SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠CBA=70°, O is the center of ⊙O. Find the measure of ∠COA.
问题：如图所示，∠CBA=70°，圆O的圆心为O。求∠COA的大小。
Construction (CDL):
Shape(BC,OCB)
Shape(CB,BA,AO,OC)
Shape(AB,OBA)
Shape(CO,OA,OAC)
Cocircular(O,BAC)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),70)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COA))
Answer: 140
Theorem sequence:
arc_property_circumference_angle_external(1,OAC,B)
arc_property_center_angle(1,OAC,O)
Problem: As shown in the diagram, ∠BAD=36°, the center of ⊙O is O, AB∥CD. Find the measure of ∠COA.
问题：如图所示，∠BAD=36°，O是⊙O的圆心，AB∥CD。求∠COA的大小。
Construction (CDL):
Shape(AB,OBA)
Shape(BA,AD,ODB)
Shape(AO,OC,CD,DA)
Shape(DC,OCD)
Shape(CO,OA,OAC)
Cocircular(O,ACDB)
Conditions (CDL):
Equal(MeasureOfAngle(BAD),36)
IsCentreOfCircle(O,O)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(COA))
Answer: 72
Theorem sequence:
parallel_property_alternate_interior_angle(1,AB,CD)
arc_property_circumference_angle_external(1,OAC,D)
arc_property_center_angle(1,OAC,O)
Problem: As shown in the diagram, ∠ACB=35°, O is the center of circle O. Find the measure of ∠BAO.
问题：如图所示，∠ACB=35°，⊙O的圆心为O。求∠BAO的大小。
Construction (CDL):
Shape(AO,OD,DA)
Shape(AD,DB,BA)
Shape(AB,OBA)
Shape(DC,CB,BD)
Shape(BC,OCB)
Shape(CD,DO,OA,OAC)
Collinear(ADC)
Collinear(ODB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(ACB),35)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BAO))
Answer: 55
Theorem sequence:
arc_property_circumference_angle_external(1,OBA,C)
arc_property_center_angle(1,OBA,O)
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,AOB)
Problem: As shown in the diagram, ∠ABD=58°, O is the center of ⊙O, AB is the diameter of ⊙O. Find the measure of ∠DCB.
问题：如图所示，∠ABD=58°，O是圆O的圆心，⊙O的直径为AB。求∠DCB的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(DA,AE,ED)
Shape(DE,EB,BD)
Shape(DB,OBD)
Shape(CE,EA,OAC)
Shape(BE,EC,CB)
Shape(BC,OCB)
Collinear(AOEB)
Collinear(DEC)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),58)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(DCB))
Answer: 32
Theorem sequence:
diameter_of_circle_property_right_angle(1,BDA,O)
triangle_property_angle_sum(1,DAB)
arc_property_circumference_angle_external(1,OBD,A)
arc_property_circumference_angle_external(1,OBD,C)
Problem: As shown in the diagram, AB=4, ∠ACB=30°, O is the center of circle O, AD is the diameter of circle O. Find the length of line AD.
问题：如图所示，AB=4，∠ACB=30°，⊙O的圆心为O，AD是⊙O的直径。求直线AD的长度。
Construction (CDL):
Shape(AB,OBA)
Shape(AE,EB,BA)
Shape(AD,DE,EA)
Shape(BE,EC,CB)
Shape(BC,OCB)
Shape(CE,ED,ODC)
Shape(DA,OAD)
Collinear(AEC)
Collinear(BED)
Collinear(AOD)
Cocircular(O,ADCB)
Conditions (CDL):
Equal(LengthOfLine(AB),4)
Equal(MeasureOfAngle(ACB),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AD,O)
Goal: Value(LengthOfLine(AD))
Answer: 8
Theorem sequence:
arc_property_circumference_angle_external(1,OBA,D)
arc_property_circumference_angle_external(1,OBA,C)
diameter_of_circle_property_right_angle(1,DBA,O)
sine_theorem(1,ADB)
Problem: As shown in the diagram, ∠ABC=50°, the center of ⊙O is O, AB is the diameter of circle O. Find the measure of ∠CDB.
