SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠CAB=40°, O is the center of circle O, AB is the tangent to circle O. Find the measure of ∠BPC.
问题：如图所示，∠CAB=40°，O是⊙O的圆心，⊙O的切线为AB。求∠BPC的大小。
Construction (CDL):
Shape(OD,DB,BO)
Shape(BD,DC,OCB)
Shape(CD,DP,PC)
Shape(CP,OPC)
Shape(CA,AB,OCB)
Shape(PD,DO,OB,OBP)
Collinear(ODCA)
Collinear(BDP)
Cocircular(O,BPC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),40)
IsCentreOfCircle(O,O)
IsTangentOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BPC))
Answer: 25
Theorem sequence:
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_external(1,OCB,P)
tangent_of_circle_property_perpendicular(2,AB,O,O)
triangle_property_angle_sum(1,OAB)
Problem: As shown in the diagram, ∠BPA=90°, the radius of ⊙O is 2, the center of circle O is O, the tangent to circle O is PA, PB is the tangent to ⊙O. Find the length of line OP.
问题：如图所示，∠BPA=90°，⊙O的半径为2，O是圆O的圆心，圆O的切线为PA，PB是⊙O的切线。求直线OP的长度。
Construction (CDL):
Shape(CP,PA,OCA)
Shape(BP,PC,OBC)
Shape(AO,OC,OCA)
Shape(CO,OB,OBC)
Collinear(PCO)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BPA),90)
Equal(RadiusOfCircle(O),2)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
IsTangentOfCircle(PB,O)
Goal: Value(LengthOfLine(OP))
Answer: 2*sqrt(2)
Theorem sequence:
angle_addition(1,BPC,CPA)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
tangent_of_circle_property_perpendicular(2,PA,O,O)
tangent_of_circle_property_perpendicular(1,PB,O,O)
mirror_congruent_triangle_judgment_hl(1,PAO,POB)
mirror_congruent_triangle_property_angle_equal(1,PAO,POB)
sine_theorem(1,OPA)
Problem: As shown in the diagram, OA=5, OC=3, the center of ⊙N is O, OC is the perpendicular bisector of AB, AC is the tangent to circle O. Find the length of line AB.
问题：如图所示，OA=5，OC=3，O是圆N的圆心，OC垂直平分AB，圆O的切线为AC。求直线AB的长度。
Construction (CDL):
Shape(BC,CA,OAB)
Shape(CO,OD,ODC)
Shape(DA,AC,ODC)
Collinear(ODA)
Collinear(ACB)
Cocircular(O,AB)
Cocircular(N,DC)
Conditions (CDL):
Equal(LengthOfLine(OA),5)
Equal(LengthOfLine(OC),3)
IsCentreOfCircle(O,N)
IsPerpendicularBisectorOfLine(OC,AB)
IsTangentOfCircle(AC,O)
Goal: Value(LengthOfLine(AB))
Answer: 8
Theorem sequence:
right_triangle_judgment_angle(1,ACO)
right_triangle_property_pythagorean(1,ACO)
line_addition(1,AC,CB)
Problem: As shown in the diagram, ∠CAB=31°, O is the center of ⊙O, the diameter of circle O is AB, PC is the tangent to ⊙O. Find the measure of ∠BPC.
问题：如图所示，∠CAB=31°，圆O的圆心为O，AB是⊙O的直径，圆O的切线为PC。求∠BPC的大小。
Construction (CDL):
Shape(CA,AO,OC)
Shape(AC,OCA)
Shape(CO,OB,OBC)
Shape(BP,PC,OBC)
Shape(BO,OA,OAB)
Collinear(AOBP)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),31)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsTangentOfCircle(PC,O)
Goal: Value(MeasureOfAngle(BPC))
Answer: 28
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
tangent_of_circle_property_perpendicular(2,PC,O,O)
triangle_property_angle_sum(1,COP)
Problem: As shown in the diagram, ∠BPA=70°, the center of ⊙O is O, the tangent to circle O is PA, the tangent to ⊙O is PB. Find the measure of ∠ACB.
