SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠OAB=65°, the center of circle O is O. Find the measure of ∠BCA.
问题：如图所示，∠OAB=65°，O是⊙O的圆心。求∠BCA的大小。
Construction (CDL):
Shape(CB,OBC)
Shape(CD,DB,BC)
Shape(BD,DA,AB)
Shape(OA,AD,DO)
Shape(BA,OAB)
Shape(OD,DC,CO)
Shape(AO,OC,OCA)
Collinear(ODB)
Collinear(ADC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(OAB),65)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 25
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, ∠ABD=50°, O is the center of circle O, the diameter of circle O is AB. Find the measure of ∠ECB.
问题：如图所示，∠ABD=50°，⊙O的圆心为O，AB是⊙O的直径。求∠ECB的大小。
Construction (CDL):
Shape(DA,AE,ED)
Shape(AD,ODA)
Shape(DE,EB,BD)
Shape(DB,OBD)
Shape(CE,EA,OAC)
Shape(BE,EC,CB)
Shape(BC,OCB)
Collinear(AOEB)
Collinear(DEC)
Cocircular(O,DACB)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),50)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ECB))
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,OBD,A)
arc_property_circumference_angle_external(1,OBD,C)
Problem: As shown in the diagram, ∠BAD=60°, ∠CDA=90°, the center of circle O is O. Find the measure of ∠DCO.
问题：如图所示，∠BAD=60°，∠CDA=90°，O是圆O的圆心。求∠DCO的大小。
Construction (CDL):
Shape(BD,DO,OB)
Shape(OD,DC,CO)
Shape(BA,AD,DB)
Shape(CD,DA,OAC)
Shape(AB,OBA)
Shape(BO,OC,OCB)
Collinear(BDC)
Collinear(ODA)
Cocircular(O,BAC)
Conditions (CDL):
Equal(MeasureOfAngle(BAD),60)
Equal(MeasureOfAngle(CDA),90)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DCO))
Answer: 30
Theorem sequence:
adjacent_complementary_angle(1,CDA,ADB)
triangle_property_angle_sum(1,DBA)
arc_property_circumference_angle_external(1,OAC,B)
arc_property_center_angle(1,OAC,O)
adjacent_complementary_angle(1,ODC,CDA)
triangle_property_angle_sum(1,ODC)
Problem: As shown in the diagram, OD=2, ∠CAO=30°, the center of ⊙O is O, OE is perpendicular to CE. Find the length of line DC.
问题：如图所示，OD=2，∠CAO=30°，⊙O的圆心为O，OE垂直于CE。求直线DC的长度。
Construction (CDL):
Shape(AC,OCA)
Shape(CE,EB,OBC)
Shape(BE,ED,ODB)
Shape(AO,OC,CA)
Shape(CO,OE,EC)
Shape(OD,DE,EO)
Shape(DO,OA,OAD)
Collinear(CED)
Collinear(AOEB)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(LengthOfLine(OD),2)
Equal(MeasureOfAngle(CAO),30)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(OE,CE)
Goal: Value(LengthOfLine(DC))
Answer: 2*sqrt(3)
Theorem sequence:
angle_addition(1,ECO,OCA)
radius_of_circle_property_length_equal(1,OA,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,OCA)
isosceles_triangle_judgment_line_equal(1,ODC)
isosceles_triangle_property_angle_equal(1,OCA)
isosceles_triangle_property_angle_equal(1,ODC)
triangle_property_angle_sum(1,CAE)
triangle_property_angle_sum(1,COD)
sine_theorem(1,COD)
Problem: As shown in the diagram, ∠BAC=35°, O is the center of circle O, the diameter of ⊙O is AB. Find the measure of ∠CDA.
