SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠DBC=32°, O is the center of circle O, AD∥OC. Find the measure of ∠ABD.
问题：如图所示，∠DBC=32°，圆O的圆心为O，AD∥OC。求∠ABD的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(DA,AE,ED)
Shape(EA,AO,OF,FE)
Shape(DE,EC,OCD)
Shape(CF,FB,BC)
Shape(CB,OBC)
Shape(FO,OB,BF)
Shape(CE,EF,FC)
Shape(BO,OA,OAB)
Collinear(DEFB)
Collinear(AEC)
Collinear(CFO)
Collinear(AOB)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(DBC),32)
IsCentreOfCircle(O,O)
ParallelBetweenLine(AD,OC)
Goal: Value(MeasureOfAngle(ABD))
Answer: 26
Theorem sequence:
angle_addition(1,DAC,CAO)
arc_property_circumference_angle_external(1,OCD,A)
arc_property_circumference_angle_external(1,OCD,B)
diameter_of_circle_judgment_pass_centre(1,AOB,O)
diameter_of_circle_property_right_angle(1,BDA,O)
parallel_property_alternate_interior_angle(1,AD,OC)
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,DAB)
Problem: As shown in the diagram, BA=2, ∠ACB=30°, the center of ⊙O is O. Find the length of line AO.
问题：如图所示，BA=2，∠ACB=30°，O是圆O的圆心。求直线AO的长度。
Construction (CDL):
Shape(CD,DO,OA,OAC)
Shape(AO,OD,DA)
Shape(AD,DB,BA)
Shape(AB,OBA)
Shape(DC,CB,BD)
Shape(BC,OCB)
Collinear(ODB)
Collinear(ADC)
Cocircular(O,ACB)
Conditions (CDL):
Equal(LengthOfLine(BA),2)
Equal(MeasureOfAngle(ACB),30)
IsCentreOfCircle(O,O)
Goal: Value(LengthOfLine(AO))
Answer: 2
Theorem sequence:
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_external(1,OBA,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,OBA)
isosceles_triangle_property_angle_equal(1,OBA)
triangle_property_angle_sum(1,AOB)
sine_theorem(1,AOB)
Problem: As shown in the diagram, DA=CD, ∠BDA=20°, O is the center of ⊙O. Find the measure of ∠DBC.
问题：如图所示，DA=CD，∠BDA=20°，O是⊙O的圆心。求∠DBC的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AD,ODA)
Shape(AB,BE,EA)
Shape(AE,ED,DA)
Shape(DE,EC,CD)
Shape(DC,OCD)
Shape(EB,BO,OC,CE)
Shape(CO,OB,OBC)
Collinear(AEC)
Collinear(DEB)
Collinear(BOC)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(LengthOfLine(DA),LengthOfLine(CD))
Equal(MeasureOfAngle(BDA),20)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DBC))
Answer: 35
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,C)
arc_property_circumference_angle_external(1,OAB,D)
arc_property_circumference_angle_external(1,ODA,B)
arc_property_circumference_angle_external(1,OCD,B)
congruent_arc_judgment_chord_equal(1,ODA,OCD)
congruent_arc_property_measure_equal(1,ODA,OCD)
angle_addition(1,ABD,DBC)
diameter_of_circle_judgment_pass_centre(1,BOC,O)
diameter_of_circle_property_right_angle(1,CAB,O)
triangle_property_angle_sum(1,ABC)
Problem: As shown in the diagram, ∠BCO=30°, the center of circle O is O. Find the measure of ∠CAB.
问题：如图所示，∠BCO=30°，⊙O的圆心为O。求∠CAB的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AB,BO,OC,CA)
Shape(AC,OCA)
Shape(OB,BC,CO)
Shape(CB,OBC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCO),30)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 60
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, ∠DOC=84°, O is the center of ⊙O. Find the measure of ∠OCD.
问题：如图所示，∠DOC=84°，⊙O的圆心为O。求∠OCD的大小。
Construction (CDL):
Shape(CO,OD,ODC)
Shape(DO,OC,CD)
Shape(DC,OCD)
Cocircular(O,CD)
Conditions (CDL):
Equal(MeasureOfAngle(DOC),84)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(OCD))
Answer: 48
Theorem sequence:
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,DOC)
Problem: As shown in the diagram, ∠CAB=25°, the measure of arc OBC is equal to the measure of ⌒ODB, the center of circle O is O. Find the measure of ∠BOD.