问题：如图所示，∠ABC=50°，O是⊙O的圆心，AB是⊙O的直径。求∠CDB的大小。
Construction (CDL):
Shape(AE,ED,ODA)
Shape(DE,EC,CD)
Shape(DC,OCD)
Shape(CE,EB,BC)
Shape(CB,OBC)
Shape(AB,BE,EA)
Shape(BA,OAB)
Collinear(AEC)
Collinear(DEB)
Collinear(AOB)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),50)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CDB))
Answer: 40
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,BCA)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_external(1,OBC,D)
Problem: As shown in the diagram, ∠DAO=70°, the center of circle O is O, the diameter of ⊙O is AB, AD∥CO. Find the measure of ∠CBO.
问题：如图所示，∠DAO=70°，O是⊙O的圆心，圆O的直径为AB，AD平行于CO。求∠CBO的大小。
Construction (CDL):
Shape(DA,AB,OBD)
Shape(AD,ODA)
Shape(CO,OA,OAC)
Shape(BO,OC,CB)
Shape(BC,OCB)
Collinear(BOA)
Cocircular(O,DACB)
Conditions (CDL):
Equal(MeasureOfAngle(DAO),70)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
ParallelBetweenLine(AD,CO)
Goal: Value(MeasureOfAngle(CBO))
Answer: 35
Theorem sequence:
parallel_property_alternate_interior_angle(1,AD,CO)
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,B)
Problem: As shown in the diagram, ∠BAC=32°, the center of ⊙O is O, the diameter of ⊙O is AC, AB∥DC. Find the measure of ∠DOA.
问题：如图所示，∠BAC=32°，⊙O的圆心为O，圆O的直径为AC，AB平行于DC。求∠DOA的大小。
Construction (CDL):
Shape(AB,OBA)
Shape(BA,AC,OCB)
Shape(DO,OA,OAD)
Shape(OD,DC,CO)
Shape(CD,ODC)
Collinear(AOC)
Cocircular(O,ADCB)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),32)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AC,O)
ParallelBetweenLine(AB,DC)
Goal: Value(MeasureOfAngle(DOA))
Answer: 64
Theorem sequence:
parallel_property_alternate_interior_angle(1,AB,DC)
arc_property_circumference_angle_external(1,OAD,C)
arc_property_center_angle(1,OAD,O)
Problem: As shown in the diagram, AB=BC, AD=6, ∠ABC=120°, the center of ⊙O is O, AD is the diameter of ⊙O. Find the length of line AB.
问题：如图所示，AB=BC，AD=6，∠ABC=120°，圆O的圆心为O，圆O的直径为AD。求直线AB的长度。
Construction (CDL):
Shape(AD,ODA)
Shape(DA,AE,ED)
Shape(DE,EC,OCD)
Shape(CE,EB,BC)
Shape(EA,AB,BE)
Shape(BA,OAB)
Shape(CB,OBC)
Collinear(AOD)
Collinear(AEC)
Collinear(DEB)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(BC))
Equal(LengthOfLine(AD),6)
Equal(MeasureOfAngle(ABC),120)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AD,O)
Goal: Value(LengthOfLine(AB))
Answer: 3
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,BCA)
isosceles_triangle_property_angle_equal(1,BCA)
triangle_property_angle_sum(1,DAE)
triangle_property_angle_sum(1,CEB)
triangle_property_angle_sum(1,BCA)
arc_property_circumference_angle_external(1,OAB,C)
arc_property_circumference_angle_external(1,OAB,D)
arc_property_circumference_angle_external(1,OCD,A)
arc_property_circumference_angle_external(1,OCD,B)
diameter_of_circle_property_right_angle(1,ABD,O)
sine_theorem(1,AED)
sine_theorem(1,BCE)
sine_theorem(1,EAB)
Problem: As shown in the diagram, the length of arc OBC is equal to the length of arc ODB, ∠CAO=25°, O is the center of circle O, AB is the diameter of circle O. Find the measure of ∠BOD.
问题：如图所示，⌒OBC的长度与弧ODB的长度相等，∠CAO=25°，圆O的圆心为O，AB是圆O的直径。求∠BOD的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(AO,OC,CA)
Shape(CO,OB,OBC)
Shape(BO,OD,ODB)
Shape(DO,OA,OAD)
Collinear(AOB)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(LengthOfArc(OBC),LengthOfArc(ODB))
Equal(MeasureOfAngle(CAO),25)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BOD))
Answer: 50
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,A)
congruent_arc_judgment_length_equal(1,ODB,OBC)
congruent_arc_property_measure_equal(1,ODB,OBC)
arc_property_center_angle(1,ODB,O)
Problem: As shown in the diagram, ∠BCA=40°, the center of circle O is O. Find the measure of ∠BOA.