问题：如图所示，∠BAC=35°，O是⊙O的圆心，圆O的直径为AB。求∠CDA的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(CA,OAC)
Shape(BO,OC,OCB)
Shape(DB,OBD)
Shape(DE,EB,BD)
Shape(AO,OE,ED,DA)
Shape(AC,CE,EO,OA)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),35)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CDA))
Answer: 55
Theorem sequence:
arc_property_circumference_angle_external(1,OCB,A)
arc_property_circumference_angle_external(1,OCB,D)
diameter_of_circle_property_right_angle(1,BDA,O)
angle_addition(1,BDC,CDA)
Problem: As shown in the diagram, ∠DAP=35°, O is the center of ⊙O, AP⊥DP. Find the measure of ∠CBP.
问题：如图所示，∠DAP=35°，O是⊙O的圆心，AP垂直于DP。求∠CBP的大小。
Construction (CDL):
Shape(CP,PA,OAC)
Shape(AP,PD,DA)
Shape(CB,BP,PC)
Shape(BC,OCB)
Shape(DP,PB,OBD)
Shape(AD,ODA)
Collinear(BPA)
Collinear(CPD)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(DAP),35)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AP,DP)
Goal: Value(MeasureOfAngle(CBP))
Answer: 55
Theorem sequence:
triangle_property_angle_sum(1,APD)
arc_property_circumference_angle_external(1,OAC,B)
arc_property_circumference_angle_external(1,OAC,D)
Problem: As shown in the diagram, ∠BCD=140°, O is the center of circle O. Find the measure of ∠DOB.
问题：如图所示，∠BCD=140°，O是圆O的圆心。求∠DOB的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(DA,AB,BO,OD)
Shape(BA,OAB)
Shape(DO,OB,BC,CD)
Shape(DC,OCD)
Shape(CB,OBC)
Cocircular(O,BCD)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),140)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DOB))
Answer: 80
Theorem sequence:
arc_property_center_angle(1,OBD,O)
arc_property_circumference_angle_internal(1,OBD,C)
Problem: As shown in the diagram, ∠CDA=35°, the center of ⊙O is O, OE⊥BE. Find the measure of ∠ABO.
问题：如图所示，∠CDA=35°，O是圆O的圆心，OE⊥BE。求∠ABO的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(OCB,BE,EC)
Shape(OAC,CE,EA)
Shape(OAC,CD,DA)
Shape(OE,EB,BO)
Shape(OBA,AO,OB)
Shape(OA,AE,EO)
Collinear(AEB)
Collinear(OEC)
Cocircular(O,DACB)
Conditions (CDL):
Equal(MeasureOfAngle(CDA),35)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(OE,BE)
Goal: Value(MeasureOfAngle(ABO))
Answer: 20
Theorem sequence:
arc_property_circumference_angle_external(1,OAC,D)
adjacent_complementary_angle(1,AEO,OEB)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
mirror_congruent_triangle_judgment_hl(1,AEO,BOE)
mirror_congruent_triangle_property_angle_equal(1,OAE,OEB)
arc_property_center_angle(1,OAC,O)
triangle_property_angle_sum(1,OEB)
Problem: As shown in the diagram, ∠BOA=40°, O is the center of ⊙O. Find the measure of ∠BPA.
问题：如图所示，∠BOA=40°，圆O的圆心为O。求∠BPA的大小。
Construction (CDL):
Shape(OAB,BO,OA)
Shape(OBA,AO,OB)
Shape(OAB,BP,PA)
Cocircular(O,PAB)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BPA))
Answer: 20
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,P)
Problem: As shown in the diagram, ∠ADC=32°, the center of circle O is O. Find the measure of ∠AOC.
问题：如图所示，∠ADC=32°，O是⊙O的圆心。求∠AOC的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CO,OB)
Shape(ODC,CD)
Shape(OCA,AC)
Shape(OA,AB,BO)
Shape(OC,CA,AO)
Shape(AD,DC,CA)
Collinear(BOC)
Cocircular(O,ABDC)
Conditions (CDL):
Equal(MeasureOfAngle(ADC),32)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AOC))
Answer: 64
Theorem sequence:
arc_property_center_angle(1,OCA,O)
arc_property_circumference_angle_external(1,OCA,D)
Problem: As shown in the diagram, ∠BCD=120°, O is the center of circle O. Find the measure of ∠DOB.
