SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠BCD=35°, the center of ⊙O is O. Find the measure of ∠DOA.
问题：如图所示，∠BCD=35°，O是⊙O的圆心。求∠DOA的大小。
Construction (CDL):
Shape(AO,OC,OCA)
Shape(AD,DO,OA)
Shape(OD,DC,CO)
Shape(BD,DA,OAB)
Shape(BC,CD,DB)
Shape(CB,OBC)
Collinear(ODB)
Collinear(ADC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),35)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DOA))
Answer: 70
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,C)
arc_property_center_angle(1,OAB,O)
Problem: As shown in the diagram, ∠ABD=50°, the center of ⊙O is O. Find the measure of ∠BCD.
问题：如图所示，∠ABD=50°，圆O的圆心为O。求∠BCD的大小。
Construction (CDL):
Shape(OA,AD,DO)
Shape(OD,DC,CO)
Shape(DA,AB,BD)
Shape(CD,DB,BC)
Shape(BA,OAB)
Shape(CB,OBC)
Shape(AO,OC,OCA)
Collinear(ODB)
Collinear(ADC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),50)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCD))
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, ∠ABD=20°, ∠BCA=90°, BE bisects ∠ABC, DC is parallel to AB. Find the measure of ∠ACD.
问题：如图所示，∠ABD=20°，∠BCA=90°，BE是∠ABC的角平分线，DC平行于AB。求∠ACD的大小。
Construction (CDL):
Shape(DE,EC,CD)
Shape(CE,EB,BC)
Shape(AB,BE,EA)
Collinear(DEB)
Collinear(AEC)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),20)
Equal(MeasureOfAngle(BCA),90)
IsBisectorOfAngle(BE,ABC)
ParallelBetweenLine(DC,AB)
Goal: Value(MeasureOfAngle(ACD))
Answer: 50
Theorem sequence:
triangle_property_angle_sum(1,CEB)
triangle_property_angle_sum(1,ABE)
adjacent_complementary_angle(1,CEB,BEA)
parallel_property_alternate_interior_angle(2,DC,AB)
Problem: As shown in the diagram, ∠AEC=28°, ∠CGO=90°, O is the center of ⊙O. Find the measure of ∠DBA.
问题：如图所示，∠AEC=28°，∠CGO=90°，O是圆O的圆心。求∠DBA的大小。
Construction (CDL):
Shape(AF,FC,OCA)
Shape(FA,AG,GF)
Shape(CF,FE,EC)
Shape(CE,OEC)
Shape(EF,FG,GB,OBE)
Shape(DG,GA,OAD)
Shape(BG,GD,DB)
Shape(BD,ODB)
Collinear(AGOB)
Collinear(CFGD)
Collinear(AFE)
Cocircular(O,ECADB)
Conditions (CDL):
Equal(MeasureOfAngle(AEC),28)
Equal(MeasureOfAngle(CGO),90)
IsCentreOfCircle(O,O)
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,OCA,E)
arc_property_circumference_angle_external(1,OAD,B)
Problem: As shown in the diagram, ∠EDC=35°, O is the center of circle O. Find the measure of ∠ECA.
问题：如图所示，∠EDC=35°，圆O的圆心为O。求∠ECA的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AB,BO,OA)
Shape(AO,OE,EA)
Shape(AE,EC,CA)
Shape(AC,OCA)
Shape(DE,EB,OBD)
Shape(CE,ED,DC)
Shape(CD,ODC)
Collinear(BOEC)
Collinear(AED)
Cocircular(O,ABDC)
Conditions (CDL):
Equal(MeasureOfAngle(EDC),35)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ECA))
Answer: 55
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,D)
arc_property_center_angle(1,OCA,O)
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,AOC)
Problem: As shown in the diagram, ∠CDB=40°, O is the center of ⊙O, the diameter of ⊙O is AB. Find the measure of ∠ABC.
