(1,FOD,DOA)
Problem: As shown in the diagram, ∠DCB=50°, O is the center of ⊙O, the diameter of ⊙O is AB. Find the measure of ∠ABD.
问题：如图所示，∠DCB=50°，圆O的圆心为O，AB是圆O的直径。求∠ABD的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(OAC,CE,EA)
Shape(DA,AE,ED)
Shape(DE,EO,OB,BD)
Shape(OBD,DB)
Shape(BO,OE,EC,CB)
Shape(BC,OCB)
Collinear(AEOB)
Collinear(DEC)
Cocircular(O,DACB)
Conditions (CDL):
Equal(MeasureOfAngle(DCB),50)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ABD))
Answer: 40
Theorem sequence:
arc_property_circumference_angle_external(1,OBD,A)
arc_property_circumference_angle_external(1,OBD,C)
diameter_of_circle_property_right_angle(1,BDA,O)
triangle_property_angle_sum(1,DAB)
Problem: As shown in the diagram, the length of arc ODA is equal to the length of ⌒OCD, ∠AOD=70°, ∠CAB=20°, the center of ⊙O is O, AB is the diameter of circle O. Find the measure of ∠DAC.
问题：如图所示，弧ODA的长度与弧OCD的长度相等，∠AOD=70°，∠CAB=20°，O是⊙O的圆心，圆O的直径为AB。求∠DAC的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(ED,DA,AE)
Shape(EA,AO,OE)
Shape(CE,EO,OC)
Shape(CD,DE,EC)
Shape(OCD,DC)
Shape(OBC,CO,OB)
Shape(BO,OA,OAB)
Collinear(DEO)
Collinear(CEA)
Collinear(AOB)
Cocircular(O,BCDA)
Conditions (CDL):
Equal(LengthOfArc(ODA),LengthOfArc(OCD))
Equal(MeasureOfAngle(AOD),70)
Equal(MeasureOfAngle(CAB),20)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(DAC))
Answer: 35
Theorem sequence:
arc_property_center_angle(1,ODA,O)
arc_property_circumference_angle_external(1,OCD,A)
congruent_arc_judgment_length_equal(1,OCD,ODA)
congruent_arc_property_measure_equal(1,OCD,ODA)
Problem: As shown in the diagram, the measure of ⌒OAB is 40, the center of circle O is O. Find the measure of ∠BCA.
问题：如图所示，弧OAB的角度为40，⊙O的圆心为O。求∠BCA的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(CA,OAB,BC)
Shape(OBC,CB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfArc(OAB),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 20
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠ABO=50°, the center of circle O is O. Find the measure of ∠BCA.
问题：如图所示，∠ABO=50°，O是圆O的圆心。求∠BCA的大小。
Construction (CDL):
Shape(OCA,AO,OE,EC)
Shape(EO,OA,AE)
Shape(EA,AB,BE)
Shape(OAB,BA)
Shape(CE,EB,BC)
Shape(OBC,CB)
Collinear(OEB)
Collinear(AEC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(ABO),50)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 40
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,OAB)
isosceles_triangle_property_angle_equal(1,OAB)
triangle_property_angle_sum(1,OAB)
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠AEB=36°, the center of circle O is O, the diameter of circle O is BE, AB and DC are opposite sides of the parallelogram BADC. Find the measure of ∠ADC.
问题：如图所示，∠AEB=36°，O是圆O的圆心，⊙O的直径为BE，AB和DC是平行四边形BADC的一组对边。求∠ADC的大小。
Construction (CDL):
Shape(OEB,BC,CO,OE)
Shape(OBA,AB)
Shape(CB,BA,AF,FC)
Shape(FA,AD,DF)
Shape(OAD,DA)
Shape(EO,OC,CF,FE)
Shape(EF,FD,ODE)
Collinear(BCOE)
Collinear(CFD)
Collinear(EFA)
Cocircular(O,BADE)
Conditions (CDL):
Equal(MeasureOfAngle(AEB),36)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BE,O)
Parallelogram(BADC)
Goal: Value(MeasureOfAngle(ADC))
Answer: 54
Theorem sequence:
diameter_of_circle_property_right_angle(1,BAE,O)
parallel_property_collinear_extend(2,BC,AD,E)
parallel_property_alternate_interior_angle(2,BE,AD)
angle_addition(1,BAF,FAD)
parallel_property_ipsilateral_internal_angle(1,AB,DC)
Problem: As shown in the diagram, ∠DOA=70°, O is the center of circle O, AO is parallel to DC. Find the measure of ∠CBA.