问题：如图所示，∠BCA=40°，O是⊙O的圆心。求∠BOA的大小。
Construction (CDL):
Shape(AO,OC,OCA)
Shape(OD,DC,CO)
Shape(CD,DB,BC)
Shape(OA,AD,DO)
Shape(DA,AB,BD)
Shape(BA,OAB)
Shape(CB,OBC)
Collinear(ODB)
Collinear(ADC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 80
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,C)
arc_property_center_angle(1,OAB,O)
Problem: As shown in the diagram, ∠BDE=15°, ∠ECA=25°, the center of circle O is O. Find the measure of ∠BOA.
问题：如图所示，∠BDE=15°，∠ECA=25°，圆O的圆心为O。求∠BOA的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AF,FC)
Shape(CF,FO,OG,GD,ODC)
Shape(DG,GB,BD)
Shape(DB,OBD)
Shape(BG,GE,OEB)
Shape(EG,GO,OF,FE)
Shape(EF,FA,OAE)
Collinear(CFE)
Collinear(AFO)
Collinear(DGE)
Collinear(OGB)
Cocircular(O,CAEBD)
Conditions (CDL):
Equal(MeasureOfAngle(BDE),15)
Equal(MeasureOfAngle(ECA),25)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 80
Theorem sequence:
arc_property_circumference_angle_external(1,OAE,C)
arc_property_circumference_angle_external(1,OEB,D)
arc_addition_measure(1,OAE,OEB)
arc_property_center_angle(1,OAB,O)
Problem: As shown in the diagram, ∠BCA=45°, the radius of ⊙O is 6, O is the center of ⊙O. Find the length of line AB.
问题：如图所示，∠BCA=45°，⊙O的半径为6，圆O的圆心为O。求直线AB的长度。
Construction (CDL):
Shape(CA,AO,OB,BC)
Shape(OA,AB,BO)
Shape(CB,OBC)
Shape(BA,OAB)
Shape(AC,OCA)
Cocircular(O,CAB)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),45)
Equal(RadiusOfCircle(O),6)
IsCentreOfCircle(O,O)
Goal: Value(LengthOfLine(AB))
Answer: 6*sqrt(2)
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
arc_property_circumference_angle_external(1,OAB,C)
arc_property_center_angle(1,OAB,O)
isosceles_triangle_judgment_line_equal(1,OAB)
isosceles_triangle_property_angle_equal(1,OAB)
triangle_property_angle_sum(1,OAB)
sine_theorem(1,ABO)
Problem: As shown in the diagram, ∠ABC=40°, AD is the angle bisector of ∠CAB, O is the center of circle O, AB is the diameter of circle O. Find the measure of ∠ABD.
问题：如图所示，∠ABC=40°，AD平分∠CAB，⊙O的圆心为O，⊙O的直径为AB。求∠ABD的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AB,BE,EA)
Shape(AE,EC,CA)
Shape(AC,OCA)
Shape(CE,ED,ODC)
Shape(EB,BD,DE)
Shape(DB,OBD)
Collinear(BEC)
Collinear(AED)
Collinear(AOB)
Cocircular(O,BDCA)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),40)
IsBisectorOfAngle(AD,CAB)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ABD))
Answer: 65
Theorem sequence:
diameter_of_circle_property_right_angle(1,BDA,O)
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,ABC)
angle_addition(1,CAD,DAB)
triangle_property_angle_sum(1,ABD)
Problem: As shown in the diagram, ∠AOC=40°, the center of ⊙O is O, AB is the diameter of ⊙O, AC∥DO. Find the measure of ∠BOD.
