SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠BOA=100°, the center of ⊙O is O. Find the measure of ∠BCA.
问题：如图所示，∠BOA=100°，⊙O的圆心为O。求∠BCA的大小。
Construction (CDL):
Shape(CA,AO,OB,BC)
Shape(BO,OA,OAB)
Shape(AC,OCA)
Shape(CB,OBC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),100)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 50
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠OAB=22°, O is the center of circle O. Find the measure of ∠ACB.
问题：如图所示，∠OAB=22°，O是⊙O的圆心。求∠ACB的大小。
Construction (CDL):
Shape(DA,AO,OB,BD)
Shape(OA,AB,BO)
Shape(BA,AC,CB)
Shape(DB,OBD)
Shape(AD,ODA)
Shape(CA,OAC)
Shape(BC,OCB)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(OAB),22)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ACB))
Answer: 112
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_internal(1,OAB,C)
Problem: As shown in the diagram, ∠EBD=59°, O is the center of ⊙O, the diameter of ⊙O is AB. Find the measure of ∠ECB.
问题：如图所示，∠EBD=59°，⊙O的圆心为O，圆O的直径为AB。求∠ECB的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(AO,OE,ED,DA)
Shape(CE,EO,OA,OAC)
Shape(DB,OBD)
Shape(BC,OCB)
Shape(DE,EB,BD)
Shape(EC,CB,BE)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(EBD),59)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ECB))
Answer: 31
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, ∠BOA=70°, the center of ⊙O is O. Find the measure of ∠BCA.
问题：如图所示，∠BOA=70°，⊙O的圆心为O。求∠BCA的大小。
Construction (CDL):
Shape(CA,AO,OB,BC)
Shape(BO,OA,OAB)
Shape(CB,OBC)
Shape(AC,OCA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),70)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 35
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠COA=160°, the center of circle O is O. Find the measure of ∠CBA.
问题：如图所示，∠COA=160°，O是圆O的圆心。求∠CBA的大小。
Construction (CDL):
Shape(BA,AO,OC,CB)
Shape(CO,OA,OAC)
Shape(BC,OCB)
Shape(AB,OBA)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(COA),160)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CBA))
Answer: 80
Theorem sequence:
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,B)
Problem: As shown in the diagram, ∠BAO=32°, O is the center of circle O, the diameter of circle O is AC, AB∥DC. Find the measure of ∠DOA.
问题：如图所示，∠BAO=32°，⊙O的圆心为O，圆O的直径为AC，AB∥DC。求∠DOA的大小。
Construction (CDL):
Shape(AB,OBA)
Shape(DO,OA,OAD)
Shape(CO,OD,DC)
Shape(CD,ODC)
Shape(BA,AO,OC,OCB)
Collinear(AOC)
Cocircular(O,ADCB)
Conditions (CDL):
Equal(MeasureOfAngle(BAO),32)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AC,O)
ParallelBetweenLine(AB,DC)
Goal: Value(MeasureOfAngle(DOA))
Answer: 64
Theorem sequence:
parallel_property_alternate_interior_angle(1,AB,DC)
arc_property_center_angle(1,OAD,O)
arc_property_circumference_angle_external(1,OAD,C)
Problem: As shown in the diagram, ∠EBA=35°, O is the center of circle O, the diameter of ⊙O is AD. Find the measure of ∠EAC.
问题：如图所示，∠EBA=35°，O是⊙O的圆心，AD是⊙O的直径。求∠EAC的大小。
Construction (CDL):
Shape(CA,OAC)
Shape(CE,EA,AC)
Shape(CD,DO,OE,EC)
Shape(DC,OCD)
Shape(BE,EO,OD,ODB)
Shape(EB,BA,AE)
Shape(AB,OBA)
Collinear(DOEA)
Collinear(CEB)
Cocircular(O,BACD)
Conditions (CDL):
Equal(MeasureOfAngle(EBA),35)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AD,O)
Goal: Value(MeasureOfAngle(EAC))
Answer: 55
Theorem sequence:
arc_property_circumference_angle_external(1,OAC,B)
arc_property_circumference_angle_external(1,OAC,D)
diameter_of_circle_property_right_angle(1,ACD,O)
triangle_property_angle_sum(1,CDA)
Problem: As shown in the diagram, ∠ACB=30°, O is the center of circle O. Find the measure of ∠AOB.