问题：如图所示，∠CDB=40°，O是⊙O的圆心，AB是圆O的直径。求∠ABC的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AE,EC)
Shape(CE,EB,BC)
Shape(CB,OBC)
Shape(DE,EA,OAD)
Shape(BE,ED,DB)
Shape(BD,ODB)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(CDB),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ABC))
Answer: 50
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_external(1,OBC,D)
triangle_property_angle_sum(1,CAB)
Problem: As shown in the diagram, ∠ABC=70°, ∠BCA=30°, O is the center of ⊙O, D bisects arc OCB. Find the measure of ∠DBC.
问题：如图所示，∠ABC=70°，∠BCA=30°，O是⊙O的圆心，D平分⌒OCB。求∠DBC的大小。
Construction (CDL):
Shape(AE,ED,ODA)
Shape(DE,EC,CD)
Shape(DC,OCD)
Shape(BA,OAB)
Shape(AB,BE,EA)
Shape(CE,EB,BC)
Shape(CB,OBC)
Collinear(AEC)
Collinear(BED)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),70)
Equal(MeasureOfAngle(BCA),30)
IsCentreOfCircle(O,O)
IsMidpointOfArc(D,OCB)
Goal: Value(MeasureOfAngle(DBC))
Answer: 50
Theorem sequence:
triangle_property_angle_sum(1,CDB)
triangle_property_angle_sum(1,ABC)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_external(1,OBC,D)
congruent_arc_judgment_length_equal(1,ODB,OCD)
congruent_arc_property_chord_equal(1,ODB,OCD)
isosceles_triangle_judgment_line_equal(1,DBC)
isosceles_triangle_property_angle_equal(1,DBC)
Problem: As shown in the diagram, AB=8, ∠CBA=30°, the center of circle O is O. Find the length of line AC.
问题：如图所示，AB=8，∠CBA=30°，O是圆O的圆心。求直线AC的长度。
Construction (CDL):
Shape(AB,OBA)
Shape(CA,OAC)
Shape(AC,CB,BA)
Shape(BC,OCB)
Collinear(AOB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(MeasureOfAngle(CBA),30)
IsCentreOfCircle(O,O)
Goal: Value(LengthOfLine(AC))
Answer: 4
Theorem sequence:
diameter_of_circle_judgment_pass_centre(1,AOB,O)
diameter_of_circle_property_right_angle(1,ACB,O)
sine_theorem(1,ACB)
Problem: As shown in the diagram, ∠CBA=25°, ∠CDE=58°, AB∥CF, CF is parallel to DE, ∠ACB is the right angle of the right triangle ACB, EC and CD are the legs of the right triangle ECD. Find the measure of ∠BCE.
问题：如图所示，∠CBA=25°，∠CDE=58°，AB∥CF，CF平行于DE，∠ACB是直角三角形ACB的直角，EC和CD是直角三角形ECD的两个直角边。求∠BCE的大小。
Construction (CDL):
Shape(AC,CB,BA)
Shape(BC,CF,FB)
Shape(FC,CE,EF)
Shape(CD,DE,EC)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),25)
Equal(MeasureOfAngle(CDE),58)
ParallelBetweenLine(AB,CF)
ParallelBetweenLine(CF,DE)
RightTriangle(ACB)
RightTriangle(ECD)
Goal: Value(MeasureOfAngle(BCE))
Answer: 57
Theorem sequence:
triangle_property_angle_sum(1,CDE)
parallel_property_alternate_interior_angle(1,CF,DE)
parallel_property_alternate_interior_angle(2,AB,CF)
angle_addition(1,BCF,FCE)
Problem: As shown in the diagram, ∠OBA=50°, the center of circle O is O. Find the measure of ∠ACB.
问题：如图所示，∠OBA=50°，圆O的圆心为O。求∠ACB的大小。
Construction (CDL):
Shape(CO,OA,OAC)
Shape(CD,DO,OC)
Shape(OD,DA,AO)
Shape(AO,OB,BA)
Shape(DC,CB,BD)
Shape(BC,OCB)
Shape(AB,OBA)
Collinear(CDA)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(OBA),50)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ACB))
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)
triangle_property_angle_sum(1,AOB)
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_external(1,OBA,C)
Problem: As shown in the diagram, ∠ACO=30°, O is the center of ⊙O. Find the measure of ∠CBA.