问题：如图所示，∠AOC=40°，O是⊙O的圆心，⊙O的直径为AB，AC∥DO。求∠BOD的大小。
Construction (CDL):
Shape(CO,OB,OBC)
Shape(AO,OC,CA)
Shape(AC,OCA)
Shape(DO,OA,OAD)
Shape(BO,OD,ODB)
Collinear(AOB)
Cocircular(O,CADB)
Conditions (CDL):
Equal(MeasureOfAngle(AOC),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
ParallelBetweenLine(AC,DO)
Goal: Value(MeasureOfAngle(BOD))
Answer: 110
Theorem sequence:
radius_of_circle_property_length_equal(1,OC,O)
radius_of_circle_property_length_equal(1,OA,O)
isosceles_triangle_judgment_line_equal(1,OCA)
isosceles_triangle_property_angle_equal(1,OCA)
triangle_property_angle_sum(1,AOC)
parallel_property_alternate_interior_angle(1,AC,DO)
adjacent_complementary_angle(1,BOD,DOA)
Problem: As shown in the diagram, ∠CAB=40°, O is the center of ⊙O, C is the midpoint of arc OBA. Find the measure of ∠ABC.
问题：如图所示，∠CAB=40°，O是⊙O的圆心，C是弧OBA的中点。求∠ABC的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AB,BC,CA)
Shape(AC,OCA)
Cocircular(O,BCA)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),40)
IsCentreOfCircle(O,O)
IsMidpointOfArc(C,OBA)
Goal: Value(MeasureOfAngle(ABC))
Answer: 40
Theorem sequence:
congruent_arc_judgment_length_equal(1,OBC,OCA)
congruent_arc_property_measure_equal(1,OBC,OCA)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_external(1,OCA,B)
Problem: As shown in the diagram, ∠ABC=72°, O is the center of circle O, AB is the diameter of circle O. Find the measure of ∠ODB.
问题：如图所示，∠ABC=72°，圆O的圆心为O，圆O的直径为AB。求∠ODB的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(AO,OC,CA)
Shape(CO,OB,BC)
Shape(CB,OBC)
Shape(BO,OD,DB)
Shape(DO,OA,OAD)
Shape(BD,ODB)
Collinear(AOB)
Collinear(COD)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),72)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ODB))
Answer: 18
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,CAB)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_external(1,OBC,D)
Problem: As shown in the diagram, ∠BAC=65°, the center of ⊙O is O, the diameter of circle O is AB, DC∥BA. Find the measure of ∠BCD.
问题：如图所示，∠BAC=65°，⊙O的圆心为O，AB是⊙O的直径，DC平行于BA。求∠BCD的大小。
Construction (CDL):
Shape(DC,OCD)
Shape(BC,CD,ODB)
Shape(BA,AC,CB)
Shape(CA,OAC)
Collinear(BOA)
Cocircular(O,DBAC)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),65)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
ParallelBetweenLine(DC,BA)
Goal: Value(MeasureOfAngle(BCD))
Answer: 25
Theorem sequence:
diameter_of_circle_property_right_angle(1,ACB,O)
triangle_property_angle_sum(1,BAC)
parallel_property_alternate_interior_angle(2,DC,BA)
Problem: As shown in the diagram, ∠CAB=50°, O is the center of circle O. Find the measure of ∠OBC.
问题：如图所示，∠CAB=50°，O是⊙O的圆心。求∠OBC的大小。
Construction (CDL):
Shape(BO,OC,CA,AB)
Shape(BA,OAB)
Shape(AC,OCA)
Shape(CB,OBC)
Shape(BC,CO,OB)
Cocircular(O,BCA)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),50)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(OBC))
Answer: 40
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,OBC,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,OBC)
isosceles_triangle_property_angle_equal(1,OBC)
triangle_property_angle_sum(1,BCO)
Problem: As shown in the diagram, ∠CDA=54°, the center of circle O is O, the diameter of circle O is BE, quadrilateral ABCD is a parallelogram. Find the measure of ∠BEA.
问题：如图所示，∠CDA=54°，O是圆O的圆心，BE是圆O的直径，AD和BC是平行四边形ABCD的一组对边。求∠BEA的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(AF,FD,DA)
Shape(DF,FE,OED)
Shape(EF,FC,CE)
Shape(AB,BC,CF,FA)
Shape(BA,OAB)
Shape(EB,OBE)
Collinear(BOCE)
Collinear(CFD)
Collinear(AFE)
Cocircular(O,ABED)
Conditions (CDL):
Equal(MeasureOfAngle(CDA),54)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BE,O)
Parallelogram(ABCD)
Goal: Value(MeasureOfAngle(BEA))
Answer: 36
Theorem sequence:
parallelogram_property_opposite_angle_equal(1,DABC)
diameter_of_circle_property_right_angle(1,EAB,O)
triangle_property_angle_sum(1,EAB)
Problem: As shown in the diagram, ∠CDB=31°, O is the center of circle O, AB is the diameter of circle O. Find the measure of ∠COB.