问题：如图所示，∠ACB=30°，⊙O的圆心为O。求∠AOB的大小。
Construction (CDL):
Shape(AO,OB,OBA)
Shape(CA,OAC)
Shape(AC,CB,BO,OA)
Shape(BC,OCB)
Cocircular(O,BAC)
Conditions (CDL):
Equal(MeasureOfAngle(ACB),30)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AOB))
Answer: 60
Theorem sequence:
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_external(1,OBA,C)
Problem: As shown in the diagram, ∠BOA=100°, the center of circle O is O. Find the measure of ∠BCA.
问题：如图所示，∠BOA=100°，⊙O的圆心为O。求∠BCA的大小。
Construction (CDL):
Shape(CA,AO,OB,BC)
Shape(AC,OCA)
Shape(CB,OBC)
Shape(BO,OA,AB)
Shape(BA,OAB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),100)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 50
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, OA=7, ∠FCO=30°, the center of ⊙O is O, AC is the diameter of circle O, F is the midpoint of segment CB, O bisects segment CA. Find the sum of the length of line GO and the length of line FH.
问题：如图所示，OA=7，∠FCO=30°，⊙O的圆心为O，AC是圆O的直径，F是线段CB的中点，O是线段CA的中点。求直线GO的长度与直线FH的长度之和。
Construction (CDL):
Shape(GO,OC,OCG)
Shape(CO,OF,FC)
Shape(CF,FH,OHC)
Shape(HF,FB,OBH)
Shape(FO,OA,AB,BF)
Shape(AO,OG,OGA)
Shape(BA,OAB)
Collinear(GOFH)
Collinear(COA)
Collinear(CFB)
Cocircular(O,GABHC)
Conditions (CDL):
Equal(LengthOfLine(OA),7)
Equal(MeasureOfAngle(FCO),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AC,O)
IsMidpointOfLine(F,CB)
IsMidpointOfLine(O,CA)
Goal: Value(Add(LengthOfLine(GO),LengthOfLine(FH)))
Answer: 21/2
Theorem sequence:
radius_of_circle_property_length_equal(1,OG,O)
radius_of_circle_property_length_equal(1,OH,O)
radius_of_circle_property_length_equal(1,OC,O)
diameter_of_circle_property_right_angle(1,ABC,O)
midsegment_of_triangle_judgment_midpoint(1,OF,CAB)
midsegment_of_triangle_property_length(1,OF,CAB)
line_addition(1,CO,OA)
line_addition(1,OF,FH)
sine_theorem(1,ABC)
Problem: As shown in the diagram, ∠FDA=70°, the center of ⊙O is O, BE is the diameter of ⊙O, ABCD is a ▱. Find the measure of ∠CEF.
问题：如图所示，∠FDA=70°，⊙O的圆心为O，⊙O的直径为BE，ABCD是平行四边形。求∠CEF的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AD,ODA)
Shape(DF,FE,OED)
Shape(FC,CE,EF)
Shape(AB,BO,OC,CF,FA)
Shape(AF,FD,DA)
Shape(EC,CO,OB,OBE)
Collinear(BOCE)
Collinear(DFC)
Collinear(AFE)
Cocircular(O,ABED)
Conditions (CDL):
Equal(MeasureOfAngle(FDA),70)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BE,O)
Parallelogram(ABCD)
Goal: Value(MeasureOfAngle(CEF))
Answer: 20
Theorem sequence:
parallelogram_property_opposite_angle_equal(1,BCDA)
diameter_of_circle_property_right_angle(1,EAB,O)
triangle_property_angle_sum(1,ABE)
Problem: As shown in the diagram, ∠BCA=35°, the center of circle O is O. Find the measure of ∠OAB.
