SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠DCB=35°, the center of ⊙O is O. Find the measure of ∠ABD.
问题：如图所示，∠DCB=35°，圆O的圆心为O。求∠ABD的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(DA,AO,OE,ED)
Shape(DE,EB,BD)
Shape(DB,OBD)
Shape(EC,CB,BE)
Shape(BC,OCB)
Shape(CE,EO,OA,OAC)
Collinear(AOEB)
Collinear(DEC)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(DCB),35)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ABD))
Answer: 55
Theorem sequence:
arc_property_circumference_angle_external(1,OBD,A)
arc_property_circumference_angle_external(1,OBD,C)
diameter_of_circle_judgment_pass_centre(1,AOB,O)
diameter_of_circle_property_right_angle(1,BDA,O)
triangle_property_angle_sum(1,DAB)
Problem: As shown in the diagram, ∠BAD=35°, O is the center of circle O, AB is parallel to CD. Find the measure of ∠COA.
问题：如图所示，∠BAD=35°，圆O的圆心为O，AB平行于CD。求∠COA的大小。
Construction (CDL):
Shape(AB,OBA)
Shape(BA,AO,OD,ODB)
Shape(CO,OA,OAC)
Shape(OC,CD,DO)
Shape(DC,OCD)
Collinear(AOD)
Cocircular(O,ACDB)
Conditions (CDL):
Equal(MeasureOfAngle(BAD),35)
IsCentreOfCircle(O,O)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(COA))
Answer: 70
Theorem sequence:
parallel_property_alternate_interior_angle(1,AB,CD)
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,D)
Problem: As shown in the diagram, the diameter of circle A is 10, CO=5, ∠COD=90°, the center of circle A is A. Find the measure of ∠OBC.
问题：如图所示，圆A的直径为10，CO=5，∠COD=90°，A是⊙A的圆心。求∠OBC的大小。
Construction (CDL):
Shape(OC,ACO)
Shape(CO,OE,EC)
Shape(CE,ED,DA,AC)
Shape(CA,AD,ADC)
Shape(EO,OB,BE)
Shape(BO,AOB)
Shape(DE,EB,ABD)
Collinear(CAD)
Collinear(CEB)
Collinear(OED)
Cocircular(A,COBD)
Conditions (CDL):
Equal(DiameterOfCircle(A),10)
Equal(LengthOfLine(CO),5)
Equal(MeasureOfAngle(COD),90)
IsCentreOfCircle(A,A)
Goal: Value(MeasureOfAngle(OBC))
Answer: 30
Theorem sequence:
diameter_of_circle_judgment_right_angle(1,COD,A)
diameter_of_circle_property_length_equal(1,DC,A)
sine_theorem(1,COD)
arc_property_circumference_angle_external(1,ACO,B)
arc_property_circumference_angle_external(1,ACO,D)
Problem: As shown in the diagram, ∠BOA=68°, O is the center of circle O. Find the measure of ∠BCA.
问题：如图所示，∠BOA=68°，圆O的圆心为O。求∠BCA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(BOA),68)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 34
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠BAO=20°, the center of ⊙O is O, AO∥CD. Find the measure of ∠OCD.
问题：如图所示，∠BAO=20°，O是⊙O的圆心，AO∥CD。求∠OCD的大小。
Construction (CDL):
Shape(AB,OBA)
Shape(OB,BA,AO)
Shape(CO,OA,OAC)
Shape(BO,OC,CD,ODB)
Shape(DC,OCD)
Collinear(BOC)
Cocircular(O,ACDB)
Conditions (CDL):
Equal(MeasureOfAngle(BAO),20)
IsCentreOfCircle(O,O)
ParallelBetweenLine(AO,CD)
Goal: Value(MeasureOfAngle(OCD))
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,OBA)
isosceles_triangle_property_angle_equal(1,OBA)
parallel_property_alternate_interior_angle(2,AO,CD)
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,B)
Problem: As shown in the diagram, ∠BAC=15°, the center of ⊙O is O. Find the measure of ∠CBO.