问题：如图所示，∠BPA=70°，O是⊙O的圆心，⊙O的切线为PA，PB是圆O的切线。求∠ACB的大小。
Construction (CDL):
Shape(OA,AC,CB,BO)
Shape(CA,OAC)
Shape(BC,OCB)
Shape(AO,OB,OBA)
Shape(BP,PA,OBA)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(BPA),70)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
IsTangentOfCircle(PB,O)
Goal: Value(MeasureOfAngle(ACB))
Answer: 55
Theorem sequence:
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_external(1,OBA,C)
tangent_of_circle_property_perpendicular(2,PA,O,O)
tangent_of_circle_property_perpendicular(1,PB,O,O)
quadrilateral_property_angle_sum(1,AOBP)
Problem: As shown in the diagram, OA=1, ∠OPA=30°, O is the center of circle O, the tangent to ⊙O is PA. Find the length of line PA.
问题：如图所示，OA=1，∠OPA=30°，O是圆O的圆心，圆O的切线为PA。求直线PA的长度。
Construction (CDL):
Shape(AO,OB,OBA)
Shape(BP,PA,OBA)
Collinear(OBP)
Cocircular(O,AB)
Conditions (CDL):
Equal(LengthOfLine(OA),1)
Equal(MeasureOfAngle(OPA),30)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
Goal: Value(LengthOfLine(PA))
Answer: sqrt(3)
Theorem sequence:
tangent_of_circle_property_perpendicular(2,PA,O,O)
sine_theorem(1,OPA)
cosine_theorem(1,AOP)
Problem: As shown in the diagram, ∠APB=70°, O is the center of circle O, the tangent to ⊙O is PA, the tangent to circle O is PB. Find the measure of ∠BCA.
问题：如图所示，∠APB=70°，O是⊙O的圆心，PA是圆O的切线，PB是⊙O的切线。求∠BCA的大小。
Construction (CDL):
Shape(AO,OB,BC,CA)
Shape(AC,OCA)
Shape(CB,OBC)
Shape(BO,OA,OAB)
Shape(AP,PB,OAB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(APB),70)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
IsTangentOfCircle(PB,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 55
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
tangent_of_circle_property_perpendicular(1,PA,O,O)
tangent_of_circle_property_perpendicular(2,PB,O,O)
quadrilateral_property_angle_sum(1,BOAP)
Problem: As shown in the diagram, ∠ACD=30°, the center of ⊙O is O, AB is the diameter of circle O, the tangent to circle O is CD. Find the measure of ∠DAC.
问题：如图所示，∠ACD=30°，O是圆O的圆心，圆O的直径为AB，圆O的切线为CD。求∠DAC的大小。
Construction (CDL):
Shape(AO,OD,DA)
Shape(AD,ODA)
Shape(DO,OB,OBD)
Shape(BC,CD,OBD)
Collinear(AOBC)
Cocircular(O,ABD)
Conditions (CDL):
Equal(MeasureOfAngle(ACD),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsTangentOfCircle(CD,O)
Goal: Value(MeasureOfAngle(DAC))
Answer: 30
Theorem sequence:
arc_property_center_angle(1,OBD,O)
arc_property_circumference_angle_external(1,OBD,A)
tangent_of_circle_property_perpendicular(2,CD,O,O)
triangle_property_angle_sum(1,DOC)
Problem: As shown in the diagram, OA=3, OP=5, O is the center of ⊙O, PA is the tangent to ⊙O. Find the length of line AP.
问题：如图所示，OA=3，OP=5，O是圆O的圆心，圆O的切线为PA。求直线AP的长度。
Construction (CDL):
Shape(AO,OB,OBA)
Shape(BP,PA,OBA)
Collinear(OBP)
Cocircular(O,AB)
Conditions (CDL):
Equal(LengthOfLine(OA),3)
Equal(LengthOfLine(OP),5)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
Goal: Value(LengthOfLine(AP))
Answer: 4
Theorem sequence:
tangent_of_circle_property_perpendicular(2,PA,O,O)
triangle_property_angle_sum(1,AOP)
sine_theorem(1,OPA)
sine_theorem(1,PAO)
Problem: As shown in the diagram, AB=AC, BF=8, CB=10, △DBF is an equilateral △, AH is the altitude of triangle ABC. Find the length of line DE.