问题：如图所示，∠BCA=35°，O是⊙O的圆心。求∠OAB的大小。
Construction (CDL):
Shape(BO,OA,AB)
Shape(BA,OAB)
Shape(BC,CA,AO,OB)
Shape(CB,OBC)
Shape(AC,OCA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),35)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(OAB))
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)
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
triangle_property_angle_sum(1,OAB)
Problem: As shown in the diagram, ∠CAO=20°, ∠OBC=20°, O is the center of ⊙O. Find the measure of ∠BOA.
问题：如图所示，∠CAO=20°，∠OBC=20°，O是圆O的圆心。求∠BOA的大小。
Construction (CDL):
Shape(CA,AO,OB,BC)
Shape(BO,OA,OAB)
Shape(CB,OBC)
Shape(AC,OCA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAO),20)
Equal(MeasureOfAngle(OBC),20)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 80
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
round_angle(1,AOB,BOA)
quadrilateral_property_angle_sum(1,CAOB)
Problem: As shown in the diagram, ∠BDE=40°, O is the center of ⊙O, the diameter of circle O is AB. Find the measure of ∠CBE.
问题：如图所示，∠BDE=40°，O是⊙O的圆心，⊙O的直径为AB。求∠CBE的大小。
Construction (CDL):
Shape(AO,OE,ED,ODA)
Shape(DE,EB,BD)
Shape(DB,OBD)
Shape(BC,OCB)
Shape(AC,CE,EO,OA)
Shape(EC,CB,BE)
Shape(CA,OAC)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(BDE),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CBE))
Answer: 50
Theorem sequence:
diameter_of_circle_property_right_angle(1,ACB,O)
triangle_property_angle_sum(1,ACB)
arc_property_circumference_angle_external(1,OCB,A)
arc_property_circumference_angle_external(1,OCB,D)
Problem: As shown in the diagram, BD=1, ∠ECB=30°, O is the center of ⊙O, AB is the diameter of circle O. Find the length of line OA.
问题：如图所示，BD=1，∠ECB=30°，O是圆O的圆心，圆O的直径为AB。求直线OA的长度。
Construction (CDL):
Shape(AD,ODA)
Shape(DA,AO,OE,ED)
Shape(DE,EB,BD)
Shape(DB,OBD)
Shape(CE,EO,OA,OAC)
Shape(EC,CB,BE)
Shape(BC,OCB)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(LengthOfLine(BD),1)
Equal(MeasureOfAngle(ECB),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(LengthOfLine(OA))
Answer: 1
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
arc_property_circumference_angle_external(1,OBD,A)
arc_property_circumference_angle_external(1,OBD,C)
diameter_of_circle_property_right_angle(1,BDA,O)
sine_theorem(1,BDA)
line_addition(1,AO,OB)
Problem: As shown in the diagram, ∠EBA=30°, O is the center of circle O, AD is the diameter of circle O. Find the measure of ∠EAC.
问题：如图所示，∠EBA=30°，O是圆O的圆心，AD是⊙O的直径。求∠EAC的大小。
Construction (CDL):
Shape(DC,OCD)
Shape(CD,DO,OE,EC)
Shape(CE,EA,AC)
Shape(CA,OAC)
Shape(BE,EO,OD,ODB)
Shape(EB,BA,AE)
Shape(AB,OBA)
Collinear(DOEA)
Collinear(BEC)
Cocircular(O,BACD)
Conditions (CDL):
Equal(MeasureOfAngle(EBA),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AD,O)
Goal: Value(MeasureOfAngle(EAC))
Answer: 60
Theorem sequence:
arc_property_circumference_angle_external(1,OAC,B)
arc_property_circumference_angle_external(1,OAC,D)
diameter_of_circle_property_right_angle(1,ACD,O)
triangle_property_angle_sum(1,CDA)
Problem: As shown in the diagram, ∠CAO=40°, ∠OBC=15°, the center of circle O is O. Find the measure of ∠BOA.