问题：如图所示，∠BAC=15°，O是圆O的圆心。求∠CBO的大小。
Construction (CDL):
Shape(CO,OD,DA,OAC)
Shape(DO,OC,CD)
Shape(DC,CB,BD)
Shape(BC,OCB)
Shape(AD,DB,BA)
Shape(AB,OBA)
Collinear(ADC)
Collinear(ODB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),15)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CBO))
Answer: 75
Theorem sequence:
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_external(1,OCB,A)
radius_of_circle_property_length_equal(1,OC,O)
radius_of_circle_property_length_equal(1,OB,O)
isosceles_triangle_judgment_line_equal(1,OCB)
isosceles_triangle_property_angle_equal(1,OCB)
triangle_property_angle_sum(1,OCB)
Problem: As shown in the diagram, ∠CAB=40°, O is the center of ⊙O, the diameter of circle O is AB. Find the measure of ∠ADC.
问题：如图所示，∠CAB=40°，⊙O的圆心为O，AB是圆O的直径。求∠ADC的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AE,EC)
Shape(CE,EO,OB,BC)
Shape(CB,OBC)
Shape(AD,DE,EA)
Shape(DA,OAD)
Shape(BO,OE,ED,ODB)
Collinear(AEOB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ADC))
Answer: 50
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, OA=2, ∠CDA=30°, O is the center of ⊙O. Find the length of line CB.
问题：如图所示，OA=2，∠CDA=30°，圆O的圆心为O。求直线CB的长度。
Construction (CDL):
Shape(AO,OB,OBA)
Shape(CA,OAC)
Shape(AC,CE,EA)
Shape(OA,AE,EB,BO)
Shape(BE,ED,ODB)
Shape(EC,CD,DE)
Shape(DC,OCD)
Collinear(AOB)
Collinear(AED)
Collinear(CEB)
Cocircular(O,ACDB)
Conditions (CDL):
Equal(LengthOfLine(OA),2)
Equal(MeasureOfAngle(CDA),30)
IsCentreOfCircle(O,O)
Goal: Value(LengthOfLine(CB))
Answer: 2*sqrt(3)
Theorem sequence:
arc_property_circumference_angle_external(1,OAC,D)
arc_property_circumference_angle_external(1,OAC,B)
diameter_of_circle_judgment_pass_centre(1,AOB,O)
circle_property_length_of_radius_and_diameter(1,O)
radius_of_circle_property_length_equal(1,OA,O)
diameter_of_circle_property_right_angle(1,ACB,O)
diameter_of_circle_property_length_equal(1,AB,O)
cosine_theorem(1,CBA)
sine_theorem(1,ACB)
Problem: As shown in the diagram, ∠CBA=30°, O is the center of circle O. Find the measure of ∠OAC.
问题：如图所示，∠CBA=30°，圆O的圆心为O。求∠OAC的大小。
Construction (CDL):
Shape(AB,OBA)
Shape(BA,AO,OC,CB)
Shape(BC,OCB)
Shape(OA,AC,CO)
Shape(CA,OAC)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),30)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(OAC))
Answer: 60
Theorem sequence:
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,B)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OC,O)
isosceles_triangle_judgment_line_equal(1,OAC)
isosceles_triangle_property_angle_equal(1,OAC)
triangle_property_angle_sum(1,OAC)
Problem: As shown in the diagram, ∠BAD=21°, O is the center of ⊙O, AB is the diameter of circle O. Find the measure of ∠DCA.
问题：如图所示，∠BAD=21°，O是⊙O的圆心，AB是⊙O的直径。求∠DCA的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,CO,OE,EC)
Shape(EO,OA,AD,DE)
Shape(BE,ED,DB)
Shape(CE,EB,OBC)
Shape(BD,ODB)
Shape(DA,OAD)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(BAD),21)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(DCA))
Answer: 69
Theorem sequence:
diameter_of_circle_property_right_angle(1,ADB,O)
triangle_property_angle_sum(1,ADB)
arc_property_circumference_angle_external(1,OAD,B)
arc_property_circumference_angle_external(1,OAD,C)
Problem: As shown in the diagram, ∠ABC=60°, O is the center of ⊙O, AB is the diameter of circle O. Find the measure of ∠CAB.
问题：如图所示，∠ABC=60°，⊙O的圆心为O，AB是圆O的直径。求∠CAB的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AO,OB,BC)
Shape(CB,OBC)
Shape(BO,OA,OAB)
Collinear(AOB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),60)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 30
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,CAB)
Problem: As shown in the diagram, ∠DAB=50°, O is the center of ⊙O, AD is the diameter of circle O. Find the measure of ∠BCA.