问题：如图所示，∠BCD=120°，O是圆O的圆心。求∠DOB的大小。
Construction (CDL):
Shape(OBC,CB)
Shape(OCD,DC)
Shape(ODA,AD)
Shape(OAB,BA)
Shape(AB,BO,OD,DA)
Shape(OB,BC,CD,DO)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),120)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DOB))
Answer: 120
Theorem sequence:
arc_property_center_angle(1,OBD,O)
arc_property_circumference_angle_internal(1,OBD,C)
Problem: As shown in the diagram, ∠CAB=34°, the center of circle O is O, the diameter of ⊙O is AC. Find the measure of ∠BDA.
问题：如图所示，∠CAB=34°，⊙O的圆心为O，AC是⊙O的直径。求∠BDA的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCD,DA,AO,OC)
Shape(DA,AB,BD)
Shape(AB,BC,CO,OA)
Collinear(AOC)
Cocircular(O,DABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),34)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AC,O)
Goal: Value(MeasureOfAngle(BDA))
Answer: 56
Theorem sequence:
diameter_of_circle_property_right_angle(1,ABC,O)
triangle_property_angle_sum(1,ABC)
arc_property_circumference_angle_external(1,OAB,C)
arc_property_circumference_angle_external(1,OAB,D)
Problem: As shown in the diagram, ∠CBA=25°, O is the center of ⊙O, AD is perpendicular to OD. Find the measure of ∠BOA.
问题：如图所示，∠CBA=25°，O是⊙O的圆心，AD⊥OD。求∠BOA的大小。
Construction (CDL):
Shape(OAC,CD,DA)
Shape(OCB,BC)
Shape(OBA,AO,OB)
Shape(OA,AD,DO)
Shape(DC,CB,BD)
Shape(OD,DB,BO)
Collinear(ADB)
Collinear(ODC)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),25)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AD,OD)
Goal: Value(MeasureOfAngle(BOA))
Answer: 100
Theorem sequence:
circle_property_chord_perpendicular_bisect_arc(1,OAB,ODC)
arc_addition_measure(1,OAC,OCB)
congruent_arc_judgment_length_equal(1,OAC,OCB)
congruent_arc_property_measure_equal(1,OAC,OCB)
arc_property_circumference_angle_external(1,OAC,B)
arc_property_center_angle(1,OAB,O)
Problem: As shown in the diagram, ∠ACD=15°, O is the center of circle O. Find the measure of ∠DAB.
问题：如图所示，∠ACD=15°，O是圆O的圆心。求∠DAB的大小。
Construction (CDL):
Shape(AC,CD,DA)
Shape(OAC,OCB,BO,OA)
Shape(OBD,ODA,AO,OB)
Shape(BD,DA,AB)
Collinear(BOA)
Cocircular(O,CBDA)
Conditions (CDL):
Equal(MeasureOfAngle(ACD),15)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DAB))
Answer: 75
Theorem sequence:
diameter_of_circle_judgment_pass_centre(1,BOA,O)
triangle_property_angle_sum(1,BDA)
diameter_of_circle_property_right_angle(1,BDA,O)
arc_property_circumference_angle_external(1,ODA,C)
arc_property_circumference_angle_external(1,ODA,B)
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的圆心为O。求∠OBC的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCA,AC)
Shape(OB,BC,CO)
Shape(OC,CA,AB,BO)
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)
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OC,O)
arc_property_center_angle(1,OBC,O)
triangle_property_angle_sum(1,OBC)
isosceles_triangle_judgment_line_equal(1,OBC)
isosceles_triangle_property_angle_equal(1,OBC)
Problem: As shown in the diagram, ∠CDA=26°, O is the center of circle O, OE⊥BE. Find the measure of ∠BOC.