问题：如图所示，AB=AC，BF=8，CB=10，△DBF为等边△，AH是三角形ABC的高。求直线DE的长度。
Construction (CDL):
Shape(AB,BD,DE,EA)
Shape(AE,EF,FC,CA)
Shape(DB,BH,HE,ED)
Shape(EH,HF,FE)
Collinear(AEH)
Collinear(DEF)
Collinear(BHFC)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(LengthOfLine(BF),8)
Equal(LengthOfLine(CB),10)
EquilateralTriangle(DBF)
IsAltitudeOfTriangle(AH,ABC)
Goal: Value(LengthOfLine(DE))
Answer: 2
Theorem sequence:
equilateral_triangle_property_angle(1,DBF)
line_addition(1,DE,EF)
line_addition(1,BH,HF)
line_addition(1,BH,HC)
triangle_property_angle_sum(1,EHF)
isosceles_triangle_judgment_line_equal(1,ABC)
isosceles_triangle_property_line_coincidence(1,ABC,H)
sine_theorem(1,FEH)
sine_theorem(1,BFD)
Problem: As shown in the diagram, OA=2, ∠CAB=30°, O is the center of circle O, AB is the diameter of ⊙O, DC is the tangent to circle O. Find the length of line OD.
问题：如图所示，OA=2，∠CAB=30°，O是⊙O的圆心，AB是圆O的直径，DC是⊙O的切线。求直线OD的长度。
Construction (CDL):
Shape(AO,OC,CA)
Shape(AC,OCA)
Shape(CO,OB,OBC)
Shape(BD,DC,OBC)
Shape(BO,OA,OAB)
Collinear(DBOA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(LengthOfLine(OA),2)
Equal(MeasureOfAngle(CAB),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsTangentOfCircle(DC,O)
Goal: Value(LengthOfLine(OD))
Answer: 4
Theorem sequence:
tangent_of_circle_property_perpendicular(2,DC,O,O)
triangle_property_angle_sum(1,AOC)
triangle_property_angle_sum(1,COD)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OC,O)
sine_theorem(1,OCA)
sine_theorem(1,CAO)
sine_theorem(1,COD)
sine_theorem(1,ODC)
sine_theorem(1,CAD)
Problem: As shown in the diagram, AB=8, OA=5, ∠AHO=90°, O is the center of ⊙O. Find the length of line HC.
问题：如图所示，AB=8，OA=5，∠AHO=90°，⊙O的圆心为O。求直线HC的长度。
Construction (CDL):
Shape(OA,AH,HO)
Shape(OH,HB,BO)
Shape(CH,HA,OAC)
Shape(BH,HC,OCB)
Collinear(AHB)
Collinear(OHC)
Cocircular(O,ACB)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(LengthOfLine(OA),5)
Equal(MeasureOfAngle(AHO),90)
IsCentreOfCircle(O,O)
Goal: Value(LengthOfLine(HC))
Answer: 2
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OC,O)
circle_property_chord_perpendicular_bisect_chord(1,O,OH,AB)
right_triangle_judgment_angle(1,AHO)
right_triangle_property_pythagorean(1,AHO)
line_addition(1,AH,HB)
line_addition(1,OH,HC)
Problem: As shown in the diagram, AB=2, BC=2, O is the center of ⊙O, the tangent to circle O is CB. Find the length of line AC.
问题：如图所示，AB=2，BC=2，O是⊙O的圆心，CB是圆O的切线。求直线AC的长度。
Construction (CDL):
Shape(DA,AO,OB,OBD)
Shape(AD,ODA)
Shape(BC,CD,OBD)
Shape(BO,OA,OAB)
Collinear(AOB)
Collinear(ADC)
Cocircular(O,ABD)
Conditions (CDL):
Equal(LengthOfLine(AB),2)
Equal(LengthOfLine(BC),2)
IsCentreOfCircle(O,O)
IsTangentOfCircle(CB,O)
Goal: Value(LengthOfLine(AC))
Answer: 2*sqrt(2)
Theorem sequence:
tangent_of_circle_property_perpendicular(1,CB,O,O)
right_triangle_judgment_angle(1,ABC)
right_triangle_property_pythagorean(1,ABC)
Problem: As shown in the diagram, ∠DAB=50°, O is the center of ⊙O, the tangent to ⊙O is AD, OB∥CD. Find the measure of ∠CBO.