问题：如图所示，∠DAB=50°，⊙O的圆心为O，AD是⊙O的直径。求∠BCA的大小。
Construction (CDL):
Shape(CE,EO,OD,ODC)
Shape(CA,AE,EC)
Shape(EA,AB,BE)
Shape(AC,OCA)
Shape(BA,OAB)
Shape(DO,OE,EB,BD)
Shape(DB,OBD)
Collinear(AEOD)
Collinear(CEB)
Cocircular(O,ABDC)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),50)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AD,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 40
Theorem sequence:
diameter_of_circle_property_right_angle(1,ABD,O)
triangle_property_angle_sum(1,ABD)
arc_property_circumference_angle_external(1,OAB,D)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠COB=100°, O is the center of circle O. Find the measure of ∠CAB.
问题：如图所示，∠COB=100°，⊙O的圆心为O。求∠CAB的大小。
Construction (CDL):
Shape(BO,OD,DA,OAB)
Shape(OB,BD,DO)
Shape(CD,DB,OBC)
Shape(AD,DC,CA)
Shape(AC,OCA)
Collinear(BDA)
Collinear(ODC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(COB),100)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 50
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠ACB=70°, ∠CBA=60°, O is the center of ⊙O. Find the measure of ∠BOD.
问题：如图所示，∠ACB=70°，∠CBA=60°，⊙O的圆心为O。求∠BOD的大小。
Construction (CDL):
Shape(DA,OAD)
Shape(OA,AD,DO)
Shape(EB,BO,OD,ODE)
Shape(DC,CE,ODE)
Shape(BE,OEB)
Shape(AO,OB,OBA)
Collinear(AOB)
Collinear(ADC)
Collinear(CEB)
Cocircular(O,ADB)
Conditions (CDL):
Equal(MeasureOfAngle(ACB),70)
Equal(MeasureOfAngle(CBA),60)
IsCentreOfCircle(O,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, CB=BD, ∠AOC=108°, O is the center of ⊙O. Find the measure of ∠BDC.
问题：如图所示，CB=BD，∠AOC=108°，⊙O的圆心为O。求∠BDC的大小。
Construction (CDL):
Shape(AO,OC,OCA)
Shape(CO,OA,AB,BC)
Shape(EC,CB,OBE)
Shape(CE,OEC)
Shape(BD,DE,OBE)
Shape(OA,OAB)
Collinear(CED)
Collinear(ABD)
Cocircular(O,ABC)
Conditions (CDL):
Equal(LengthOfLine(CB),LengthOfLine(BD))
Equal(MeasureOfAngle(AOC),108)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BDC))
Answer: 27
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, ∠ACB=40°, O is the center of circle O. Find the measure of ∠AOB.
问题：如图所示，∠ACB=40°，O是⊙O的圆心。求∠AOB的大小。
Construction (CDL):
Shape(BO,OD,DC,OCB)
Shape(OB,BD,DO)
Shape(CD,DA,AC)
Shape(CA,OAC)
Shape(AD,DB,OBA)
Collinear(ODA)
Collinear(BDC)
Cocircular(O,BAC)
Conditions (CDL):
Equal(MeasureOfAngle(ACB),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AOB))
Answer: 80
Theorem sequence:
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_external(1,OBA,C)
Problem: As shown in the diagram, ∠BOA=40°, O is the center of circle O. Find the measure of ∠BCA.
问题：如图所示，∠BOA=40°，⊙O的圆心为O。求∠BCA的大小。
Construction (CDL):
Shape(AO,OD,DC,OCA)
Shape(OA,AD,DO)
Shape(CD,DB,BC)
Shape(CB,OBC)
Shape(BD,DA,OAB)
Collinear(ODB)
Collinear(ADC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 20
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠BAC=44°, the center of circle O is O. Find the measure of ∠BOC.
问题：如图所示，∠BAC=44°，⊙O的圆心为O。求∠BOC的大小。
Construction (CDL):
Shape(CO,OD,DA,OAC)
Shape(AD,DB,BA)
Shape(AB,OBA)
Shape(BD,DC,OCB)
Shape(OC,CD,DO)
Collinear(ODB)
Collinear(CDA)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),44)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOC))
Answer: 88
Theorem sequence:
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_external(1,OCB,A)
Problem: As shown in the diagram, ∠COB=110°, O is the center of circle O. Find the measure of ∠CAB.
问题：如图所示，∠COB=110°，⊙O的圆心为O。求∠CAB的大小。
Construction (CDL):
Shape(CO,OD,DA,OAB)
Shape(AD,DC,CA)
Shape(AC,OCA)
Shape(CD,DB,OBC)
Shape(OB,BD,DO)
Collinear(ODC)
Collinear(BDA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(COB),110)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 55
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠CDB=30°, the radius of ⊙O is 3, O is the center of circle O, OE⊥CE. Find the length of line OE.