问题：如图所示，∠CDA=26°，⊙O的圆心为O，OE⊥BE。求∠BOC的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(OCB,BE,EC)
Shape(OAC,CE,EA)
Shape(OAC,CD,DA)
Shape(OE,EB,BO)
Shape(OBA,AO,OB)
Shape(OA,AE,EO)
Collinear(AEB)
Collinear(OEC)
Cocircular(O,DACB)
Conditions (CDL):
Equal(MeasureOfAngle(CDA),26)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(OE,BE)
Goal: Value(MeasureOfAngle(BOC))
Answer: 52
Theorem sequence:
arc_property_circumference_angle_external(1,OAC,D)
adjacent_complementary_angle(1,AEO,OEB)
circle_property_chord_perpendicular_bisect_arc(1,OAB,OEC)
congruent_arc_judgment_length_equal(1,OAC,OCB)
congruent_arc_property_measure_equal(1,OAC,OCB)
arc_property_center_angle(1,OCB,O)
Problem: As shown in the diagram, AB=10, BC=6, the center of ⊙O is O, AD is perpendicular to OD. Find the length of line OD.
问题：如图所示，AB=10，BC=6，圆O的圆心为O，AD垂直于OD。求直线OD的长度。
Construction (CDL):
Shape(OBA,AO,OB)
Shape(OAC,CD,DA)
Shape(OCB,BC)
Shape(OA,AD,DO)
Shape(OD,DC,CB,BO)
Collinear(AOB)
Collinear(ADC)
Cocircular(O,ACB)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(LengthOfLine(BC),6)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AD,OD)
Goal: Value(LengthOfLine(OD))
Answer: 3
Theorem sequence:
circle_property_chord_perpendicular_bisect_chord(1,O,OD,AC)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
midsegment_of_triangle_judgment_midpoint(1,DO,ACB)
midsegment_of_triangle_property_length(1,DO,ACB)
Problem: As shown in the diagram, ∠ADC=26°, O is the center of ⊙O. Find the measure of ∠CAB.
问题：如图所示，∠ADC=26°，O是圆O的圆心。求∠CAB的大小。
Construction (CDL):
Shape(OAD,DA)
Shape(ODB,BC,CD)
Shape(OBC,CB)
Shape(OCA,AC)
Shape(CA,AD,DC)
Shape(CA,AB,BC)
Collinear(AOB)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(ADC),26)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 64
Theorem sequence:
diameter_of_circle_judgment_pass_centre(1,AOB,O)
diameter_of_circle_property_right_angle(1,BCA,O)
arc_property_circumference_angle_external(1,OCA,D)
arc_property_circumference_angle_external(1,OCA,B)
triangle_property_angle_sum(1,ABC)
Problem: As shown in the diagram, CD=AD, ∠ABC=70°, the center of circle O is O. Find the measure of ∠BAD.
问题：如图所示，CD=AD，∠ABC=70°，圆O的圆心为O。求∠BAD的大小。
Construction (CDL):
Shape(OAD,DA)
Shape(ODB,BO,OA,AD)
Shape(OBC,CB)
Shape(OCA,AC)
Shape(AD,DC,CA)
Shape(CA,AO,OB,BC)
Collinear(AOB)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(LengthOfLine(CD),LengthOfLine(AD))
Equal(MeasureOfAngle(ABC),70)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BAD))
Answer: 35
Theorem sequence:
triangle_property_angle_sum(1,ADC)
triangle_property_angle_sum(1,ABC)
diameter_of_circle_judgment_pass_centre(1,AOB,O)
diameter_of_circle_property_right_angle(1,BCA,O)
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_external(1,OCA,D)
isosceles_triangle_judgment_line_equal(1,DCA)
isosceles_triangle_property_angle_equal(1,DCA)
angle_addition(1,CAB,BAD)
Problem: As shown in the diagram, ∠AOC=120°, the center of ⊙O is O. Find the measure of ∠CDB.