问题：如图所示，∠DAB=50°，⊙O的圆心为O，⊙O的切线为AD，OB平行于CD。求∠CBO的大小。
Construction (CDL):
Shape(DA,AB,ODB)
Shape(BE,ED,ODB)
Shape(EC,CD,DE)
Shape(DC,OCD)
Shape(OE,EB,BO)
Shape(CE,EO,OB,OBC)
Collinear(CEB)
Collinear(OED)
Collinear(OBA)
Cocircular(O,CDB)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),50)
IsCentreOfCircle(O,O)
IsTangentOfCircle(AD,O)
ParallelBetweenLine(OB,CD)
Goal: Value(MeasureOfAngle(CBO))
Answer: 20
Theorem sequence:
tangent_of_circle_property_perpendicular(1,AD,O,O)
triangle_property_angle_sum(1,ODA)
arc_property_center_angle(1,ODB,O)
arc_property_circumference_angle_external(1,ODB,C)
parallel_property_alternate_interior_angle(2,OB,CD)
Problem: As shown in the diagram, the area of △AEF is 6, AE=1, EB=2, AD and BC are opposite sides of the ▱ ABCD. Find the area of △CDF.
问题：如图所示，三角形AEF的面积为6，AE=1，EB=2，ABCD是平行四边形。求三角形CDF的面积。
Construction (CDL):
Shape(DA,AF,FD)
Shape(FA,AE,EF)
Shape(DF,FC,CD)
Shape(CF,FE,EB,BC)
Collinear(DFE)
Collinear(AFC)
Collinear(AEB)
Conditions (CDL):
Equal(AreaOfTriangle(AEF),6)
Equal(LengthOfLine(AE),1)
Equal(LengthOfLine(EB),2)
Parallelogram(ABCD)
Goal: Value(AreaOfTriangle(CDF))
Answer: 54
Theorem sequence:
parallelogram_property_opposite_line_equal(1,ABCD)
line_addition(1,AE,EB)
parallel_property_alternate_interior_angle(2,BA,CD)
vertical_angle(1,EFA,DFC)
similar_triangle_judgment_aa(1,EFA,DFC)
similar_triangle_property_line_ratio(1,FAE,FCD)
similar_triangle_property_area_square_ratio(1,EFA,DFC)
Problem: As shown in the diagram, ∠BCA=20°, ∠COB=Mul(MeasureOfAngle(BOA),3)°, the center of circle O is O. Find the measure of ∠DAB.
问题：如图所示，∠BCA=20°，∠COB=Mul(MeasureOfAngle(BOA),3)°，O是⊙O的圆心。求∠DAB的大小。
Construction (CDL):
Shape(AO,OC,OCA)
Shape(OA,AD,DO)
Shape(OD,DC,CO)
Shape(AB,BD,DA)
Shape(DB,BC,CD)
Shape(BA,OAB)
Shape(CB,OBC)
Collinear(ADC)
Collinear(ODB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),20)
Equal(MeasureOfAngle(COB),Mul(MeasureOfAngle(BOA),3))
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DAB))
Answer: 60
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,C)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,OAB,O)
arc_property_center_angle(1,OBC,O)
Problem: As shown in the diagram, ∠EAB=40°, BD is the diameter of circle O. Find the measure of ∠DBC.
问题：如图所示，∠EAB=40°，圆O的直径为BD。求∠DBC的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(CB,OBC)
Shape(DC,OCD)
Shape(AE,EO,OD,ODA)
Shape(AB,BE,EA)
Shape(EB,BC,CE)
Shape(OE,EC,CO)
Shape(DO,OC,CD)
Collinear(AEC)
Collinear(BEOD)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(EAB),40)
IsDiameterOfCircle(BD,O)
Goal: Value(MeasureOfAngle(DBC))
Answer: 50
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCD,O)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_external(1,OBC,D)
triangle_property_angle_sum(1,DBC)
Problem: As shown in the diagram, ∠DAE=70°, the center of ⊙O is O, AB is the diameter of ⊙O, AE is perpendicular to DE. Find the measure of ∠BOC.
问题：如图所示，∠DAE=70°，O是圆O的圆心，圆O的直径为AB，AE垂直于DE。求∠BOC的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(DO,OB,OBD)
Shape(DE,EO,OD)
Shape(DA,AE,ED)
Shape(CE,EA,OAC)
Shape(OE,EC,CO)
Shape(BO,OC,OCB)
Collinear(AEOB)
Collinear(DEC)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(DAE),70)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
PerpendicularBetweenLine(AE,DE)
Goal: Value(MeasureOfAngle(BOC))
Answer: 140
Theorem sequence:
triangle_property_angle_sum(1,DAE)
arc_property_circumference_angle_external(1,OAC,D)
arc_property_center_angle(1,OAC,O)
flat_angle(1,BOE)
angle_addition(1,BOC,COE)
Problem: As shown in the diagram, ∠BAC=30°, the center of ⊙O is O. Find the measure of ∠BOC.