问题：如图所示，∠CAB=25°，弧OBC的角度与弧ODB的角度相等，圆O的圆心为O。求∠BOD的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AO,OC)
Shape(CO,OB,OBC)
Shape(BO,OD,ODB)
Shape(DO,OA,OAD)
Collinear(AOB)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),25)
Equal(MeasureOfArc(OBC),MeasureOfArc(ODB))
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOD))
Answer: 50
Theorem sequence:
arc_property_center_angle(1,ODB,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠CDA=52°, O is the center of ⊙O, BA and CD are opposite sides of the ▱ ABCD. Find the measure of ∠BEA.
问题：如图所示，∠CDA=52°，O是圆O的圆心，AD和BC是▱ABCD的一组对边。求∠BEA的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AB,BO,OC,CF,FA)
Shape(AF,FD,DA)
Shape(AD,ODA)
Shape(DF,FE,OEF)
Shape(FC,CE,EF)
Shape(EC,CO,OB,OBE)
Collinear(AFE)
Collinear(DFC)
Collinear(BOCE)
Cocircular(O,ABED)
Conditions (CDL):
Equal(MeasureOfAngle(CDA),52)
IsCentreOfCircle(O,O)
Parallelogram(ABCD)
Goal: Value(MeasureOfAngle(BEA))
Answer: 38
Theorem sequence:
parallelogram_property_opposite_angle_equal(1,BCDA)
diameter_of_circle_judgment_pass_centre(1,BOE,O)
diameter_of_circle_property_right_angle(1,EAB,O)
triangle_property_angle_sum(1,ABE)
Problem: As shown in the diagram, ∠COB=70°, the center of circle O is O. Find the measure of ∠CAB.
问题：如图所示，∠COB=70°，O是圆O的圆心。求∠CAB的大小。
Construction (CDL):
Shape(BO,OA,OAB)
Shape(AO,OC,CA)
Shape(AC,OCA)
Shape(OB,BC,CO)
Shape(CB,OBC)
Collinear(AOB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(COB),70)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 35
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, AB=5, CA=4, ∠ODA=90°, O is the center of ⊙O, AB is the diameter of ⊙O. Find the length of line DO.
问题：如图所示，AB=5，CA=4，∠ODA=90°，O是⊙O的圆心，AB是圆O的直径。求直线DO的长度。
Construction (CDL):
Shape(AD,DC,OCA)
Shape(DA,AO,OD)
Shape(CD,DO,OB,BC)
Shape(CB,OBC)
Shape(BO,OA,OAB)
Collinear(ADC)
Collinear(AOB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(LengthOfLine(AB),5)
Equal(LengthOfLine(CA),4)
Equal(MeasureOfAngle(ODA),90)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(LengthOfLine(DO))
Answer: 3/2
Theorem sequence:
diameter_of_circle_property_length_equal(1,AB,O)
diameter_of_circle_property_right_angle(1,BCA,O)
right_triangle_judgment_angle(1,BCA)
right_triangle_property_pythagorean(1,BCA)
similar_triangle_judgment_aa(1,ODA,BCA)
circle_property_length_of_radius_and_diameter(1,O)
radius_of_circle_property_length_equal(1,OA,O)
similar_triangle_property_line_ratio(1,DAO,CAB)
similar_triangle_property_line_ratio(1,AOD,ABC)
Problem: As shown in the diagram, ∠ACE=70°, the center of circle O is O. Find the measure of ∠AOE.
问题：如图所示，∠ACE=70°，O是圆O的圆心。求∠AOE的大小。
Construction (CDL):
Shape(AO,OE,OEA)
Shape(EO,OD,OE)
Shape(ED,DB,OBE)
Shape(DO,OA,AC,CD)
Shape(BD,DC,CB)
Shape(CA,OAC)
Shape(BC,OCB)
Collinear(AODB)
Collinear(EDC)
Cocircular(O,ACBE)
Conditions (CDL):
Equal(MeasureOfAngle(ACE),70)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AOE))
Answer: 140
Theorem sequence:
arc_property_center_angle(1,OEA,O)
arc_property_circumference_angle_external(1,OEA,C)
Problem: As shown in the diagram, ∠ABC=25°, O is the center of ⊙O, the diameter of ⊙O is BC. Find the measure of ∠BDA.