问题：如图所示，∠AOC=120°，圆O的圆心为O。求∠CDB的大小。
Construction (CDL):
Shape(ODB,BD)
Shape(OBC,CO,OB)
Shape(OBC,CD,DB)
Shape(OCA,AO,OC)
Shape(OAD,ODB,BO,OA)
Collinear(BOA)
Cocircular(O,BCAD)
Conditions (CDL):
Equal(MeasureOfAngle(AOC),120)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CDB))
Answer: 30
Theorem sequence:
adjacent_complementary_angle(1,AOC,COB)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,D)
Problem: As shown in the diagram, ∠BOC=30°, the center of circle O is O, CE is perpendicular to OE. Find the measure of ∠DAB.
问题：如图所示，∠BOC=30°，O是⊙O的圆心，CE⊥OE。求∠DAB的大小。
Construction (CDL):
Shape(OCB,BE,EC)
Shape(OBD,DE,EB)
Shape(ODA,AD)
Shape(OAC,CO,OA)
Shape(OC,CE,EO)
Shape(AE,ED,DA)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,CBDA)
Conditions (CDL):
Equal(MeasureOfAngle(BOC),30)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(CE,OE)
Goal: Value(MeasureOfAngle(DAB))
Answer: 15
Theorem sequence:
circle_property_chord_perpendicular_bisect_arc(1,OCD,OEB)
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_external(1,OBD,A)
congruent_arc_judgment_length_equal(1,OBD,OCB)
congruent_arc_property_measure_equal(1,OBD,OCB)
Problem: As shown in the diagram, ∠CAB=25°, the center of circle O is O, BO is parallel to CD. Find the measure of ∠DOB.
问题：如图所示，∠CAB=25°，O是⊙O的圆心，BO平行于CD。求∠DOB的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CO,OB)
Shape(OCD,DC)
Shape(ODA,AB,BO,OD)
Shape(AB,BO,OC,CA)
Shape(OC,CD,DO)
Cocircular(O,BCDA)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),25)
IsCentreOfCircle(O,O)
ParallelBetweenLine(BO,CD)
Goal: Value(MeasureOfAngle(DOB))
Answer: 130
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,OBC,O)
angle_addition(1,DOC,COB)
radius_of_circle_property_length_equal(1,OC,O)
radius_of_circle_property_length_equal(1,OD,O)
isosceles_triangle_judgment_line_equal(1,OCD)
isosceles_triangle_property_angle_equal(1,OCD)
triangle_property_angle_sum(1,OCD)
parallel_property_alternate_interior_angle(2,BO,CD)
Problem: As shown in the diagram, ∠AOC=80°, O is the center of circle O. Find the measure of ∠BCO.
问题：如图所示，∠AOC=80°，圆O的圆心为O。求∠BCO的大小。
Construction (CDL):
Shape(OCA,AO,OC)
Shape(OAB,BO,OA)
Shape(OBC,CB)
Shape(CO,OB,BC)
Collinear(AOB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(AOC),80)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCO))
Answer: 40
Theorem sequence:
arc_property_center_angle(1,OCA,O)
arc_property_circumference_angle_external(1,OCA,B)
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)
Problem: As shown in the diagram, ∠COA=140°, the center of ⊙O is O. Find the measure of ∠ABC.
问题：如图所示，∠COA=140°，⊙O的圆心为O。求∠ABC的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(ODA,AD)
Shape(OA,AB,BC,CO)
Shape(DO,OC,CD)
Shape(DA,AO,OD)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(COA),140)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ABC))
Answer: 110
Theorem sequence:
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_internal(1,OAC,B)
Problem: As shown in the diagram, ∠ABD=15°, O is the center of ⊙O, A bisects arc ODC. Find the measure of ∠BOC.