问题：如图所示，∠CDB=30°，圆O的半径为3，⊙O的圆心为O，OE⊥CE。求直线OE的长度。
Construction (CDL):
Shape(AO,OC,OCA)
Shape(CO,OE,EC)
Shape(CE,EB,BC)
Shape(CB,OBC)
Shape(BE,ED,DB)
Shape(BD,ODB)
Shape(DE,EO,OA,OAD)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(CDB),30)
Equal(RadiusOfCircle(O),3)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(OE,CE)
Goal: Value(LengthOfLine(OE))
Answer: 3/2
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,D)
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,COB)
isosceles_triangle_judgment_angle_equal(1,COB)
altitude_of_triangle_judgment(1,CE,COB)
isosceles_triangle_property_line_coincidence(1,COB,E)
line_addition(1,AO,OE)
line_addition(1,AO,OB)
line_addition(1,AE,EB)
Problem: As shown in the diagram, ∠BCA=30°, the center of ⊙O is O. Find the measure of ∠ABO.
问题：如图所示，∠BCA=30°，⊙O的圆心为O。求∠ABO的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AO,OB,BC)
Shape(OA,AB,BO)
Shape(BA,OAB)
Shape(CB,OBC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),30)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ABO))
Answer: 60
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
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)
Problem: As shown in the diagram, ∠ABC=40°, O is the center of ⊙O. Find the measure of ∠AOC.
问题：如图所示，∠ABC=40°，圆O的圆心为O。求∠AOC的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AB,BC,CO,OA)
Shape(AO,OC,OCA)
Shape(BC,OBC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AOC))
Answer: 80
Theorem sequence:
arc_property_center_angle(1,OCA,O)
arc_property_circumference_angle_external(1,OCA,B)
Problem: As shown in the diagram, ∠BCO=35°, ∠CAO=25°, the center of circle O is O. Find the measure of ∠BOA.
问题：如图所示，∠BCO=35°，∠CAO=25°，⊙O的圆心为O。求∠BOA的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AO,OC)
Shape(CO,OB,BC)
Shape(CB,OBC)
Shape(DO,OA,OAD)
Shape(BO,OD,ODB)
Collinear(COD)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(BCO),35)
Equal(MeasureOfAngle(CAO),25)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 120
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)
angle_addition(1,BCO,OCA)
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠CBA=50°, the center of ⊙O is O, AB is parallel to CD. Find the measure of ∠BOD.
问题：如图所示，∠CBA=50°，O是⊙O的圆心，AB平行于CD。求∠BOD的大小。
Construction (CDL):
Shape(AB,OBA)
Shape(CB,BA,OAC)
Shape(BC,CD,DO,OB)
Shape(BO,OD,ODB)
Shape(DC,OCD)
Cocircular(O,ACDB)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),50)
IsCentreOfCircle(O,O)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(BOD))
Answer: 100
Theorem sequence:
parallel_property_alternate_interior_angle(2,AB,CD)
arc_property_center_angle(1,ODB,O)
arc_property_circumference_angle_external(1,ODB,C)
Problem: As shown in the diagram, ∠AEC=28°, the center of circle O is O, AG is perpendicular to CG, BG is perpendicular to DG, CG is perpendicular to BG, DG⊥AG. Find the measure of ∠DBA.
问题：如图所示，∠AEC=28°，O是圆O的圆心，AG⊥CG，BG⊥DG，CG⊥BG，DG垂直于AG。求∠DBA的大小。
Construction (CDL):
Shape(CE,OEC)
Shape(EC,CF,FE)
Shape(FA,AG,GF)
Shape(AF,FC,OCA)
Shape(EF,FG,GO,OB,OBE)
Shape(DG,GA,OAD)
Shape(GD,DB,BO,OG)
Shape(BD,ODB)
Collinear(CFGD)
Collinear(AFE)
Collinear(AGOB)
Cocircular(O,ADBEC)
Conditions (CDL):
Equal(MeasureOfAngle(AEC),28)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AG,CG)
PerpendicularBetweenLine(BG,DG)
PerpendicularBetweenLine(CG,BG)
PerpendicularBetweenLine(DG,AG)
Goal: Value(MeasureOfAngle(DBA))
Answer: 28
Theorem sequence:
circle_property_chord_perpendicular_bisect_arc(1,OCD,OGA)
congruent_arc_judgment_length_equal(1,OCA,OAD)
congruent_arc_property_measure_equal(1,OCA,OAD)
arc_property_circumference_angle_external(1,OAD,B)
arc_property_circumference_angle_external(1,OCA,E)
Problem: As shown in the diagram, ∠COA=50°, the center of ⊙O is O. Find the measure of ∠CBA.