问题：如图所示，∠ACO=30°，O是圆O的圆心。求∠CBA的大小。
Construction (CDL):
Shape(CA,OAC)
Shape(AB,OBA)
Shape(BC,OCB)
Shape(CO,OA,AC)
Shape(CB,BO,OC)
Shape(OB,BA,AO)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(ACO),30)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CBA))
Answer: 60
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,OAC)
isosceles_triangle_property_angle_equal(1,OAC)
triangle_property_angle_sum(1,COA)
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,B)
Problem: As shown in the diagram, ∠CBA=25°, O is the center of ⊙O, the diameter of circle O is AD. Find the measure of ∠DAC.
问题：如图所示，∠CBA=25°，圆O的圆心为O，AD是圆O的直径。求∠DAC的大小。
Construction (CDL):
Shape(DC,OCD)
Shape(CD,OAC)
Shape(AB,OBA)
Shape(DE,EB,ODB)
Shape(CD,DE,EC)
Shape(EB,BA,AE)
Shape(CE,EA,AC)
Collinear(DOEA)
Collinear(CEB)
Cocircular(O,DBAC)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),25)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AD,O)
Goal: Value(MeasureOfAngle(DAC))
Answer: 65
Theorem sequence:
diameter_of_circle_property_right_angle(1,ACD,O)
arc_property_circumference_angle_external(1,OAC,D)
arc_property_circumference_angle_external(1,OAC,B)
triangle_property_angle_sum(1,CDA)
Problem: As shown in the diagram, ∠AOC=130°, O is the center of ⊙O. Find the measure of ∠EDB.
问题：如图所示，∠AOC=130°，O是⊙O的圆心。求∠EDB的大小。
Construction (CDL):
Shape(DB,BE,ED)
Shape(BD,ODB)
Shape(DE,EA,OAD)
Shape(CE,EB,OBC)
Shape(OE,EC,CO)
Shape(AO,OC,OCA)
Collinear(BEOA)
Collinear(DEC)
Cocircular(O,DBCA)
Conditions (CDL):
Equal(MeasureOfAngle(AOC),130)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(EDB))
Answer: 25
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, ∠DEA=90°, ∠EBC=60°, the center of ⊙O is O. Find the measure of ∠EAD.
问题：如图所示，∠DEA=90°，∠EBC=60°，圆O的圆心为O。求∠EAD的大小。
Construction (CDL):
Shape(AE,EC,OCA)
Shape(DA,OAD)
Shape(CB,OBC)
Shape(AD,DE,EA)
Shape(CE,EB,BC)
Shape(BE,ED,ODB)
Collinear(AEB)
Collinear(DEC)
Cocircular(O,CADB)
Conditions (CDL):
Equal(MeasureOfAngle(DEA),90)
Equal(MeasureOfAngle(EBC),60)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(EAD))
Answer: 30
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,D)
arc_property_circumference_angle_external(1,OCA,B)
triangle_property_angle_sum(1,ADE)
Problem: As shown in the diagram, the measure of arc ODA is 56, O is the center of ⊙O, triangle ACB is a right triangle. Find the measure of ∠DCB.
问题：如图所示，⌒ODA的角度为56，O是圆O的圆心，∠ACB是直角△ACB的直角。求∠DCB的大小。
Construction (CDL):
Shape(AO,OD,ODA)
Shape(OA,AC,CO)
Shape(DO,OB,OBD)
Shape(OC,CB,BO)
Shape(BC,CA,OAB)
Collinear(AOB)
Collinear(DOC)
Cocircular(O,BDAC)
Conditions (CDL):
Equal(MeasureOfArc(ODA),56)
IsCentreOfCircle(O,O)
RightTriangle(ACB)
Goal: Value(MeasureOfAngle(DCB))
Answer: 62
Theorem sequence:
arc_property_circumference_angle_external(1,ODA,C)
angle_addition(1,ACD,DCB)
Problem: As shown in the diagram, the measure of ⌒OAB is 88-30, O is the center of circle O. Find the measure of ∠BCA.