问题：如图所示，∠CAO=40°，∠OBC=15°，O是圆O的圆心。求∠BOA的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(CA,AO,OC)
Shape(OA,AB,BO)
Shape(OAB,BA)
Shape(OB,BC,CO)
Shape(OBC,CB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAO),40)
Equal(MeasureOfAngle(OBC),15)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 110
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,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,OCA)
isosceles_triangle_judgment_line_equal(1,OBC)
isosceles_triangle_property_angle_equal(1,OCA)
isosceles_triangle_property_angle_equal(1,OBC)
angle_addition(1,BCO,OCA)
arc_property_circumference_angle_external(1,OAB,C)
arc_property_center_angle(1,OAB,O)
Problem: As shown in the diagram, ∠CBA=62°, O is the center of ⊙O. Find the measure of ∠OAC.
问题：如图所示，∠CBA=62°，O是圆O的圆心。求∠OAC的大小。
Construction (CDL):
Shape(OAC,CA)
Shape(AC,CO,OA)
Shape(AO,OC,CB,BA)
Shape(OBA,AB)
Shape(OCB,BC)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),62)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(OAC))
Answer: 28
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,ACO)
Problem: As shown in the diagram, AB=3, AB=AC, ∠BAC=120°, the center of circle O is O, the diameter of ⊙O is BD. Find the length of line AD.
问题：如图所示，AB=3，AB=AC，∠BAC=120°，⊙O的圆心为O，BD是⊙O的直径。求直线AD的长度。
Construction (CDL):
Shape(BA,AC,CB)
Shape(OBA,AB)
Shape(OAC,CA)
Shape(OAC,OCD,DA)
Shape(ODB,BO,OD)
Shape(DO,OB,BA,AD)
Collinear(BOD)
Cocircular(O,BACD)
Conditions (CDL):
Equal(LengthOfLine(AB),3)
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(MeasureOfAngle(BAC),120)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BD,O)
Goal: Value(LengthOfLine(AD))
Answer: 3*sqrt(3)
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,ACB)
isosceles_triangle_property_angle_equal(1,ACB)
triangle_property_angle_sum(1,BAC)
triangle_property_angle_sum(1,DBA)
diameter_of_circle_property_right_angle(1,BAD,O)
arc_property_circumference_angle_external(1,OBA,C)
arc_property_circumference_angle_external(1,OBA,D)
sine_theorem(1,ADB)
Problem: As shown in the diagram, ∠ACD=42°, O is the center of ⊙O, AB is the diameter of circle O. Find the measure of ∠DAB.
问题：如图所示，∠ACD=42°，O是圆O的圆心，⊙O的直径为AB。求∠DAB的大小。
Construction (CDL):
Shape(OBD,DB)
Shape(ODA,AD)
Shape(OAC,CA)
Shape(OAC,OCB,BO,OA)
Shape(BD,DA,AO,OB)
Shape(DA,AC,CD)
Collinear(AOB)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(ACD),42)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(DAB))
Answer: 48
Theorem sequence:
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, ∠CAP=40°, ∠DPA=75°, the center of ⊙O is O. Find the measure of ∠DBP.
问题：如图所示，∠CAP=40°，∠DPA=75°，O是圆O的圆心。求∠DBP的大小。
Construction (CDL):
Shape(CA,AP,PC)
Shape(BP,PD,DB)
Shape(OCA,AC)
Shape(OAD,DP,PA)
Shape(ODB,BD)
Shape(OBC,CP,PB)
Collinear(CPD)
Collinear(APB)
Cocircular(O,CADB)
Conditions (CDL):
Equal(MeasureOfAngle(CAP),40)
Equal(MeasureOfAngle(DPA),75)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DBP))
Answer: 35
Theorem sequence:
adjacent_complementary_angle(1,BPD,DPA)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_external(1,OBC,D)
triangle_property_angle_sum(1,BPD)
Problem: As shown in the diagram, ∠CBA=130°, O is the center of circle O. Find the measure of ∠AOC.