问题：如图所示，∠BAC=30°，圆O的圆心为O。求∠BOC的大小。
Construction (CDL):
Shape(CA,OAC)
Shape(AB,OBA)
Shape(AC,CO,OB,BA)
Shape(BO,OC,OCB)
Cocircular(O,CBA)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),30)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOC))
Answer: 60
Theorem sequence:
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_external(1,OCB,A)
Problem: As shown in the diagram, ∠BOA=60°, O is the center of circle O. Find the measure of ∠BPA.
问题：如图所示，∠BOA=60°，O是⊙O的圆心。求∠BPA的大小。
Construction (CDL):
Shape(AP,OPA)
Shape(PB,OBP)
Shape(BP,PA,AO,OB)
Shape(BO,OA,OAB)
Cocircular(O,PAB)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),60)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BPA))
Answer: 30
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,P)
Problem: As shown in the diagram, AB=AD, ∠BOE=124°, O is the center of circle O. Find the measure of ∠DCE.
问题：如图所示，AB=AD，∠BOE=124°，O是圆O的圆心。求∠DCE的大小。
Construction (CDL):
Shape(DA,OAD)
Shape(DE,EA,AD)
Shape(AE,EO,OA)
Shape(AO,OB,BA)
Shape(AB,OBA)
Shape(OE,EC,CB,BO)
Shape(ED,DC,CE)
Shape(CD,ODC)
Shape(BC,OCB)
Collinear(DEO)
Collinear(CEA)
Cocircular(O,DCBA)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AD))
Equal(MeasureOfAngle(BOE),124)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DCE))
Answer: 90
Theorem sequence:
arc_property_center_angle(1,ODA,O)
arc_property_center_angle(1,OAD,O)
arc_property_circumference_angle_external(1,OAD,C)
round_angle(1,EOA,AOE)
congruent_arc_judgment_chord_equal(1,ODA,OAD)
congruent_arc_property_measure_equal(1,ODA,OAD)
Problem: As shown in the diagram, ∠BOC=116°, the center of ⊙O is O, AD is parallel to CO. Find the measure of ∠AOD.
问题：如图所示，∠BOC=116°，O是⊙O的圆心，AD平行于CO。求∠AOD的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(DA,AO,OD)
Shape(DO,OB,OBD)
Shape(CO,OA,OAC)
Shape(BO,OC,OCB)
Collinear(AOB)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(BOC),116)
IsCentreOfCircle(O,O)
ParallelBetweenLine(AD,CO)
Goal: Value(MeasureOfAngle(AOD))
Answer: 52
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OD,O)
flat_angle(1,BOA)
angle_addition(1,BOC,COA)
parallel_property_alternate_interior_angle(1,AD,CO)
isosceles_triangle_judgment_line_equal(1,ODA)
isosceles_triangle_property_angle_equal(1,ODA)
triangle_property_angle_sum(1,ODA)
Problem: As shown in the diagram, ∠CAB=50°, the center of ⊙O is O, D is the midpoint of segment BC, OD is perpendicular to BD. Find the measure of ∠COD.
问题：如图所示，∠CAB=50°，O是⊙O的圆心，D平分线段BC，OD垂直于BD。求∠COD的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(AB,BO,OC,CA)
Shape(BA,OAB)
Shape(OB,BD,DO)
Shape(OD,DC,CO)
Shape(CD,DB,OBC)
Collinear(BDC)
Cocircular(O,BCA)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),50)
IsCentreOfCircle(O,O)
IsMidpointOfLine(D,BC)
PerpendicularBetweenLine(OD,BC)
Goal: Value(MeasureOfAngle(COD))
Answer: 50
Theorem sequence:
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)
arc_property_circumference_angle_external(1,OBC,A)
mirror_congruent_triangle_judgment_sss(1,OBD,ODC)
mirror_congruent_triangle_property_angle_equal(1,OBD,ODC)
angle_addition(1,COD,DOB)
Problem: As shown in the diagram, ∠OAC=30°, O is the center of circle O, AB is the diameter of ⊙O. Find the measure of ∠CBA.