问题：如图所示，弧OAB的角度为88-30，O是圆O的圆心。求∠BCA的大小。
Construction (CDL):
Shape(DB,BO,OD)
Shape(CD,DO,OC)
Shape(BD,DA,OAB)
Shape(AD,DC,CA)
Shape(AC,OCA)
Shape(CO,OB,OBC)
Collinear(ADC)
Collinear(BDC)
Cocircular(O,CAB)
Conditions (CDL):
Equal(MeasureOfArc(OAB),88-30)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 29
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠OBD=50°, O is the center of circle O, AO∥BD. Find the measure of ∠OCA.
问题：如图所示，∠OBD=50°，O是圆O的圆心，AO平行于BD。求∠OCA的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AO,OC)
Shape(CO,OD,ODC)
Shape(DO,OB,BD)
Shape(DB,OBD)
Shape(BO,OA,OAB)
Collinear(COB)
Cocircular(O,CABD)
Conditions (CDL):
Equal(MeasureOfAngle(OBD),50)
IsCentreOfCircle(O,O)
ParallelBetweenLine(AO,BD)
Goal: Value(MeasureOfAngle(OCA))
Answer: 25
Theorem sequence:
parallel_property_alternate_interior_angle(2,AO,BD)
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠EAD=50°. Find the sum of the measure of ∠EDC and the measure of ∠BED.
问题：如图所示，∠EAD=50°。求∠EDC的大小与∠BED的大小之和。
Construction (CDL):
Shape(EA,AD,DE)
Shape(BE,ED,DC,CB)
Collinear(BEA)
Collinear(ADC)
Conditions (CDL):
Equal(MeasureOfAngle(EAD),50)
Goal: Value(Add(MeasureOfAngle(EDC),MeasureOfAngle(BED)))
Answer: 230
Theorem sequence:
triangle_property_angle_sum(1,BAC)
quadrilateral_property_angle_sum(1,BEDC)
Problem: As shown in the diagram, ∠CBA=40°, O is the center of ⊙O, AB is parallel to CD. Find the measure of ∠BOD.
问题：如图所示，∠CBA=40°，圆O的圆心为O，AB∥CD。求∠BOD的大小。
Construction (CDL):
Shape(AB,OBA)
Shape(BO,OD,ODB)
Shape(BC,CD,DO,OB)
Shape(DC,OCD)
Shape(CB,BA,OAC)
Cocircular(O,ACDB)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),40)
IsCentreOfCircle(O,O)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(BOD))
Answer: 80
Theorem sequence:
parallel_property_alternate_interior_angle(2,AB,CD)
arc_property_circumference_angle_external(1,ODB,C)
arc_property_center_angle(1,ODB,O)
Problem: As shown in the diagram, ∠CBA=40°, O is the center of circle O, the diameter of ⊙O is DA. Find the measure of ∠DAC.
问题：如图所示，∠CBA=40°，O是⊙O的圆心，DA是⊙O的直径。求∠DAC的大小。
Construction (CDL):
Shape(DC,OCD)
Shape(CD,DE,EC)
Shape(CE,EA,AC)
Shape(CA,OAC)
Shape(AE,EB,BA)
Shape(AB,OBA)
Shape(BE,ED,ODB)
Collinear(DOEA)
Collinear(CEB)
Cocircular(O,CDBA)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(DA,O)
Goal: Value(MeasureOfAngle(DAC))
Answer: 50
Theorem sequence:
arc_property_circumference_angle_external(1,OAC,D)
arc_property_circumference_angle_external(1,OAC,B)
diameter_of_circle_property_right_angle(1,ACD,O)
triangle_property_angle_sum(1,CDA)
Problem: As shown in the diagram, ∠CBA=30°, the center of ⊙O is O, AB is the diameter of circle O. Find the measure of ∠BAC.
问题：如图所示，∠CBA=30°，O是圆O的圆心，圆O的直径为AB。求∠BAC的大小。
Construction (CDL):
Shape(AO,OB,OBA)
Shape(CA,OAC)
Shape(AC,CB,BO,OA)
Shape(BC,OCB)
Collinear(AOB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BAC))
Answer: 60
Theorem sequence:
diameter_of_circle_property_right_angle(1,ACB,O)
triangle_property_angle_sum(1,ACB)
Problem: As shown in the diagram, ∠ABC=70°, O is the center of circle O. Find the measure of ∠AOC.