问题：如图所示，∠CBA=130°，圆O的圆心为O。求∠AOC的大小。
Construction (CDL):
Shape(AO,OC,CB,BA)
Shape(OBA,AB)
Shape(OCB,BC)
Shape(OAC,CO,OA)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),130)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AOC))
Answer: 100
Theorem sequence:
arc_property_circumference_angle_internal(1,OCA,B)
arc_property_center_angle(1,OCA,O)
Problem: As shown in the diagram, AB=2, ∠BCA=45°, the center of ⊙O is O, the diameter of ⊙O is AD. Find the radius of ⊙O.
问题：如图所示，AB=2，∠BCA=45°，O是⊙O的圆心，⊙O的直径为AD。求⊙O的半径。
Construction (CDL):
Shape(ODC,CE,EO,OD)
Shape(OCA,AC)
Shape(CA,AE,EC)
Shape(EA,AB,BE)
Shape(OAB,BA)
Shape(OBD,DB)
Shape(DO,OE,EB,BD)
Collinear(AEOD)
Collinear(CEB)
Cocircular(O,ABDC)
Conditions (CDL):
Equal(LengthOfLine(AB),2)
Equal(MeasureOfAngle(BCA),45)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AD,O)
Goal: Value(RadiusOfCircle(O))
Answer: sqrt(2)
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,D)
arc_property_circumference_angle_external(1,OAB,C)
diameter_of_circle_property_right_angle(1,ABD,O)
triangle_property_angle_sum(1,ABD)
right_triangle_judgment_angle(1,ABD)
isosceles_triangle_judgment_angle_equal(1,BDA)
right_triangle_property_pythagorean(1,ABD)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OD,O)
line_addition(1,AO,OD)
Problem: As shown in the diagram, AB=4, AB=AC, ∠BAC=120°, the center of circle O is O, BD is the diameter of ⊙O. Find the length of line BD.
问题：如图所示，AB=4，AB=AC，∠BAC=120°，O是圆O的圆心，BD是⊙O的直径。求直线BD的长度。
Construction (CDL):
Shape(OBA,AB)
Shape(OAC,CA)
Shape(OAC,OCD,DA)
Shape(ODB,BO,OD)
Shape(OB,BA,AO)
Shape(DO,OA,AD)
Shape(CB,BA,AC)
Collinear(BOD)
Cocircular(O,BACD)
Conditions (CDL):
Equal(LengthOfLine(AB),4)
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(MeasureOfAngle(BAC),120)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BD,O)
Goal: Value(LengthOfLine(BD))
Answer: 8
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,ACB)
isosceles_triangle_property_angle_equal(1,ACB)
triangle_property_angle_sum(1,CBA)
arc_property_circumference_angle_external(1,OBA,C)
arc_property_circumference_angle_external(1,OBA,D)
diameter_of_circle_property_right_angle(1,BAD,O)
sine_theorem(1,BAD)
Problem: As shown in the diagram, BC=3, ∠CAB=30°, the center of circle O is O. Find the diameter of ⊙O.
问题：如图所示，BC=3，∠CAB=30°，⊙O的圆心为O。求圆O的直径。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OB,BC,CO)
Shape(CA,AB,BC)
Shape(OCA,AC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(LengthOfLine(BC),3)
Equal(MeasureOfAngle(CAB),30)
IsCentreOfCircle(O,O)
Goal: Value(DiameterOfCircle(O))
Answer: 6
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)
isosceles_triangle_judgment_angle_equal(1,COB)
circle_property_length_of_radius_and_diameter(1,O)
Problem: As shown in the diagram, the measure of arc OCA is 60, the measure of arc ODB is 100, the center of ⊙O is O. Find the measure of ∠AEC.