问题：如图所示，∠ABC=25°，O是圆O的圆心，圆O的直径为BC。求∠BDA的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(BA,OAB)
Shape(CA,AE,EC)
Shape(CE,ED,ODC)
Shape(EO,OB,BD,DE)
Shape(AB,BO,OE,EA)
Shape(DB,OBD)
Collinear(AED)
Collinear(CEOB)
Cocircular(O,ABDC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),25)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BC,O)
Goal: Value(MeasureOfAngle(BDA))
Answer: 65
Theorem sequence:
diameter_of_circle_property_right_angle(1,CAB,O)
triangle_property_angle_sum(1,CAB)
arc_property_circumference_angle_external(1,OAB,D)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠CAB=35°, O is the center of ⊙O, AB is the diameter of circle O. Find the measure of ∠ADC.
问题：如图所示，∠CAB=35°，O是圆O的圆心，AB是圆O的直径。求∠ADC的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AO,OE,EC)
Shape(CE,EB,BC)
Shape(CB,OBC)
Shape(EO,OA,AD,DE)
Shape(DA,OAD)
Shape(BE,ED,ODB)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),35)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ADC))
Answer: 55
Theorem sequence:
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, ∠AOB=60°, O is the center of circle O. Find the measure of ∠ACB.
问题：如图所示，∠AOB=60°，O是圆O的圆心。求∠ACB的大小。
Construction (CDL):
Shape(CD,DO,OA,OAC)
Shape(AO,OD,DA)
Shape(AD,DB,OBA)
Shape(DC,CB,BD)
Shape(BC,OCB)
Collinear(ODB)
Collinear(ADC)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(AOB),60)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ACB))
Answer: 30
Theorem sequence:
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_external(1,OBA,C)
Problem: As shown in the diagram, ∠AOB=50°, ∠BFO=90°, the center of ⊙O is O. Find the measure of ∠CDA.
问题：如图所示，∠AOB=50°，∠BFO=90°，O是⊙O的圆心。求∠CDA的大小。
Construction (CDL):
Shape(CE,EA,OAC)
Shape(AE,EF,FA)
Shape(AF,FB,BA)
Shape(AB,OBA)
Shape(CD,DE,EC)
Shape(BF,FO,OB)
Shape(DC,OCD)
Shape(BO,OF,FE,ED,ODB)
Collinear(CEFB)
Collinear(AED)
Collinear(AFO)
Cocircular(O,ACDB)
Conditions (CDL):
Equal(MeasureOfAngle(AOB),50)
Equal(MeasureOfAngle(BFO),90)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CDA))
Answer: 25
Theorem sequence:
arc_property_center_angle(1,OBA,O)
circle_property_chord_perpendicular_bisect_arc(1,OBC,OFA)
congruent_arc_judgment_length_equal(1,OBA,OAC)
congruent_arc_property_measure_equal(1,OBA,OAC)
arc_property_circumference_angle_external(1,OAC,D)
Problem: As shown in the diagram, ∠BED=90°, ∠COB=70°, O is the center of circle O. Find the measure of ∠DBA.
问题：如图所示，∠BED=90°，∠COB=70°，圆O的圆心为O。求∠DBA的大小。
Construction (CDL):
Shape(AO,OC,OCA)
Shape(CO,OE,EC)
Shape(CE,EB,BC)
Shape(CB,OBC)
Shape(DE,EO,OA,OAD)
Shape(BE,ED,DB)
Shape(BD,ODB)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(BED),90)
Equal(MeasureOfAngle(COB),70)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DBA))
Answer: 55
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,D)
triangle_property_angle_sum(1,BED)
Problem: As shown in the diagram, CB=1, ∠BAC=30°, O is the center of circle O. Find the length of line BO.
问题：如图所示，CB=1，∠BAC=30°，O是⊙O的圆心。求直线BO的长度。
Construction (CDL):
Shape(AD,DO,OB,OBA)
Shape(BO,OD,DB)
Shape(BD,DC,CB)
Shape(BC,OCB)
Shape(DA,AC,CD)
Shape(CA,OAC)
Collinear(ADB)
Collinear(ODC)
Cocircular(O,ACB)
Conditions (CDL):
Equal(LengthOfLine(CB),1)
Equal(MeasureOfAngle(BAC),30)
IsCentreOfCircle(O,O)
Goal: Value(LengthOfLine(BO))
Answer: 1
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,BOC)
sine_theorem(1,BOC)
Problem: As shown in the diagram, ∠COD=50°, ∠ODC=90°, O is the center of circle O. Find the measure of ∠CAB.