问题：如图所示，∠DOA=70°，圆O的圆心为O，AO平行于DC。求∠CBA的大小。
Construction (CDL):
Shape(OBA,AB)
Shape(OAD,DO,OA)
Shape(ODC,CD)
Shape(OD,DC,CO)
Shape(BA,AO,OC,CB)
Shape(OCB,BC)
Cocircular(O,BADC)
Conditions (CDL):
Equal(MeasureOfAngle(DOA),70)
IsCentreOfCircle(O,O)
ParallelBetweenLine(AO,DC)
Goal: Value(MeasureOfAngle(CBA))
Answer: 55
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, ∠BAO=25°, the center of circle O is O, AC∥OB. Find the measure of ∠COB.
问题：如图所示，∠BAO=25°，O是圆O的圆心，AC平行于OB。求∠COB的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(OBC,CE,EB)
Shape(CA,AE,EC)
Shape(EA,AO,OE)
Shape(BE,EO,OB)
Shape(OAB,BO,OA)
Collinear(CEO)
Collinear(AEB)
Cocircular(O,CAB)
Conditions (CDL):
Equal(MeasureOfAngle(BAO),25)
IsCentreOfCircle(O,O)
ParallelBetweenLine(AC,OB)
Goal: Value(MeasureOfAngle(COB))
Answer: 50
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)
parallel_property_alternate_interior_angle(1,AC,OB)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠CBA=60°, ∠DCB=70°, the center of ⊙O is O, AB is the diameter of circle O. Find the measure of ∠BOD.
问题：如图所示，∠CBA=60°，∠DCB=70°，O是⊙O的圆心，AB是⊙O的直径。求∠BOD的大小。
Construction (CDL):
Shape(OBA,AO,OB)
Shape(OAD,DA)
Shape(OA,AD,DO)
Shape(OD,ODE,EB,BO)
Shape(OEB,BE)
Shape(DC,CE,ODE)
Collinear(CEB)
Collinear(AOB)
Collinear(ADC)
Cocircular(O,ADEB)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),60)
Equal(MeasureOfAngle(DCB),70)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BOD))
Answer: 100
Theorem sequence:
triangle_property_angle_sum(1,ACB)
arc_property_center_angle(1,ODB,O)
arc_property_circumference_angle_external(1,ODB,A)
Problem: As shown in the diagram, ∠BOA=100°, O is the center of circle O. Find the measure of ∠ACB.
问题：如图所示，∠BOA=100°，O是⊙O的圆心。求∠ACB的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(OBD,DB)
Shape(BD,DA,AO,OB)
Shape(BO,OA,AC,CB)
Shape(OAC,CA)
Shape(OCB,BC)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),100)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ACB))
Answer: 130
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_internal(1,OAB,C)
Problem: As shown in the diagram, ∠BOA=100°, O is the center of circle O. Find the measure of ∠BCA.
问题：如图所示，∠BOA=100°，O是⊙O的圆心。求∠BCA的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(CA,AO,OB,BC)
Shape(OBC,CB)
Shape(OAB,BO,OA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),100)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 50
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠CBD=20°, O is the center of circle O, BD is the diameter of circle O. Find the measure of ∠BAC.
问题：如图所示，∠CBD=20°，⊙O的圆心为O，圆O的直径为BD。求∠BAC的大小。
Construction (CDL):
Shape(OBA,AB)
Shape(BA,AE,EO,OB)
Shape(BO,OE,EC,CB)
Shape(OCB,BC)
Shape(EA,AD,DE)
Shape(OAD,DA)
Shape(ODC,CE,ED)
Collinear(BOED)
Collinear(AEC)
Cocircular(O,BADC)
Conditions (CDL):
Equal(MeasureOfAngle(CBD),20)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BD,O)
Goal: Value(MeasureOfAngle(BAC))
Answer: 70
Theorem sequence:
diameter_of_circle_property_right_angle(1,BAD,O)
arc_property_circumference_angle_external(1,ODC,B)
arc_property_circumference_angle_external(1,ODC,A)
angle_addition(1,BAE,EAD)
Problem: As shown in the diagram, ∠ADC=35°, O is the center of circle O, OP⊥AP. Find the measure of ∠DAO.