问题：如图所示，∠OAC=30°，⊙O的圆心为O，AB是⊙O的直径。求∠CBA的大小。
Construction (CDL):
Shape(BO,OA,AC,CB)
Shape(CA,OAC)
Shape(BC,OCB)
Shape(AO,OB,OBA)
Collinear(AOB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(OAC),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CBA))
Answer: 60
Theorem sequence:
diameter_of_circle_property_right_angle(1,ACB,O)
triangle_property_angle_sum(1,BAC)
Problem: As shown in the diagram, ∠COB=76°, O is the center of circle O. Find the measure of ∠CAB.
问题：如图所示，∠COB=76°，O是⊙O的圆心。求∠CAB的大小。
Construction (CDL):
Shape(CO,OB,OBC)
Shape(CA,AB,BO,OC)
Shape(AC,OCA)
Shape(BA,OAB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(COB),76)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 38
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠ABC=110°, O is the center of ⊙O. Find the measure of ∠COA.
问题：如图所示，∠ABC=110°，圆O的圆心为O。求∠COA的大小。
Construction (CDL):
Shape(OA,AB,BC,CO)
Shape(AO,OC,OCA)
Shape(CB,OBC)
Shape(BA,OAB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),110)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COA))
Answer: 140
Theorem sequence:
arc_property_center_angle(1,OCA,O)
arc_property_circumference_angle_external(1,OCA,B)
round_angle(1,COA,AOC)
Problem: As shown in the diagram, ∠OBC=50°, O is the center of ⊙O, the diameter of ⊙O is AB. Find the measure of ∠EDB.
问题：如图所示，∠OBC=50°，圆O的圆心为O，AB是圆O的直径。求∠EDB的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AE,EC)
Shape(CE,EO,OB,BC)
Shape(CB,OBC)
Shape(DE,EA,OAD)
Shape(BO,OE,ED,DB)
Shape(BD,ODB)
Collinear(CED)
Collinear(AEOB)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(OBC),50)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(EDB))
Answer: 40
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,CAB)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_external(1,OBC,D)
Problem: As shown in the diagram, ∠BCA=35°, O is the center of circle O. Find the measure of ∠BOA.
问题：如图所示，∠BCA=35°，O是⊙O的圆心。求∠BOA的大小。
Construction (CDL):
Shape(CA,AO,OB,BC)
Shape(CB,OBC)
Shape(BO,OA,OAB)
Shape(AC,OCA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),35)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 70
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠EBC=70°, O is the center of ⊙O, AB is the diameter of circle O. Find the measure of ∠EDB.
问题：如图所示，∠EBC=70°，⊙O的圆心为O，⊙O的直径为AB。求∠EDB的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AO,OE,EC)
Shape(CE,EB,BC)
Shape(CB,OBC)
Shape(DE,EO,OA,OAD)
Shape(BE,ED,DB)
Shape(BD,ODB)
Collinear(CED)
Collinear(AOEB)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(EBC),70)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(EDB))
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,A)
arc_property_circumference_angle_external(1,OBC,D)
Problem: As shown in the diagram, AC=8, ∠EBA=60°, the center of circle O is O, the diameter of ⊙O is AD. Find the length of line AD.
问题：如图所示，AC=8，∠EBA=60°，O是圆O的圆心，⊙O的直径为AD。求直线AD的长度。
Construction (CDL):
Shape(AB,OBA)
Shape(BA,AO,OE,EB)
Shape(BE,ED,ED,ODB)
Shape(DC,OCD)
Shape(DE,EC,CD)
Shape(EO,OA,AC,CE)
Shape(CA,OAC)
Collinear(AOED)
Collinear(BEC)
Cocircular(O,ACDB)
Conditions (CDL):
Equal(LengthOfLine(AC),8)
Equal(MeasureOfAngle(EBA),60)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AD,O)
Goal: Value(LengthOfLine(AD))
Answer: 16*sqrt(3)/3
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)
sine_theorem(1,ACD)
Problem: As shown in the diagram, ∠CAO=20°, the center of ⊙O is O, AB is the diameter of ⊙O, E is the midpoint of segment DC. Find the measure of ∠EOD.