问题：如图所示，∠ABC=70°，⊙O的圆心为O。求∠AOC的大小。
Construction (CDL):
Shape(AB,BC,CO,OA)
Shape(AO,OC,CA)
Shape(BA,OAB)
Shape(CB,OBC)
Shape(AC,OCA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),70)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AOC))
Answer: 140
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,B)
arc_property_center_angle(1,OCA,O)
Problem: As shown in the diagram, ∠CDA=30°, ∠DAB=90°, ∠DCB=142°. Find the measure of ∠ABC.
问题：如图所示，∠CDA=30°，∠DAB=90°，∠DCB=142°。求∠ABC的大小。
Construction (CDL):
Shape(DA,AE,ED)
Shape(CE,EB,BC)
Shape(DC,CB,BD)
Collinear(DCE)
Collinear(AEB)
Conditions (CDL):
Equal(MeasureOfAngle(CDA),30)
Equal(MeasureOfAngle(DAB),90)
Equal(MeasureOfAngle(DCB),142)
Goal: Value(MeasureOfAngle(ABC))
Answer: 22
Theorem sequence:
triangle_property_angle_sum(1,DAE)
triangle_property_angle_sum(1,CEB)
adjacent_complementary_angle(1,DCB,BCE)
adjacent_complementary_angle(1,AED,DEB)
Problem: As shown in the diagram, ∠CAB=35°, O is the center of circle O, the diameter of circle O is AB. Find the measure of ∠ADC.
问题：如图所示，∠CAB=35°，⊙O的圆心为O，圆O的直径为AB。求∠ADC的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(AE,EC,CA)
Shape(CE,EB,OBC)
Shape(BE,ED,DB)
Shape(BD,ODB)
Shape(DE,EA,AD)
Shape(DA,OAD)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,CADB)
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,ADB,O)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_external(1,OBC,D)
angle_addition(1,ADE,EDB)
Problem: As shown in the diagram, ∠AOB=70°, ∠BGO=90°, O is the center of ⊙O. Find the measure of ∠CDA.
问题：如图所示，∠AOB=70°，∠BGO=90°，⊙O的圆心为O。求∠CDA的大小。
Construction (CDL):
Shape(BO,OD,ODB)
Shape(OE,ED,DO)
Shape(DE,EC,CD)
Shape(DC,OCD)
Shape(CE,EF,FC)
Shape(OG,GF,FE,EO)
Shape(OB,BG,GO)
Shape(AG,GB,OBA)
Shape(FG,GA,AF)
Shape(CF,FA,OAC)
Collinear(DEFA)
Collinear(OGA)
Collinear(OEC)
Collinear(BGFC)
Cocircular(O,DBAC)
Conditions (CDL):
Equal(MeasureOfAngle(AOB),70)
Equal(MeasureOfAngle(BGO),90)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CDA))
Answer: 35
Theorem sequence:
arc_property_center_angle(1,OBA,O)
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_circumference_angle_external(1,OAC,D)
Problem: As shown in the diagram, ∠ADC=35°, the center of circle O is O, BC is the diameter of circle O. Find the measure of ∠CAO.
问题：如图所示，∠ADC=35°，O是圆O的圆心，BC是圆O的直径。求∠CAO的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AB,BO,OA)
Shape(AO,OE,EA)
Shape(AE,EC,CA)
Shape(AC,OCA)
Shape(DE,EB,OBD)
Shape(CE,ED,DC)
Shape(CD,ODC)
Collinear(BOEC)
Collinear(AED)
Cocircular(O,ABDC)
Conditions (CDL):
Equal(MeasureOfAngle(ADC),35)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BC,O)
Goal: Value(MeasureOfAngle(CAO))
Answer: 55
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,D)
arc_property_center_angle(1,OCA,O)
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,AOC)
Problem: As shown in the diagram, ∠AOC=100°, O is the center of circle O. Find the measure of ∠EDB.