问题：如图所示，∠ABD=15°，O是圆O的圆心，A是⌒ODC的中点。求∠BOC的大小。
Construction (CDL):
Shape(OAC,CO,OA)
Shape(OCB,BO,OC)
Shape(OBD,DB)
Shape(ODA,AD)
Shape(DA,AO,OD)
Shape(DO,OB,BD)
Collinear(AOB)
Cocircular(O,DACB)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),15)
IsCentreOfCircle(O,O)
IsMidpointOfArc(A,ODC)
Goal: Value(MeasureOfAngle(BOC))
Answer: 150
Theorem sequence:
adjacent_complementary_angle(1,BOC,COA)
congruent_arc_judgment_length_equal(1,ODA,OAC)
congruent_arc_property_measure_equal(1,ODA,OAC)
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,ODA,B)
Problem: As shown in the diagram, ∠EAD=70°, the center of circle O is O. Find the measure of ∠DOE.
问题：如图所示，∠EAD=70°，⊙O的圆心为O。求∠DOE的大小。
Construction (CDL):
Shape(OCE,EC)
Shape(OED,DO,OE)
Shape(OED,EA,AD)
Shape(ODB,BD)
Shape(BO,OD,DB)
Shape(OC,CE,EO)
Shape(CD,DB,BO,OC)
Shape(CA,AD,DC)
Collinear(BOC)
Collinear(BDA)
Collinear(AEC)
Cocircular(O,CEDB)
Conditions (CDL):
Equal(MeasureOfAngle(EAD),70)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DOE))
Answer: 40
Theorem sequence:
triangle_property_angle_sum(1,ADC)
arc_property_circumference_angle_external(1,OED,C)
arc_property_center_angle(1,OED,O)
diameter_of_circle_judgment_pass_centre(1,BOC,O)
diameter_of_circle_property_right_angle(1,CDB,O)
adjacent_complementary_angle(1,ADC,CDB)
Problem: As shown in the diagram, ∠ABC=20°, O is the center of ⊙O, OD is perpendicular to AD. Find the measure of ∠BAO.
问题：如图所示，∠ABC=20°，⊙O的圆心为O，OD⊥AD。求∠BAO的大小。
Construction (CDL):
Shape(OAB,BO,OA)
Shape(OBC,CB)
Shape(CD,DB,BC)
Shape(DO,OB,BD)
Shape(DA,AO,OD)
Shape(OCA,AD,DC)
Collinear(ADB)
Collinear(ODC)
Cocircular(O,BCA)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),20)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(OD,AD)
Goal: Value(MeasureOfAngle(BAO))
Answer: 50
Theorem sequence:
triangle_property_angle_sum(1,DAO)
arc_property_center_angle(1,OCA,O)
arc_property_circumference_angle_external(1,OCA,B)
Problem: As shown in the diagram, ∠COA=126°, O is the center of circle O. Find the measure of ∠BDC.
问题：如图所示，∠COA=126°，⊙O的圆心为O。求∠BDC的大小。
Construction (CDL):
Shape(OCB,BO,OC)
Shape(OCB,BD,DC)
Shape(OBD,DB)
Shape(ODA,AO,OB,BD)
Shape(OAC,CO,OA)
Collinear(AOB)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(COA),126)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BDC))
Answer: 27
Theorem sequence:
adjacent_complementary_angle(1,BOC,COA)
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_external(1,OCB,D)
Problem: As shown in the diagram, ∠AOC=60°, ∠CDO=70°, O is the center of ⊙O. Find the measure of ∠DCA.
问题：如图所示，∠AOC=60°，∠CDO=70°，O是⊙O的圆心。求∠DCA的大小。
Construction (CDL):
Shape(OAB,BO,OD,DA)
Shape(OBC,CO,OB)
Shape(OCA,AC)
Shape(CA,AD,DC)
Shape(CD,DO,OC)
Collinear(ADOB)
Cocircular(O,BCA)
Conditions (CDL):
Equal(MeasureOfAngle(AOC),60)
Equal(MeasureOfAngle(CDO),70)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DCA))
Answer: 10
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)
triangle_property_angle_sum(1,OCA)
triangle_property_angle_sum(1,DCA)
adjacent_complementary_angle(1,ADC,CDO)
Problem: As shown in the diagram, ∠DOA=132°, O is the center of circle O. Find the measure of ∠BCD.