问题：如图所示，∠COD=50°，∠ODC=90°，O是⊙O的圆心。求∠CAB的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AB,BO,OC,CA)
Shape(AC,OCA)
Shape(OB,BD,DO)
Shape(OD,DC,CO)
Shape(CD,DB,OBC)
Collinear(BDC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(COD),50)
Equal(MeasureOfAngle(ODC),90)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 50
Theorem sequence:
triangle_property_angle_sum(1,ODC)
triangle_property_angle_sum(1,OBC)
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)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠BAC=30°, ∠DAB=45°, the center of ⊙O is O, the diameter of ⊙O is AB. Find the measure of ∠BEC.
问题：如图所示，∠BAC=30°，∠DAB=45°，O是圆O的圆心，AB是圆O的直径。求∠BEC的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(CA,OAC)
Shape(AC,CE,EO,OA)
Shape(AO,OE,ED,DA)
Shape(EC,CB,BE)
Shape(EB,BD,DE)
Shape(DB,OBD)
Shape(BC,OCB)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),30)
Equal(MeasureOfAngle(DAB),45)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BEC))
Answer: 75
Theorem sequence:
diameter_of_circle_property_right_angle(1,ACB,O)
triangle_property_angle_sum(1,ECB)
triangle_property_angle_sum(1,ACB)
arc_property_circumference_angle_external(1,OBD,A)
arc_property_circumference_angle_external(1,OBD,C)
Problem: As shown in the diagram, ∠ABC=71°, ∠CAB=53°, the center of ⊙O is O. Find the measure of ∠BDA.
问题：如图所示，∠ABC=71°，∠CAB=53°，⊙O的圆心为O。求∠BDA的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AD,ODA)
Shape(AE,ED,DA)
Shape(AB,BE,EA)
Shape(EB,BC,CE)
Shape(DE,EC,OCD)
Shape(CB,OBC)
Collinear(AEC)
Collinear(BED)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),71)
Equal(MeasureOfAngle(CAB),53)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BDA))
Answer: 56
Theorem sequence:
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, ∠CAB=20°, O is the center of circle O. Find the measure of ∠COB.
问题：如图所示，∠CAB=20°，O是⊙O的圆心。求∠COB的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AO,OC)
Shape(CO,OB,OBC)
Shape(BO,OA,OAB)
Collinear(AOB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),20)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COB))
Answer: 40
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠ABO=35°, O is the center of circle O. Find the measure of ∠BCA.
问题：如图所示，∠ABO=35°，⊙O的圆心为O。求∠BCA的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AO,OB,BC)
Shape(CB,OBC)
Shape(OA,AB,BO)
Shape(BA,OAB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(ABO),35)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 55
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, ∠EAC=40°, ∠EBD=30°, O is the center of circle O. Find the measure of ∠AED.
问题：如图所示，∠EAC=40°，∠EBD=30°，⊙O的圆心为O。求∠AED的大小。
Construction (CDL):
Shape(CA,OAC)
Shape(EA,AC,CE)
Shape(AE,ED,ODA)
Shape(DE,EB,BD)
Shape(DB,OBD)
Shape(BE,EC,OCB)
Collinear(CED)
Collinear(AEB)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(EAC),40)
Equal(MeasureOfAngle(EBD),30)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AED))
Answer: 70
Theorem sequence:
arc_property_circumference_angle_external(1,OCB,A)
arc_property_circumference_angle_external(1,OCB,D)
triangle_property_angle_sum(1,DEB)
adjacent_complementary_angle(1,AED,DEB)
Problem: As shown in the diagram, BO=8, CO=6, ∠COB=90°, the center of ⊙A is A, the diameter of circle A is BC. Find the length of line CA.
问题：如图所示，BO=8，CO=6，∠COB=90°，A是圆A的圆心，BC是圆A的直径。求直线CA的长度。
Construction (CDL):
Shape(CB,ABC)
Shape(CO,OB,BC)
Shape(OC,ACO)
Shape(BO,AOB)
Collinear(CAB)
Cocircular(A,COB)
Conditions (CDL):
Equal(LengthOfLine(BO),8)
Equal(LengthOfLine(CO),6)
Equal(MeasureOfAngle(COB),90)
IsCentreOfCircle(A,A)
IsDiameterOfCircle(BC,A)
Goal: Value(LengthOfLine(CA))
Answer: 5
Theorem sequence:
right_triangle_judgment_angle(1,COB)
right_triangle_property_pythagorean(1,COB)
radius_of_circle_property_length_equal(1,AC,A)
radius_of_circle_property_length_equal(1,AB,A)
line_addition(1,CA,AB)
Problem: As shown in the diagram, ∠AOC=110°, the center of ⊙O is O, AB is the diameter of circle O. Find the measure of ∠EDB.