问题：如图所示，∠AOC=100°，O是圆O的圆心。求∠EDB的大小。
Construction (CDL):
Shape(BD,ODB)
Shape(DB,BE,ED)
Shape(DE,EA,OAD)
Shape(AO,OC,OCA)
Shape(OE,EC,CO)
Shape(CE,EB,OBC)
Collinear(BEOA)
Collinear(DEC)
Cocircular(O,DBCA)
Conditions (CDL):
Equal(MeasureOfAngle(AOC),100)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(EDB))
Answer: 40
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, ∠CAB=40°, O is the center of ⊙O. Find the measure of ∠BCO.
问题：如图所示，∠CAB=40°，圆O的圆心为O。求∠BCO的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(CB,OBC)
Shape(AC,OCA)
Shape(AB,BO,OC,CA)
Shape(OB,BC,CO)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCO))
Answer: 50
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,OBC)
Problem: As shown in the diagram, ∠EOA=60°, the center of circle O is O. Find the measure of ∠BCD.
问题：如图所示，∠EOA=60°，O是圆O的圆心。求∠BCD的大小。
Construction (CDL):
Shape(AD,DC,OCA)
Shape(CD,DB,BC)
Shape(CB,OBC)
Shape(ED,DA,OAE)
Shape(OD,DE,EO)
Shape(BO,OE,OEB)
Collinear(ADOB)
Collinear(CDE)
Cocircular(O,CAEB)
Conditions (CDL):
Equal(MeasureOfAngle(EOA),60)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCD))
Answer: 60
Theorem sequence:
adjacent_complementary_angle(1,BOE,EOA)
arc_property_center_angle(1,OEB,O)
arc_property_circumference_angle_external(1,OEB,C)
Problem: As shown in the diagram, ∠CBD=36°, O is the center of ⊙O. Find the measure of ∠COA.
问题：如图所示，∠CBD=36°，⊙O的圆心为O。求∠COA的大小。
Construction (CDL):
Shape(AO,OD,DB,OBA)
Shape(OA,AD,DO)
Shape(BD,DC,CB)
Shape(BC,OCB)
Shape(CD,DA,OAC)
Collinear(ADB)
Collinear(ODC)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(CBD),36)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COA))
Answer: 72
Theorem sequence:
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,B)
Problem: As shown in the diagram, ∠CED=20°, ∠DAC=20°. Find the measure of ∠ADF.
问题：如图所示，∠CED=20°，∠DAC=20°。求∠ADF的大小。
Construction (CDL):
Shape(AC,CD,DA)
Shape(DC,CE,ED)
Shape(BD,DE)
Shape(FD,DB)
Shape(AD,DF)
Collinear(ACE)
Collinear(ADB)
Collinear(FDE)
Conditions (CDL):
Equal(MeasureOfAngle(CED),20)
Equal(MeasureOfAngle(DAC),20)
Goal: Value(MeasureOfAngle(ADF))
Answer: 40
Theorem sequence:
triangle_property_angle_sum(1,DAE)
adjacent_complementary_angle(1,EDA,ADF)
Problem: As shown in the diagram, ∠ABC=70°, O is the center of ⊙O. Find the measure of ∠AOC.
问题：如图所示，∠ABC=70°，⊙O的圆心为O。求∠AOC的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AB,BC,CO,OA)
Shape(AO,OC,CA)
Shape(AC,OCA)
Shape(CB,OBC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),70)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AOC))
Answer: 140
Theorem sequence:
arc_property_center_angle(1,OCA,O)
arc_property_circumference_angle_external(1,OCA,B)
Problem: As shown in the diagram, ∠AOB=100°, the center of circle O is O. Find the measure of ∠BCA.
问题：如图所示，∠AOB=100°，O是圆O的圆心。求∠BCA的大小。
Construction (CDL):
Shape(BO,OA,OAB)
Shape(AC,OCA)
Shape(CB,OBC)
Shape(AO,OB,BC,CA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(AOB),100)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 130
Theorem sequence:
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_external(1,OAB,C)
round_arc(1,OAB,OBA)
Problem: As shown in the diagram, ∠ABC=55°, the diameter of ⊙O is AB. Find the measure of ∠CAB.