问题：如图所示，∠DOA=132°，圆O的圆心为O。求∠BCD的大小。
Construction (CDL):
Shape(ODB,BO,OD)
Shape(ODB,BC,CD)
Shape(OBC,CB)
Shape(OCA,AO,OB,BC)
Shape(OAD,DO,OA)
Collinear(AOB)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(DOA),132)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCD))
Answer: 24
Theorem sequence:
adjacent_complementary_angle(1,BOD,DOA)
arc_property_center_angle(1,ODB,O)
arc_property_circumference_angle_external(1,ODB,C)
Problem: As shown in the diagram, ∠ABC=130°, the center of circle O is O. Find the measure of ∠COA.
问题：如图所示，∠ABC=130°，⊙O的圆心为O。求∠COA的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(ODA,AD)
Shape(OA,AB,BC,CO)
Shape(DO,OC,CD)
Shape(DA,AO,OD)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),130)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COA))
Answer: 100
Theorem sequence:
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_internal(1,OAC,B)
Problem: As shown in the diagram, ∠ABO=40°, O is the center of ⊙O. Find the measure of ∠BCA.
问题：如图所示，∠ABO=40°，O是圆O的圆心。求∠BCA的大小。
Construction (CDL):
Shape(AB,BO,OA)
Shape(OAB,BA)
Shape(OBC,CB)
Shape(AB,BC,CA)
Shape(OCA,AC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(ABO),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
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,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, ∠ADC=35°, O is the center of ⊙O. Find the measure of ∠ABC.
问题：如图所示，∠ADC=35°，⊙O的圆心为O。求∠ABC的大小。
Construction (CDL):
Shape(OAD,DA)
Shape(ODB,BC,CD)
Shape(OBC,CB)
Shape(OCA,AC)
Shape(CA,AD,DC)
Shape(CA,AB,BC)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(ADC),35)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ABC))
Answer: 35
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_external(1,OCA,D)
Problem: As shown in the diagram, BO=4, ∠ABD=30°, the center of circle O is O. Find the length of line AD.
问题：如图所示，BO=4，∠ABD=30°，⊙O的圆心为O。求直线AD的长度。
Construction (CDL):
Shape(OBD,DB)
Shape(ODA,AD)
Shape(OAB,BO,OA)
Shape(BD,DA,AO,OB)
Collinear(AOB)
Cocircular(O,ABD)
Conditions (CDL):
Equal(LengthOfLine(BO),4)
Equal(MeasureOfAngle(ABD),30)
IsCentreOfCircle(O,O)
Goal: Value(LengthOfLine(AD))
Answer: 4
Theorem sequence:
diameter_of_circle_judgment_pass_centre(1,BOA,O)
diameter_of_circle_property_right_angle(1,BDA,O)
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OA,O)
line_addition(1,BO,OA)
sine_theorem(1,ABD)
Problem: As shown in the diagram, ∠OAB=50°, the center of circle O is O. Find the measure of ∠BCA.
问题：如图所示，∠OAB=50°，O是⊙O的圆心。求∠BCA的大小。
Construction (CDL):
Shape(OBC,OCA,AO,OB)
Shape(OBC,CB)
Shape(OAB,BA)
Shape(OA,AB,BO)
Shape(BC,CA,AB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(OAB),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, ∠COB=40°, O is the center of ⊙O. Find the measure of ∠CAB.
问题：如图所示，∠COB=40°，圆O的圆心为O。求∠CAB的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CO,OB)
Shape(OA,AB,BO)
Collinear(COA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(COB),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 20
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)