问题：如图所示，∠AOC=110°，圆O的圆心为O，AB是圆O的直径。求∠EDB的大小。
Construction (CDL):
Shape(BD,ODB)
Shape(BE,ED,DB)
Shape(DE,EO,OA,OAD)
Shape(AO,OC,OCA)
Shape(EC,CO,OE)
Shape(CE,EB,OBC)
Collinear(BEOA)
Collinear(DEC)
Cocircular(O,DBCA)
Conditions (CDL):
Equal(MeasureOfAngle(AOC),110)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(EDB))
Answer: 35
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, BC=2, triangle ABC is an equilateral triangle, O is the center of ⊙O, the diameter of circle O is BC. Find the length of line CD.
问题：如图所示，BC=2，△ABC为等边△，⊙O的圆心为O，BC是圆O的直径。求直线CD的长度。
Construction (CDL):
Shape(OED,EA,AD)
Shape(DC,CE,OED)
Shape(EC,OCE)
Shape(CD,DB,BO,OC)
Shape(BD,ODB)
Shape(CO,OB,OBC)
Collinear(BOC)
Collinear(ADB)
Collinear(AEC)
Cocircular(O,DBCE)
Conditions (CDL):
Equal(LengthOfLine(BC),2)
EquilateralTriangle(ABC)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BC,O)
Goal: Value(LengthOfLine(CD))
Answer: sqrt(3)
Theorem sequence:
triangle_property_angle_sum(1,ABC)
diameter_of_circle_property_right_angle(1,CDB,O)
sine_theorem(1,CDB)
sine_theorem(1,ABC)
sine_theorem(1,BCA)
Problem: As shown in the diagram, ∠CBD=130°, O is the center of ⊙O. Find the measure of ∠AOC.
问题：如图所示，∠CBD=130°，⊙O的圆心为O。求∠AOC的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(AE,EC,CA)
Shape(CE,EB,BC)
Shape(CB,OBC)
Shape(BE,EO,OA,OAB)
Shape(BD,DC,OBC)
Shape(EA,AO,OE)
Collinear(AEBD)
Collinear(CEO)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CBD),130)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AOC))
Answer: 100
Theorem sequence:
adjacent_complementary_angle(1,ABC,CBD)
arc_property_circumference_angle_external(1,OCA,B)
arc_property_center_angle(1,OCA,O)
Problem: As shown in the diagram, ∠CBA=25°, O is the center of circle O, AB is the diameter of ⊙O. Find the measure of ∠BDO.
问题：如图所示，∠CBA=25°，O是⊙O的圆心，⊙O的直径为AB。求∠BDO的大小。
Construction (CDL):
Shape(AO,OD,ODA)
Shape(DO,OB,BD)
Shape(DB,OBD)
Shape(BO,OC,CB)
Shape(CO,OA,AC)
Shape(CA,OAC)
Shape(BC,OCB)
Collinear(COD)
Collinear(AOB)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),25)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BDO))
Answer: 65
Theorem sequence:
diameter_of_circle_property_right_angle(1,ACB,O)
triangle_property_angle_sum(1,CBA)
arc_property_circumference_angle_external(1,OCB,A)
arc_property_circumference_angle_external(1,OCB,D)
Problem: As shown in the diagram, ∠OBC=50°, the center of circle O is O, the diameter of ⊙O is AB. Find the measure of ∠CAO.
问题：如图所示，∠OBC=50°，O是圆O的圆心，AB是⊙O的直径。求∠CAO的大小。
Construction (CDL):
Shape(BO,OA,OAB)
Shape(AO,OC,CA)
Shape(AC,OCA)
Shape(CO,OB,BC)
Shape(CB,OBC)
Collinear(AOB)
Cocircular(O,BCA)
Conditions (CDL):
Equal(MeasureOfAngle(OBC),50)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CAO))
Answer: 40
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,CAB)
Problem: As shown in the diagram, ∠ABD=58°, the center of circle O is O, the diameter of circle O is AB. Find the measure of ∠DCB.
问题：如图所示，∠ABD=58°，圆O的圆心为O，⊙O的直径为AB。求∠DCB的大小。
Construction (CDL):
Shape(CD,ODC)
Shape(CE,ED,DC)
Shape(DE,EB,BD)
Shape(DB,OBD)
Shape(EA,AB,BE)
Shape(AE,EC,OCA)
Shape(BA,OAB)
Collinear(CEB)
Collinear(DEA)
Collinear(AOB)
Cocircular(O,CABD)
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,BDA)
arc_property_circumference_angle_external(1,OBD,C)
arc_property_circumference_angle_external(1,OBD,A)
Problem: As shown in the diagram, ∠DOB=130°, the center of ⊙O is O. Find the measure of ∠BCD.