问题：如图所示，∠CDB=31°，O是⊙O的圆心，圆O的直径为AB。求∠COB的大小。
Construction (CDL):
Shape(DC,OCD)
Shape(DE,EC,CD)
Shape(CE,EB,OBC)
Shape(BE,EO,OB)
Shape(AO,OE,ED,ODA)
Collinear(AOB)
Collinear(DEB)
Collinear(CEO)
Cocircular(O,DABC)
Conditions (CDL):
Equal(MeasureOfAngle(CDB),31)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(COB))
Answer: 62
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,D)
arc_property_center_angle(1,OBC,O)
Problem: As shown in the diagram, ∠ODE=50°, the center of ⊙O is O, AO is parallel to DE. Find the measure of ∠CAO.
问题：如图所示，∠ODE=50°，⊙O的圆心为O，AO平行于DE。求∠CAO的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AO,OC)
Shape(DO,OA,OAD)
Shape(CD,DE,OEC)
Shape(ED,ODE)
Collinear(COD)
Cocircular(O,CADE)
Conditions (CDL):
Equal(MeasureOfAngle(ODE),50)
IsCentreOfCircle(O,O)
ParallelBetweenLine(AO,DE)
Goal: Value(MeasureOfAngle(CAO))
Answer: 25
Theorem sequence:
parallel_property_alternate_interior_angle(2,AO,DE)
arc_property_center_angle(1,OAD,O)
arc_property_circumference_angle_external(1,OAD,C)
radius_of_circle_property_length_equal(1,OC,O)
radius_of_circle_property_length_equal(1,OA,O)
isosceles_triangle_judgment_line_equal(1,OCA)
isosceles_triangle_property_angle_equal(1,OCA)
Problem: As shown in the diagram, ∠COA=110°, O is the center of circle O. Find the measure of ∠ABC.
问题：如图所示，∠COA=110°，圆O的圆心为O。求∠ABC的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(DC,OCD)
Shape(DA,AO,OC,CD)
Shape(OA,AB,BC,CO)
Shape(CB,OBC)
Shape(BA,OAB)
Cocircular(O,DABC)
Conditions (CDL):
Equal(MeasureOfAngle(COA),110)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ABC))
Answer: 125
Theorem sequence:
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_internal(1,OAC,B)
Problem: As shown in the diagram, ∠DCB=40°, the center of ⊙O is O. Find the measure of ∠DOB.
问题：如图所示，∠DCB=40°，O是圆O的圆心。求∠DOB的大小。
Construction (CDL):
Shape(DO,OA,AD)
Shape(DA,AB,BD)
Shape(DB,OBD)
Shape(BA,AC,CB)
Shape(BC,OCB)
Shape(CA,AO,OD,ODC)
Collinear(DAC)
Collinear(OAB)
Cocircular(O,DCB)
Conditions (CDL):
Equal(MeasureOfAngle(DCB),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DOB))
Answer: 80
Theorem sequence:
arc_property_circumference_angle_external(1,OBD,C)
arc_property_center_angle(1,OBD,O)
Problem: As shown in the diagram, ∠BAC=35°, ∠CBA=110°, the center of circle O is O. Find the measure of ∠AOB.
问题：如图所示，∠BAC=35°，∠CBA=110°，圆O的圆心为O。求∠AOB的大小。
Construction (CDL):
Shape(CD,DO,OA,OAC)
Shape(OD,DA,AO)
Shape(AD,DB,BA)
Shape(DC,CB,BD)
Shape(AB,OBA)
Shape(BC,OCB)
Collinear(CDA)
Collinear(ODB)
Cocircular(O,CBA)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),35)
Equal(MeasureOfAngle(CBA),110)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AOB))
Answer: 70
Theorem sequence:
triangle_property_angle_sum(1,BAC)
arc_property_circumference_angle_external(1,OBA,C)
arc_property_center_angle(1,OBA,O)
Problem: As shown in the diagram, ∠BCD=50°, the center of ⊙O is O, the diameter of ⊙O is CD. Find the measure of ∠CAB.