问题：如图所示，∠COA=50°，O是⊙O的圆心。求∠CBA的大小。
Construction (CDL):
Shape(AO,OD,DB,OBA)
Shape(OA,AD,DO)
Shape(CD,DA,OAC)
Shape(BD,DC,CB)
Shape(BC,OCB)
Collinear(ODC)
Collinear(ADB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(COA),50)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CBA))
Answer: 25
Theorem sequence:
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,B)
Problem: As shown in the diagram, ∠ACO=20°, ∠OBA=40°, the center of ⊙O is O. Find the measure of ∠BOC.
问题：如图所示，∠ACO=20°，∠OBA=40°，⊙O的圆心为O。求∠BOC的大小。
Construction (CDL):
Shape(CA,OAC)
Shape(CO,OA,AC)
Shape(AO,OB,BA)
Shape(BO,OC,OCB)
Shape(AB,OBA)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(ACO),20)
Equal(MeasureOfAngle(OBA),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOC))
Answer: 120
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,OB,O)
isosceles_triangle_judgment_line_equal(1,OAC)
isosceles_triangle_judgment_line_equal(1,OBA)
isosceles_triangle_property_angle_equal(1,OAC)
isosceles_triangle_property_angle_equal(1,OBA)
angle_addition(1,BAO,OAC)
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_external(1,OCB,A)
Problem: As shown in the diagram, ∠CBA=70°, the center of circle O is O. Find the measure of ∠OAC.
问题：如图所示，∠CBA=70°，O是圆O的圆心。求∠OAC的大小。
Construction (CDL):
Shape(CA,OAC)
Shape(AC,CE,EO,OA)
Shape(AO,OE,EB,BA)
Shape(AB,OBA)
Shape(DE,EC,CD)
Shape(DC,OCD)
Shape(BE,ED,ODB)
Collinear(AOED)
Collinear(CEB)
Cocircular(O,ACDB)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),70)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(OAC))
Answer: 20
Theorem sequence:
diameter_of_circle_judgment_pass_centre(1,AOD,O)
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,ACD)
Problem: As shown in the diagram, ∠FCD=25°, the center of ⊙O is O, G bisects segment EF. Find the measure of ∠GOE.
问题：如图所示，∠FCD=25°，O是⊙O的圆心，G平分线段EF。求∠GOE的大小。
Construction (CDL):
Shape(EO,OC,OCE)
Shape(OE,EG,GO)
Shape(CO,OG,GF,FC)
Shape(CF,OFC)
Shape(DG,GE,OED)
Shape(FG,GD,ODF)
Collinear(EGF)
Collinear(COGD)
Cocircular(O,CEDF)
Conditions (CDL):
Equal(MeasureOfAngle(FCD),25)
IsCentreOfCircle(O,O)
IsMidpointOfLine(G,EF)
Goal: Value(MeasureOfAngle(GOE))
Answer: 50
Theorem sequence:
circle_property_chord_perpendicular_bisect_arc(2,OEF,OGD)
congruent_arc_judgment_length_equal(1,OED,ODF)
congruent_arc_property_measure_equal(1,OED,ODF)
arc_property_center_angle(1,OED,O)
arc_property_circumference_angle_external(1,ODF,C)
Problem: As shown in the diagram, ∠CBA=40°, O is the center of circle O. Find the measure of ∠BAC.
问题：如图所示，∠CBA=40°，⊙O的圆心为O。求∠BAC的大小。
Construction (CDL):
Shape(AO,OB,OBA)
Shape(BO,OA,AC,CB)
Shape(CA,OAC)
Shape(BC,OCB)
Collinear(AOB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BAC))
Answer: 50
Theorem sequence:
diameter_of_circle_judgment_pass_centre(1,AOB,O)
diameter_of_circle_property_right_angle(1,ACB,O)
triangle_property_angle_sum(1,BAC)
Problem: As shown in the diagram, ∠ADB=28°, O is the center of ⊙O, AG is perpendicular to BG, BG⊥OG, CG is perpendicular to AG, OG⊥CG. Find the measure of ∠COA.