问题：如图所示，∠ABC=55°，圆O的直径为AB。求∠CAB的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CB,OBC)
Shape(BO,OA,OAB)
Shape(CA,AO,OB,BC)
Collinear(AOB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),55)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 35
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,CAB)
Problem: As shown in the diagram, ∠BDC=64°, the center of circle O is O, CE is perpendicular to AE. Find the measure of ∠ACD.
问题：如图所示，∠BDC=64°，⊙O的圆心为O，CE⊥AE。求∠ACD的大小。
Construction (CDL):
Shape(CA,OAC)
Shape(AE,ED,ODA)
Shape(DB,OBD)
Shape(BE,EC,OCB)
Shape(AC,CE,EA)
Shape(DE,EB,BD)
Collinear(AEB)
Collinear(CED)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(BDC),64)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(CE,AE)
Goal: Value(MeasureOfAngle(ACD))
Answer: 26
Theorem sequence:
vertical_angle(1,CEA,DEB)
triangle_property_angle_sum(1,DEB)
arc_property_circumference_angle_external(1,ODA,C)
arc_property_circumference_angle_external(1,ODA,B)
Problem: As shown in the diagram, AC=6, ∠BDO=90°, O is the center of circle O. Find the length of line OD.
问题：如图所示，AC=6，∠BDO=90°，O是圆O的圆心。求直线OD的长度。
Construction (CDL):
Shape(AC,OCA)
Shape(CD,DB,OBC)
Shape(OB,OA,OAB)
Shape(AO,OD,DC,CA)
Shape(DO,OB,BD)
Collinear(CDB)
Collinear(AOB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(LengthOfLine(AC),6)
Equal(MeasureOfAngle(BDO),90)
IsCentreOfCircle(O,O)
Goal: Value(LengthOfLine(OD))
Answer: 3
Theorem sequence:
diameter_of_circle_judgment_pass_centre(1,AOB,O)
diameter_of_circle_property_right_angle(1,BCA,O)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
parallel_judgment_corresponding_angle(1,DO,CA,B)
midsegment_of_triangle_judgment_parallel(2,DO,BCA)
midsegment_of_triangle_property_length(1,DO,BCA)
Problem: As shown in the diagram, ∠BOA=38°, the center of ⊙O is O, AO is parallel to BC. Find the measure of ∠OAC.
问题：如图所示，∠BOA=38°，圆O的圆心为O，AO平行于BC。求∠OAC的大小。
Construction (CDL):
Shape(OA,AD,DO)
Shape(CD,DB,BC)
Shape(DA,OAB,BD)
Shape(OBC,CB)
Shape(OD,DC,OCA,AO)
Collinear(ADC)
Collinear(ODB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),38)
IsCentreOfCircle(O,O)
ParallelBetweenLine(AO,BC)
Goal: Value(MeasureOfAngle(OAC))
Answer: 19
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
parallel_property_alternate_interior_angle(1,AO,BC)
Problem: As shown in the diagram, ∠ABC=70°, O is the center of ⊙O, AB is the diameter of ⊙O. Find the measure of ∠CDB.
问题：如图所示，∠ABC=70°，⊙O的圆心为O，⊙O的直径为AB。求∠CDB的大小。
Construction (CDL):
Shape(CA,AO,OB,BC)
Shape(OCA,AC)
Shape(OBC,CB)
Shape(OA,OAB,BO)
Shape(BC,CD,DB)
Collinear(AOB)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),70)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CDB))
Answer: 20
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,CAB)
arc_property_circumference_angle_external(1,OBC,D)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠APD=80°, ∠CBP=30°. Find the measure of ∠PCB.
问题：如图所示，∠APD=80°，∠CBP=30°。求∠PCB的大小。
Construction (CDL):
Shape(PC,CB,BP)
Shape(AP,PD,DA)
Shape(CP,PA,OAC)
Shape(OCB,BC)
Shape(PB,OBD,DP)
Shape(ODA,AD)
Collinear(CPD)
Collinear(APB)
Cocircular(P,CBDA)
Conditions (CDL):
Equal(MeasureOfAngle(APD),80)
Equal(MeasureOfAngle(CBP),30)
Goal: Value(MeasureOfAngle(PCB))
Answer: 70
Theorem sequence:
vertical_angle(1,APD,BPC)
triangle_property_angle_sum(1,PCB)