问题：如图所示，∠BCD=50°，O是圆O的圆心，CD是⊙O的直径。求∠CAB的大小。
Construction (CDL):
Shape(BD,ODB)
Shape(DB,BE,ED)
Shape(DE,EA,OAD)
Shape(BC,CE,EB)
Shape(AE,EC,CA)
Shape(AC,OCA)
Shape(CB,OBC)
Collinear(DOEC)
Collinear(BEA)
Cocircular(O,DBCA)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),50)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(CD,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 40
Theorem sequence:
diameter_of_circle_property_right_angle(1,DBC,O)
triangle_property_angle_sum(1,DBC)
arc_property_circumference_angle_external(1,OBC,D)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠BOA=110°, the center of circle O is O. Find the measure of ∠BCA.
问题：如图所示，∠BOA=110°，圆O的圆心为O。求∠BCA的大小。
Construction (CDL):
Shape(CB,OBC)
Shape(BC,CA,AO,OB)
Shape(BO,OA,AB)
Shape(AC,OCA)
Shape(BA,OAB)
Cocircular(O,BCA)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),110)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 55
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠CAB=30°, the center of circle O is O. Find the measure of arc OBC.
问题：如图所示，∠CAB=30°，圆O的圆心为O。求弧OBC的角度。
Construction (CDL):
Shape(BO,OD,DA,OAB)
Shape(OB,BD,DO)
Shape(CD,DB,OBC)
Shape(AD,DC,CA)
Shape(AC,OCA)
Collinear(ODC)
Collinear(ADB)
Cocircular(O,BCA)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),30)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfArc(OBC))
Answer: 60
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠COB=76°, the center of circle O is O. Find the measure of ∠CAB.
问题：如图所示，∠COB=76°，⊙O的圆心为O。求∠CAB的大小。
Construction (CDL):
Shape(DO,OB,BD)
Shape(AD,DC,CA)
Shape(AC,OCA)
Shape(CD,DB,OBC)
Shape(BO,OD,DA,OAB)
Collinear(ODC)
Collinear(BDA)
Cocircular(O,BCA)
Conditions (CDL):
Equal(MeasureOfAngle(COB),76)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 38
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠BDC=20°, ∠DOA=110°, the center of ⊙O is O. Find the measure of ∠CPB.
问题：如图所示，∠BDC=20°，∠DOA=110°，O是⊙O的圆心。求∠CPB的大小。
Construction (CDL):
Shape(CP,PB,OCB)
Shape(BD,DC,OCB)
Shape(DB,BA,AO,OD)
Shape(CD,ODC)
Shape(AB,OBA)
Shape(DO,OA,OAD)
Collinear(PBA)
Collinear(PCD)
Cocircular(O,CBAD)
Conditions (CDL):
Equal(MeasureOfAngle(BDC),20)
Equal(MeasureOfAngle(DOA),110)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CPB))
Answer: 35
Theorem sequence:
arc_property_center_angle(1,OAD,O)
arc_property_circumference_angle_external(1,OAD,B)
adjacent_complementary_angle(1,PBD,DBA)
triangle_property_angle_sum(1,PBD)
Problem: As shown in the diagram, the length of ⌒OCB is equal to the length of ⌒OBD, ∠COA=140°, O is the center of circle O. Find the measure of ∠DAB.
问题：如图所示，弧OCB的长度与弧OBD的长度相等，∠COA=140°，O是⊙O的圆心。求∠DAB的大小。
Construction (CDL):
Shape(CO,OA,OAC)
Shape(BO,OC,OCB)
Shape(DA,AB,OBD)
Shape(AD,ODA)
Collinear(AOB)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(LengthOfArc(OCB),LengthOfArc(OBD))
Equal(MeasureOfAngle(COA),140)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DAB))
Answer: 20
Theorem sequence:
adjacent_complementary_angle(1,BOC,COA)
congruent_arc_judgment_length_equal(1,OCB,OBD)
arc_property_center_angle(1,OCB,O)
congruent_arc_property_measure_equal(1,OCB,OBD)
arc_property_circumference_angle_external(1,OBD,A)
Problem: As shown in the diagram, AB=10, BC=6, O is the center of ⊙O, AB is the diameter of ⊙O, BD⊥OD. Find the length of line OD.