问题：如图所示，∠ADC=35°，⊙O的圆心为O，OP垂直于AP。求∠DAO的大小。
Construction (CDL):
Shape(OAB,BO,OA)
Shape(AO,OP,PA)
Shape(AP,PC,OCA)
Shape(ODC,CD)
Shape(CP,PD,DC)
Shape(PO,OB,OBD,DP)
Collinear(BOPC)
Collinear(APD)
Cocircular(O,ABDC)
Conditions (CDL):
Equal(MeasureOfAngle(ADC),35)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(OP,AP)
Goal: Value(MeasureOfAngle(DAO))
Answer: 20
Theorem sequence:
arc_property_center_angle(1,OCA,O)
arc_property_circumference_angle_external(1,OCA,D)
triangle_property_angle_sum(1,AOP)
Problem: As shown in the diagram, BC=6, BC=CD, ∠ABD=30°, ∠BEO=90°, O is the center of ⊙O. Find the area of the ⊙O.
问题：如图所示，BC=6，BC=CD，∠ABD=30°，∠BEO=90°，圆O的圆心为O。求圆O的面积。
Construction (CDL):
Shape(OAB,BO,OA)
Shape(OBC,CB)
Shape(CE,EB,BC)
Shape(EO,OB,BE)
Shape(DE,EC,CD)
Shape(OCD,DC)
Shape(ODA,AO,OE,ED)
Collinear(CEO)
Collinear(DEB)
Collinear(AOB)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(LengthOfLine(BC),6)
Equal(LengthOfLine(BC),LengthOfLine(CD))
Equal(MeasureOfAngle(ABD),30)
Equal(MeasureOfAngle(BEO),90)
IsCentreOfCircle(O,O)
Goal: Value(AreaOfCircle(O))
Answer: 36*pi
Theorem sequence:
triangle_property_angle_sum(1,EOB)
triangle_property_angle_sum(1,BCO)
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)
isosceles_triangle_judgment_angle_equal(1,COB)
circle_area_formula(1,O)
Problem: As shown in the diagram, ∠OBA=35°, O is the center of ⊙O, AO∥DC. Find the measure of ∠ACB.
问题：如图所示，∠OBA=35°，圆O的圆心为O，AO平行于DC。求∠ACB的大小。
Construction (CDL):
Shape(OAC,CE,EO,OA)
Shape(AO,OE,EA)
Shape(AE,EB,BA)
Shape(EC,CB,BE)
Shape(OCB,BC)
Shape(OBA,AB)
Collinear(AEC)
Collinear(OEB)
Cocircular(O,CBA)
Conditions (CDL):
Equal(MeasureOfAngle(OBA),35)
IsCentreOfCircle(O,O)
ParallelBetweenLine(AO,DC)
Goal: Value(MeasureOfAngle(ACB))
Answer: 55
Theorem sequence:
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OA,O)
isosceles_triangle_judgment_line_equal(1,OBA)
isosceles_triangle_property_angle_equal(1,OBA)
triangle_property_angle_sum(1,AOB)
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_external(1,OBA,C)
Problem: As shown in the diagram, ∠ABC=74°, ∠BCA=46°, the measure of arc ODB is equal to the measure of arc OCD, the center of circle O is O. Find the measure of ⌒ODA.
问题：如图所示，∠ABC=74°，∠BCA=46°，⌒ODB的角度与⌒OCD的角度相等，O是圆O的圆心。求弧ODA的角度。
Construction (CDL):
Shape(OAB,BA)
Shape(ODA,AE,ED)
Shape(OCD,DE,EC)
Shape(AB,BF,FE,EA)
Shape(EF,FC,CE)
Shape(OBC,CF,FB)
Collinear(DEF)
Collinear(AEC)
Collinear(BFC)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),74)
Equal(MeasureOfAngle(BCA),46)
Equal(MeasureOfArc(ODB),MeasureOfArc(OCD))
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfArc(ODA))
Answer: 28
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,C)
arc_property_circumference_angle_external(1,OCA,B)
arc_addition_measure(1,ODA,OAB)
arc_addition_measure(1,OCD,ODA)
Problem: As shown in the diagram, BC=BA, ∠BCA=25°, the center of ⊙O is O, AD is the diameter of ⊙O. Find the measure of ∠DAC.