问题：如图所示，弧OCA的角度为60，弧ODB的角度为100，O是圆O的圆心。求∠AEC的大小。
Construction (CDL):
Shape(OBC,CE,EB)
Shape(OCA,AE,EC)
Shape(AD,DE,EA)
Shape(ODB,BE,ED)
Shape(OAD,DA)
Collinear(DEC)
Collinear(AEB)
Cocircular(O,DBCA)
Conditions (CDL):
Equal(MeasureOfArc(OCA),60)
Equal(MeasureOfArc(ODB),100)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AEC))
Answer: 80
Theorem sequence:
arc_property_circumference_angle_external(1,ODB,A)
arc_property_circumference_angle_external(1,OCA,D)
triangle_property_angle_sum(1,ADE)
adjacent_complementary_angle(1,DEA,AEC)
Problem: As shown in the diagram, ∠COB=56°, O is the center of ⊙O, OE is perpendicular to CE. Find the measure of ∠BDA.
问题：如图所示，∠COB=56°，O是圆O的圆心，OE⊥CE。求∠BDA的大小。
Construction (CDL):
Shape(OE,EC,CO)
Shape(OAB,BE,EA)
Shape(OBC,CE,EB)
Shape(OAB,BD,DA)
Shape(OCA,AE,EO,OC)
Collinear(AEC)
Collinear(OEB)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(COB),56)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(OE,CE)
Goal: Value(MeasureOfAngle(BDA))
Answer: 28
Theorem sequence:
adjacent_complementary_angle(1,AEO,OEC)
circle_property_chord_perpendicular_bisect_arc(1,OAC,OEB)
congruent_arc_judgment_length_equal(1,OAB,OBC)
congruent_arc_property_measure_equal(1,OAB,OBC)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OAB,D)
Problem: As shown in the diagram, ∠BDC=65°, the center of circle O is O, the diameter of ⊙O is AB. Find the measure of ∠CBA.
问题：如图所示，∠BDC=65°，O是圆O的圆心，AB是圆O的直径。求∠CBA的大小。
Construction (CDL):
Shape(CB,BO,OA,AC)
Shape(BD,DC,CB)
Shape(OAC,CA)
Shape(OCB,BC)
Shape(OBD,DB)
Shape(OBD,ODA,AO,OB)
Collinear(BOA)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(BDC),65)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CBA))
Answer: 25
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,ACB,O)
triangle_property_angle_sum(1,CBA)
Problem: As shown in the diagram, ∠DOC=80°, O is the center of ⊙O. Find the sum of the measure of ∠ABD and the measure of ∠OCA.
问题：如图所示，∠DOC=80°，圆O的圆心为O。求∠ABD的大小与∠OCA的大小之和。
Construction (CDL):
Shape(OC,CD,DO)
Shape(AO,OC,CA)
Shape(AB,BD,ODA)
Shape(OAB,BA)
Shape(OBC,CO,OD,DB)
Shape(OCD,DC)
Shape(ODA,AC,CD)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(DOC),80)
IsCentreOfCircle(O,O)
Goal: Value(Add(MeasureOfAngle(ABD),MeasureOfAngle(OCA)))
Answer: 50
Theorem sequence:
triangle_property_angle_sum(1,OCD)
arc_property_circumference_angle_external(1,ODA,B)
arc_property_circumference_angle_external(1,ODA,C)
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)
angle_addition(1,OCA,ACD)
Problem: As shown in the diagram, ∠BCA=30°, the radius of circle O is 3, O is the center of ⊙O. Find the length of line AB.
问题：如图所示，∠BCA=30°，圆O的半径为3，O是⊙O的圆心。求直线AB的长度。
Construction (CDL):
Shape(OA,AB,BO)
Shape(AB,BC,CA)
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCA,AC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),30)
Equal(RadiusOfCircle(O),3)
IsCentreOfCircle(O,O)
Goal: Value(LengthOfLine(AB))
Answer: 3
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,C)
arc_property_center_angle(1,OAB,O)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
isosceles_triangle_judgment_line_equal(1,OAB)
isosceles_triangle_property_angle_equal(1,OAB)
triangle_property_angle_sum(1,OAB)
isosceles_triangle_judgment_angle_equal(1,ABO)
Problem: As shown in the diagram, ∠COB=80°, O is the center of ⊙O. Find the measure of ∠CAB.