问题：如图所示，AB=10，BC=6，O是⊙O的圆心，AB是圆O的直径，BD垂直于OD。求直线OD的长度。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AO,OD,DC)
Shape(DO,OB,BD)
Shape(CB,OBC)
Shape(BA,OAB)
Collinear(AOB)
Collinear(CDB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(LengthOfLine(BC),6)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
PerpendicularBetweenLine(BD,OD)
Goal: Value(LengthOfLine(OD))
Answer: 4
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
right_triangle_judgment_angle(1,BCA)
right_triangle_property_pythagorean(1,BCA)
adjacent_complementary_angle(1,BDO,ODC)
parallel_judgment_ipsilateral_internal_angle(1,DO,CA)
circle_property_chord_perpendicular_bisect_chord(1,O,OD,BC)
midsegment_of_triangle_judgment_parallel(1,DO,BCA)
midsegment_of_triangle_property_length(1,DO,BCA)
Problem: As shown in the diagram, ∠CAB=30°, O is the center of circle O, AB is the diameter of circle O. Find the measure of ∠ADE.
问题：如图所示，∠CAB=30°，O是圆O的圆心，⊙O的直径为AB。求∠ADE的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AE,EC)
Shape(CE,EB,BC)
Shape(CB,OBC)
Shape(AD,DE,EA)
Shape(BE,ED,ODB)
Shape(DA,OAD)
Collinear(CED)
Collinear(AOEB)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ADE))
Answer: 60
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,CAB)
arc_property_circumference_angle_external(1,OCA,D)
arc_property_circumference_angle_external(1,OCA,B)
Problem: As shown in the diagram, Div(LengthOfLine(AC)=LengthOfLine(BC)), AB=10, O is the center of ⊙O, the diameter of circle O is AB, BD⊥OD. Find the length of line OD.
问题：如图所示，Div(LengthOfLine(AC)=LengthOfLine(BC))，AB=10，圆O的圆心为O，AB是圆O的直径，BD⊥OD。求直线OD的长度。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AO,OD,DC)
Shape(DO,OB,BD)
Shape(CB,OBC)
Collinear(CDB)
Collinear(AOB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(Div(LengthOfLine(AC),LengthOfLine(BC)),4/3)
Equal(LengthOfLine(AB),10)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
PerpendicularBetweenLine(BD,OD)
Goal: Value(LengthOfLine(OD))
Answer: 4
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
right_triangle_judgment_angle(1,BCA)
right_triangle_property_pythagorean(1,BCA)
parallel_judgment_corresponding_angle(1,DO,CA,B)
circle_property_chord_perpendicular_bisect_chord(1,O,OD,BC)
midsegment_of_triangle_judgment_parallel(1,DO,BCA)
midsegment_of_triangle_property_length(1,DO,BCA)
Problem: As shown in the diagram, ∠ABC=50°, O is the center of circle O, CD is the diameter of circle O. Find the measure of ∠DCA.
问题：如图所示，∠ABC=50°，O是圆O的圆心，CD是⊙O的直径。求∠DCA的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AE,EC)
Shape(CE,EB,BC)
Shape(CB,OBC)
Shape(AD,DE,EA)
Shape(DA,OAD)
Shape(BE,ED,ODB)
Collinear(COED)
Collinear(AEB)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),50)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(CD,O)
Goal: Value(MeasureOfAngle(DCA))
Answer: 40
Theorem sequence:
diameter_of_circle_property_right_angle(1,CAD,O)
arc_property_circumference_angle_external(1,OCA,D)
arc_property_circumference_angle_external(1,OCA,B)
triangle_property_angle_sum(1,CAD)
Problem: As shown in the diagram, ∠AOC=30°, the center of ⊙O is O, the diameter of circle O is AB. Find the measure of ∠DBA.
问题：如图所示，∠AOC=30°，O是⊙O的圆心，⊙O的直径为AB。求∠DBA的大小。
Construction (CDL):
Shape(BD,ODB)
Shape(BE,ED,DB)
Shape(CE,EB,OBC)
Shape(DO,OA,OAD)
Shape(AO,OE,EA)
Shape(AE,EC,OCA)
Collinear(DOEC)
Collinear(BEA)
Cocircular(O,DBCA)
Conditions (CDL):
Equal(MeasureOfAngle(AOC),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(DBA))
Answer: 75
Theorem sequence:
adjacent_complementary_angle(1,DOA,AOE)
arc_property_center_angle(1,OAD,O)
arc_property_circumference_angle_external(1,OAD,B)