问题：如图所示，BC=BA，∠BCA=25°，圆O的圆心为O，⊙O的直径为AD。求∠DAC的大小。
Construction (CDL):
Shape(ODA,AO,OD)
Shape(OAB,BA)
Shape(EA,AB,BE)
Shape(DO,OA,AE,ED)
Shape(EB,BC,CE)
Shape(OBC,CB)
Shape(OCD,DE,EC)
Collinear(AOD)
Collinear(AEC)
Collinear(DEB)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(LengthOfLine(BC),LengthOfLine(BA))
Equal(MeasureOfAngle(BCA),25)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AD,O)
Goal: Value(MeasureOfAngle(DAC))
Answer: 40
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,BCA)
isosceles_triangle_property_angle_equal(1,BCA)
diameter_of_circle_property_right_angle(1,ABD,O)
arc_property_circumference_angle_external(1,OAB,C)
arc_property_circumference_angle_external(1,OAB,D)
triangle_property_angle_sum(1,ABD)
angle_addition(1,DAE,EAB)
Problem: As shown in the diagram, ∠ABD=60°, O is the center of ⊙O, AO is parallel to DC. Find the measure of ∠COA.
问题：如图所示，∠ABD=60°，O是圆O的圆心，AO平行于DC。求∠COA的大小。
Construction (CDL):
Shape(ODA,AO,OD)
Shape(OA,AE,EO)
Shape(BD,DO,OE,EB)
Shape(OBD,DB)
Shape(OAC,CE,EA)
Shape(OCB,BE,EC)
Collinear(DOEC)
Collinear(AEB)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),60)
IsCentreOfCircle(O,O)
ParallelBetweenLine(AO,DC)
Goal: Value(MeasureOfAngle(COA))
Answer: 60
Theorem sequence:
arc_property_center_angle(1,ODA,O)
arc_property_circumference_angle_external(1,ODA,B)
adjacent_complementary_angle(1,EOA,AOD)
Problem: As shown in the diagram, ∠CAB=40°, ∠DPA=75°, O is the center of ⊙O. Find the measure of ∠DBA.
问题：如图所示，∠CAB=40°，∠DPA=75°，圆O的圆心为O。求∠DBA的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(PC,CA,AP)
Shape(OBC,CP,PB)
Shape(OAD,DP,PA)
Shape(BP,PD,DB)
Shape(ODB,BD)
Collinear(CPD)
Collinear(BPA)
Cocircular(O,CADB)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),40)
Equal(MeasureOfAngle(DPA),75)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DBA))
Answer: 35
Theorem sequence:
adjacent_complementary_angle(1,DPA,APC)
triangle_property_angle_sum(1,PCA)
arc_property_circumference_angle_external(1,OAD,C)
arc_property_circumference_angle_external(1,OAD,B)
Problem: As shown in the diagram, ∠BCO=50°, the center of ⊙O is O. Find the measure of ∠CAB.
问题：如图所示，∠BCO=50°，O是圆O的圆心。求∠CAB的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(AB,BO,OC,CA)
Shape(OCA,AC)
Shape(OB,BC,CO)
Shape(OBC,CB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCO),50)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
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)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠CAB=35°, the center of circle O is O, the diameter of ⊙O is AB. Find the measure of ∠ABC.
问题：如图所示，∠CAB=35°，O是⊙O的圆心，⊙O的直径为AB。求∠ABC的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(OBC,CB)
Shape(BC,CA,AO,OB)
Shape(OAB,BO,OA)
Collinear(AOB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),35)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ABC))
Answer: 55
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,BCA)
Problem: As shown in the diagram, ∠ABD=48°, O is the center of circle O, the diameter of circle O is AB. Find the measure of ∠DCB.
问题：如图所示，∠ABD=48°，O是⊙O的圆心，AB是圆O的直径。求∠DCB的大小。
Construction (CDL):
Shape(OAC,CE,EO,OA)
Shape(OCB,BC)
Shape(EC,CB,BE)
Shape(OBD,DB)
Shape(EB,BD,DE)
Shape(AO,OE,ED,DA)
Shape(ODA,AD)
Collinear(AOEB)
Collinear(DEC)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),48)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(DCB))
Answer: 42
Theorem sequence:
diameter_of_circle_property_right_angle(1,BDA,O)
triangle_property_angle_sum(1,BDA)
arc_property_circumference_angle_external(1,OBD,A)
arc_property_circumference_angle_external(1,OBD,C)
Problem: As shown in the diagram, ∠BCA=40°, ∠CAB=60°, the center of circle O is O. Find the measure of ∠CDA.