问题：如图所示，∠COB=80°，O是圆O的圆心。求∠CAB的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(OB,BC,CO)
Shape(OBC,CB)
Shape(OCA,AC)
Shape(OAB,BA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(COB),80)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 40
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,OBC,O)
Problem: As shown in the diagram, ∠OAB=28°, the center of ⊙O is O. Find the measure of ∠BCA.
问题：如图所示，∠OAB=28°，O是圆O的圆心。求∠BCA的大小。
Construction (CDL):
Shape(CA,AB,BC)
Shape(OA,AB,BO)
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCA,AC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(OAB),28)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 62
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, AB=20, ∠ACP=30°, O is the center of ⊙O, the center of circle Z is Z, the diameter of circle O is AC, BC is the diameter of circle Z. Find the length of line PQ.
问题：如图所示，AB=20，∠ACP=30°，圆O的圆心为O，圆Z的圆心为Z，AC是圆O的直径，⊙Z的直径为BC。求直线PQ的长度。
Construction (CDL):
Shape(OPA,AP)
Shape(PA,AB,ZQB,QP)
Shape(OCP,PQ,ZCQ)
Shape(ZQB,BQ)
Shape(QB,BO,OZ,ZC,CQ)
Shape(ZCQ,QC)
Collinear(ABOZC)
Collinear(PQC)
Cocircular(O,CPA)
Cocircular(Z,CQB)
Conditions (CDL):
Equal(LengthOfLine(AB),20)
Equal(MeasureOfAngle(ACP),30)
IsCentreOfCircle(O,O)
IsCentreOfCircle(Z,Z)
IsDiameterOfCircle(AC,O)
IsDiameterOfCircle(BC,Z)
Goal: Value(LengthOfLine(PQ))
Answer: 10*sqrt(3)
Theorem sequence:
diameter_of_circle_property_right_angle(1,CPA,O)
diameter_of_circle_property_right_angle(1,CQB,Z)
triangle_property_angle_sum(1,QBC)
triangle_property_angle_sum(1,PAC)
sine_theorem(1,QBC)
sine_theorem(1,BCQ)
sine_theorem(1,PAC)
sine_theorem(1,ACP)
line_addition(1,AB,BC)
line_addition(1,PQ,QC)
Problem: As shown in the diagram, AB=AC, AB=AD, ∠DBC=23°, A is the center of circle A. Find the measure of ∠DAC.
问题：如图所示，AB=AC，AB=AD，∠DBC=23°，A是⊙A的圆心。求∠DAC的大小。
Construction (CDL):
Shape(DA,AE,ED)
Shape(CD,DE,EC)
Shape(EA,AB,BE)
Shape(CE,EB,BC)
Shape(ABC,CB)
Shape(ACD,DC)
Collinear(DEB)
Collinear(AEC)
Cocircular(A,BCD)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(LengthOfLine(AB),LengthOfLine(AD))
Equal(MeasureOfAngle(DBC),23)
IsCentreOfCircle(A,A)
Goal: Value(MeasureOfAngle(DAC))
Answer: 46
Theorem sequence:
arc_property_center_angle(1,ACD,A)
arc_property_circumference_angle_external(1,ACD,B)
Problem: As shown in the diagram, ∠CDA=80°, O is the center of ⊙O, the diameter of circle O is CD. Find the measure of ∠ABD.
问题：如图所示，∠CDA=80°，圆O的圆心为O，CD是圆O的直径。求∠ABD的大小。
Construction (CDL):
Shape(AC,CO,OD,DA)
Shape(DA,AB,BD)
Shape(OAC,CA)
Shape(ODA,AD)
Shape(OBD,DB)
Shape(OCB,OBD,DO,OC)
Collinear(COD)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(CDA),80)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(CD,O)
Goal: Value(MeasureOfAngle(ABD))
Answer: 10
Theorem sequence:
triangle_property_angle_sum(1,ACD)
diameter_of_circle_property_right_angle(1,DAC,O)
arc_property_circumference_angle_external(1,ODA,C)
arc_property_circumference_angle_external(1,ODA,B)