问题：如图所示，∠CAO=20°，⊙O的圆心为O，⊙O的直径为AB，E平分线段DC。求∠EOD的大小。
Construction (CDL):
Shape(AC,OAC)
Shape(AO,OC,CA)
Shape(OE,EC,CO)
Shape(CE,EB,OBC)
Shape(DO,OA,OAD)
Shape(OD,DE,EO)
Shape(BE,ED,ODB)
Collinear(AOEB)
Collinear(DEC)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(CAO),20)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsMidpointOfLine(E,DC)
Goal: Value(MeasureOfAngle(EOD))
Answer: 40
Theorem sequence:
radius_of_circle_property_length_equal(1,OD,O)
radius_of_circle_property_length_equal(1,OC,O)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
mirror_congruent_triangle_judgment_sss(1,OEC,ODE)
mirror_congruent_triangle_property_angle_equal(1,OEC,ODE)
Problem: As shown in the diagram, ∠ECB=38°, the center of circle O is O. Find the measure of ∠AOD.
问题：如图所示，∠ECB=38°，O是⊙O的圆心。求∠AOD的大小。
Construction (CDL):
Shape(AO,OD,ODA)
Shape(DO,OE,ED)
Shape(DE,EB,OBD)
Shape(BC,OCB)
Shape(BE,EC,CB)
Shape(CE,EO,OA,OAC)
Collinear(DEC)
Collinear(AOEB)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(ECB),38)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AOD))
Answer: 104
Theorem sequence:
arc_property_center_angle(1,OBD,O)
arc_property_circumference_angle_external(1,OBD,C)
flat_angle(1,AOE)
angle_addition(1,AOD,DOE)
Problem: As shown in the diagram, ∠BPA=40°, O is the center of ⊙O. Find the measure of ∠BOA.
问题：如图所示，∠BPA=40°，圆O的圆心为O。求∠BOA的大小。
Construction (CDL):
Shape(PA,AO,OP)
Shape(PO,OB,BP)
Shape(OA,AB,BO)
Shape(AP,OPA)
Shape(BA,OAB)
Shape(PB,OBP)
Cocircular(O,ABP)
Conditions (CDL):
Equal(MeasureOfAngle(BPA),40)
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,P)
Problem: As shown in the diagram, ∠BCO=40°, O is the center of circle O. Find the measure of ∠CAB.
问题：如图所示，∠BCO=40°，O是圆O的圆心。求∠CAB的大小。
Construction (CDL):
Shape(OB,BC,CO)
Shape(AB,BO,OC,CA)
Shape(CB,OBC)
Shape(BA,OAB)
Shape(AC,OCA)
Cocircular(O,BCA)
Conditions (CDL):
Equal(MeasureOfAngle(BCO),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 50
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, ∠COB=80°, the center of circle O is O. Find the measure of ∠CAB.
问题：如图所示，∠COB=80°，O是⊙O的圆心。求∠CAB的大小。
Construction (CDL):
Shape(AB,BO,OC,CA)
Shape(CO,OB,OBC)
Shape(AC,OCA)
Shape(BA,OAB)
Cocircular(O,BCA)
Conditions (CDL):
Equal(MeasureOfAngle(COB),80)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
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, ∠COA=150°, the center of ⊙O is O. Find the measure of ∠ABC.
问题：如图所示，∠COA=150°，O是圆O的圆心。求∠ABC的大小。
Construction (CDL):
Shape(DA,AO,OC,CD)
Shape(OA,AB,BC,CO)
Shape(DC,OCD)
Shape(CB,OBC)
Shape(BA,OAB)
Shape(AD,ODA)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(COA),150)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ABC))
Answer: 105
Theorem sequence:
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_internal(1,OAC,B)
Problem: As shown in the diagram, ∠AOC=140°, the center of circle O is O, the diameter of circle O is BA. Find the measure of ∠EDB.
问题：如图所示，∠AOC=140°，圆O的圆心为O，BA是圆O的直径。求∠EDB的大小。
Construction (CDL):
Shape(BD,ODB)
Shape(DB,BE,ED)
Shape(DE,EO,OA,OAD)
Shape(AO,OC,OCA)
Shape(CO,OE,EC)
Shape(CE,EB,OBC)
Collinear(DEC)
Collinear(BEOA)
Cocircular(O,BCAD)
Conditions (CDL):
Equal(MeasureOfAngle(AOC),140)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BA,O)
Goal: Value(MeasureOfAngle(EDB))
Answer: 20
Theorem sequence:
flat_angle(1,AOE)
angle_addition(1,AOC,COE)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,D)