问题：如图所示，∠BCA=40°，∠CAB=60°，O是圆O的圆心。求∠CDA的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(OAB,BA)
Shape(ED,DA,AE)
Shape(EA,AB,BE)
Shape(OBC,CB)
Shape(EB,BC,CE)
Shape(OCD,DC)
Shape(EC,CD,DE)
Collinear(DEB)
Collinear(AEC)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),40)
Equal(MeasureOfAngle(CAB),60)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CDA))
Answer: 100
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,C)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_external(1,OAB,D)
arc_property_circumference_angle_external(1,OBC,D)
angle_addition(1,CDE,EDA)
Problem: As shown in the diagram, ∠CBA=26°, the center of ⊙O is O, the diameter of circle O is AD. Find the measure of ∠DAC.
问题：如图所示，∠CBA=26°，圆O的圆心为O，AD是圆O的直径。求∠DAC的大小。
Construction (CDL):
Shape(OCD,DC)
Shape(CD,DO,OE,EC)
Shape(OAC,CA)
Shape(CE,EA,AC)
Shape(OBA,AB)
Shape(EB,BA,AE)
Shape(ODB,BE,EO,OD)
Collinear(DOEA)
Collinear(CEB)
Cocircular(O,ACDB)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),26)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AD,O)
Goal: Value(MeasureOfAngle(DAC))
Answer: 64
Theorem sequence:
diameter_of_circle_property_right_angle(1,ACD,O)
arc_property_circumference_angle_external(1,OAC,D)
arc_property_circumference_angle_external(1,OAC,B)
triangle_property_angle_sum(1,CDA)
Problem: As shown in the diagram, ∠ACD=30°, the center of circle O is O. Find the measure of ∠ABD.
问题：如图所示，∠ACD=30°，圆O的圆心为O。求∠ABD的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(ODA,AE,ED)
Shape(AB,BE,EA)
Shape(DE,EC,CD)
Shape(OCD,DC)
Shape(OSC,CE,EB)
Collinear(AEC)
Collinear(BED)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(ACD),30)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ABD))
Answer: 30
Theorem sequence:
arc_property_circumference_angle_external(1,ODA,B)
arc_property_circumference_angle_external(1,ODA,C)
Problem: As shown in the diagram, ∠DBC=31°, O is the center of ⊙O, AD is parallel to OC. Find the measure of ∠ABD.
问题：如图所示，∠DBC=31°，O是⊙O的圆心，AD平行于OC。求∠ABD的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(OCD,DE,EC)
Shape(OD,DA,AO)
Shape(ED,DO,OE)
Shape(EO,OB,BE)
Shape(CE,EB,BC)
Shape(OBC,CB)
Collinear(AOB)
Collinear(DEB)
Collinear(CEO)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(DBC),31)
IsCentreOfCircle(O,O)
ParallelBetweenLine(AD,OC)
Goal: Value(MeasureOfAngle(ABD))
Answer: 28
Theorem sequence:
arc_property_center_angle(1,ODA,O)
arc_property_center_angle(1,OCD,O)
arc_property_circumference_angle_external(1,ODA,B)
arc_property_circumference_angle_external(1,OCD,B)
parallel_property_alternate_interior_angle(2,AD,OC)
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,ODA)
Problem: As shown in the diagram, ∠DAB=50°, the center of ⊙O is O, the diameter of ⊙O is AB. Find the measure of ∠ACD.
问题：如图所示，∠DAB=50°，O是圆O的圆心，AB是圆O的直径。求∠ACD的大小。
Construction (CDL):
Shape(OAC,CA)
Shape(ODA,AD)
Shape(AC,CE,EA)
Shape(DA,AE,ED)
Shape(OCB,BO,OE,EC)
Shape(EO,OB,BD,DE)
Shape(OBD,DB)
Collinear(AEOB)
Collinear(CED)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),50)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ACD))
Answer: 40
Theorem sequence:
diameter_of_circle_property_right_angle(1,BDA,O)
triangle_property_angle_sum(1,DAB)
arc_property_circumference_angle_external(1,ODA,C)
arc_property_circumference_angle_external(1,ODA,B)
Problem: As shown in the diagram, ∠OCA=15°, the center of circle O is O. Find the measure of ∠COB.