问题：如图所示，∠DOB=130°，⊙O的圆心为O。求∠BCD的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AD,ODA)
Shape(AB,BO,OD,DA)
Shape(BC,CD,DO,OB)
Shape(CB,OBC)
Shape(DC,OCD)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(DOB),130)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCD))
Answer: 115
Theorem sequence:
arc_property_center_angle(1,OBD,O)
arc_property_circumference_angle_internal(1,OBD,C)
Problem: As shown in the diagram, ∠DBC=18°, the center of circle O is O, the diameter of ⊙O is BD. Find the measure of ∠CAB.
问题：如图所示，∠DBC=18°，O是圆O的圆心，圆O的直径为BD。求∠CAB的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AE,ED,ODA)
Shape(DE,EC,OCD)
Shape(EO,OC,CE)
Shape(CO,OB,BC)
Shape(AB,BE,EA)
Shape(CB,OBC)
Collinear(BOED)
Collinear(AEC)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(DBC),18)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BD,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 72
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,COB)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠COA=70°, O is the center of ⊙O, AB∥CD. Find the measure of ∠BAD.
问题：如图所示，∠COA=70°，圆O的圆心为O，AB∥CD。求∠BAD的大小。
Construction (CDL):
Shape(AB,OBA)
Shape(BA,AD,ODB)
Shape(CO,OA,OAC)
Shape(DA,AO,OC,CD)
Shape(DC,OCD)
Cocircular(O,ACDB)
Conditions (CDL):
Equal(MeasureOfAngle(COA),70)
IsCentreOfCircle(O,O)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(BAD))
Answer: 35
Theorem sequence:
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,D)
parallel_property_alternate_interior_angle(1,AB,CD)
Problem: As shown in the diagram, ∠BCO=35°, the center of circle O is O. Find the measure of ∠CAB.
问题：如图所示，∠BCO=35°，O是圆O的圆心。求∠CAB的大小。
Construction (CDL):
Shape(BO,OA,OAB)
Shape(OD,DA,AO)
Shape(OB,BD,DO)
Shape(DB,BC,CD)
Shape(AD,DC,CA)
Shape(AC,OCA)
Shape(CB,OBC)
Collinear(BDA)
Collinear(ODC)
Cocircular(O,BCA)
Conditions (CDL):
Equal(MeasureOfAngle(BCO),35)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 55
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, ∠CDA=25°, the center of circle O is O, BF is perpendicular to OF. Find the measure of ∠AOB.
问题：如图所示，∠CDA=25°，O是圆O的圆心，BF垂直于OF。求∠AOB的大小。
Construction (CDL):
Shape(CE,EA,OAC)
Shape(AE,EF,FA)
Shape(AF,FB,OBA)
Shape(BF,FO,OB)
Shape(EC,CD,DE)
Shape(DC,OCD)
Shape(BO,OF,FE,ED,ODB)
Collinear(CEFB)
Collinear(AFO)
Collinear(AED)
Cocircular(O,ACDB)
Conditions (CDL):
Equal(MeasureOfAngle(CDA),25)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(BF,OF)
Goal: Value(MeasureOfAngle(AOB))
Answer: 50
Theorem sequence:
circle_property_chord_perpendicular_bisect_arc(1,OBC,OFA)
congruent_arc_judgment_length_equal(1,OAC,OBA)
congruent_arc_property_measure_equal(1,OAC,OBA)
arc_property_circumference_angle_external(1,OAC,D)
arc_property_center_angle(1,OBA,O)
Problem: As shown in the diagram, ∠CAB=60°, the center of circle O is O, AB is the diameter of ⊙O. Find the measure of ∠ADC.
问题：如图所示，∠CAB=60°，⊙O的圆心为O，AB是圆O的直径。求∠ADC的大小。
Construction (CDL):
Shape(EA,AO,OB,BE)
Shape(DE,EB,OBD)
Shape(CD,ODC)
Shape(CE,ED,DC)
Shape(CA,AE,EC)
Shape(AC,OCA)
Collinear(CEB)
Collinear(AED)
Collinear(AOB)
Cocircular(O,BDCA)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),60)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ADC))
Answer: 30
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,ABC)
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_external(1,OCA,D)