问题：如图所示，∠ADB=28°，圆O的圆心为O，AG⊥BG，BG垂直于OG，CG垂直于AG，OG⊥CG。求∠COA的大小。
Construction (CDL):
Shape(BD,ODB)
Shape(DE,EG,GO,OC,OCD)
Shape(CO,OG,GC)
Shape(DB,BE,ED)
Shape(EA,AG,GE)
Shape(CE,EB,OBA)
Shape(CG,GA,OAC)
Collinear(BEGC)
Collinear(OGA)
Collinear(DEA)
Cocircular(O,ACDB)
Conditions (CDL):
Equal(MeasureOfAngle(ADB),28)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AG,BG)
PerpendicularBetweenLine(BG,OG)
PerpendicularBetweenLine(CG,AG)
PerpendicularBetweenLine(OG,CG)
Goal: Value(MeasureOfAngle(COA))
Answer: 56
Theorem sequence:
circle_property_chord_perpendicular_bisect_arc(1,OBC,OGA)
congruent_arc_judgment_length_equal(1,OBA,OAC)
congruent_arc_property_measure_equal(1,OBA,OAC)
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OBA,D)
Problem: As shown in the diagram, ∠CDB=20°, A is the center of ⊙A. Find the measure of ∠CAB.
问题：如图所示，∠CDB=20°，A是⊙A的圆心。求∠CAB的大小。
Construction (CDL):
Shape(BA,AD,ADB)
Shape(AB,BE,EA)
Shape(AE,ED,DA)
Shape(EB,BC,CE)
Shape(EC,CD,DE)
Shape(CB,ABC)
Shape(DC,ACD)
Collinear(AEC)
Collinear(BED)
Cocircular(A,BCD)
Conditions (CDL):
Equal(MeasureOfAngle(CDB),20)
IsCentreOfCircle(A,A)
Goal: Value(MeasureOfAngle(CAB))
Answer: 40
Theorem sequence:
arc_property_center_angle(1,ABC,A)
arc_property_circumference_angle_external(1,ABC,D)
Problem: As shown in the diagram, ∠CAB=52°, O is the center of ⊙O. Find the measure of ∠ABC.
问题：如图所示，∠CAB=52°，O是⊙O的圆心。求∠ABC的大小。
Construction (CDL):
Shape(BO,OA,OAB)
Shape(AO,OB,BC,CA)
Shape(AC,OCA)
Shape(BC,OBC)
Collinear(AOB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),52)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ABC))
Answer: 38
Theorem sequence:
diameter_of_circle_judgment_pass_centre(1,AOB,O)
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,ABC)
Problem: As shown in the diagram, ∠ACD=20°, ∠DBC=Mul(MeasureOfAngle(ABD),2)°, the center of circle O is O. Find the measure of ∠DAC.
问题：如图所示，∠ACD=20°，∠DBC=Mul(MeasureOfAngle(ABD),2)°，O是⊙O的圆心。求∠DAC的大小。
Construction (CDL):
Shape(DC,OCD)
Shape(DE,EC,CD)
Shape(AD,ODA)
Shape(DA,AE,ED)
Shape(CE,EB,BC)
Shape(CB,OBC)
Shape(BA,OAB)
Shape(EA,AB,BE)
Collinear(DEB)
Collinear(AEC)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(ACD),20)
Equal(MeasureOfAngle(DBC),Mul(MeasureOfAngle(ABD),2))
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DAC))
Answer: 40
Theorem sequence:
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,OCD,B)
Problem: As shown in the diagram, ∠BCA=35°, O is the center of circle O. Find the measure of ∠BOA.
问题：如图所示，∠BCA=35°，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(CB,OBC)
Shape(BD,DA,OAB)
Collinear(ODB)
Collinear(ADC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),35)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 70
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠BEC=90°, ∠CDA=50°, O is the center of ⊙O. Find the measure of ∠DCB.
问题：如图所示，∠BEC=90°，∠CDA=50°，O是圆O的圆心。求∠DCB的大小。
Construction (CDL):
Shape(BC,OCB)
Shape(CB,BE,EC)
Shape(CE,EA,OAC)
Shape(DE,EB,OBD)
Shape(ED,DA,AE)
Shape(AD,ODA)
Collinear(BEA)
Collinear(CED)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(BEC),90)
Equal(MeasureOfAngle(CDA),50)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DCB))
Answer: 40
Theorem sequence:
arc_property_circumference_angle_external(1,OAC,B)
arc_property_circumference_angle_external(1,OAC,D)
triangle_property_angle_sum(1,CBE)