问题：如图所示，∠OCA=15°，⊙O的圆心为O。求∠COB的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(OC,CA,AO)
Shape(OBC,CO,OB)
Shape(OAB,BO,OA)
Collinear(AOB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(OCA),15)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COB))
Answer: 30
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OC,O)
isosceles_triangle_judgment_line_equal(1,OCA)
isosceles_triangle_property_angle_equal(1,OCA)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, BD=BC, ∠AOC=104°, O is the center of ⊙O. Find the measure of ∠ADC.
问题：如图所示，BD=BC，∠AOC=104°，O是⊙O的圆心。求∠ADC的大小。
Construction (CDL):
Shape(OCA,AO,OC)
Shape(AB,BC,CO,OA)
Shape(OEC,CE)
Shape(OBE,EC,CB)
Shape(BD,DE,OBE)
Collinear(ABD)
Collinear(CED)
Cocircular(O,ABEC)
Conditions (CDL):
Equal(LengthOfLine(BD),LengthOfLine(BC))
Equal(MeasureOfAngle(AOC),104)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ADC))
Answer: 26
Theorem sequence:
arc_property_center_angle(1,OCA,O)
arc_property_circumference_angle_external(1,OCA,B)
adjacent_complementary_angle(1,ABC,CBD)
isosceles_triangle_judgment_line_equal(1,BDC)
isosceles_triangle_property_angle_equal(1,BDC)
triangle_property_angle_sum(1,CBD)
Problem: As shown in the diagram, ∠CDB=40°, O is the center of circle O. Find the measure of ∠AOC.
问题：如图所示，∠CDB=40°，O是圆O的圆心。求∠AOC的大小。
Construction (CDL):
Shape(OBC,CE,EB)
Shape(EC,CO,OE)
Shape(OCA,AO,OC)
Shape(OAD,DE,EO,OA)
Shape(BE,ED,DB)
Shape(ODB,BD)
Collinear(CED)
Collinear(AOEB)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(CDB),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AOC))
Answer: 100
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,D)
adjacent_complementary_angle(1,AOC,COE)
Problem: As shown in the diagram, ∠CDA=35°, ∠CEA=105°, O is the center of ⊙O. Find the measure of ∠DCB.
问题：如图所示，∠CDA=35°，∠CEA=105°，圆O的圆心为O。求∠DCB的大小。
Construction (CDL):
Shape(OCB,BC)
Shape(CB,BE,EC)
Shape(OAC,CE,EA)
Shape(ODA,AD)
Shape(AE,ED,DA)
Shape(OBD,DE,EB)
Collinear(CED)
Collinear(AEB)
Cocircular(O,CBDA)
Conditions (CDL):
Equal(MeasureOfAngle(CDA),35)
Equal(MeasureOfAngle(CEA),105)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DCB))
Answer: 70
Theorem sequence:
adjacent_complementary_angle(1,CEA,AED)
triangle_property_angle_sum(1,AED)
arc_property_circumference_angle_external(1,OBD,C)
arc_property_circumference_angle_external(1,OBD,A)
Problem: As shown in the diagram, ∠BCD=34°, the center of ⊙O is O. Find the measure of ∠BOA.
问题：如图所示，∠BCD=34°，圆O的圆心为O。求∠BOA的大小。
Construction (CDL):
Shape(AO,OD,DC,OCA)
Shape(OA,AD,DO)
Shape(CD,DB,BC)
Shape(BD,DA,OAB)
Shape(CB,OBC)
Collinear(ADC)
Collinear(ODB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),34)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 68
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠BOA=60°, O is the center of ⊙O. Find the measure of ∠BPA.
问题：如图所示，∠BOA=60°，圆O的圆心为O。求∠BPA的大小。
Construction (CDL):
Shape(AP,OPA)
Shape(PA,AO,OB,BP)
Shape(BO,OA,OAB)
Shape(PB,OBP)
Cocircular(O,ABP)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),60)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BPA))
Answer: 30
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,P)
Problem: As shown in the diagram, ∠BCD=40°, O is the center of ⊙O. Find the measure of ∠BOA.
问题：如图所示，∠BCD=40°，圆O的圆心为O。求∠BOA的大小。
Construction (CDL):
Shape(AO,OD,DC,OCA)
Shape(OA,AD,DO)
Shape(CD,DB,BC)
Shape(BD,DA,OAB)
Shape(CB,OBC)
Collinear(ADC)
Collinear(ODB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 80
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)