SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, AB=2, the measure of ⌒OBE is 60, the radius of ⊙O is 1, O is the center of ⊙O, the diameter of ⊙O is AB, the diameter of circle O is CD, CO is perpendicular to BO. Find the sum of the length of line EA and the length of line BE.
问题：如图所示，AB=2，弧OBE的角度为60，⊙O的半径为1，圆O的圆心为O，⊙O的直径为AB，⊙O的直径为CD，CO⊥BO。求直线EA的长度与直线BE的长度之和。
Construction (CDL):
Shape(AP,PC,OCA)
Shape(CP,PE,OEC)
Shape(PA,AO,OP)
Shape(PO,OE,EP)
Shape(EO,OB,BE)
Shape(EB,OBE)
Shape(DO,OA,OAD)
Shape(BO,OD,ODB)
Collinear(AOB)
Collinear(CPOD)
Collinear(EPA)
Cocircular(O,ADBEC)
Conditions (CDL):
Equal(LengthOfLine(AB),2)
Equal(MeasureOfArc(OBE),60)
Equal(RadiusOfCircle(O),1)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsDiameterOfCircle(CD,O)
PerpendicularBetweenLine(CO,BO)
Goal: Value(Add(LengthOfLine(EA),LengthOfLine(BE)))
Answer: 1+sqrt(3)
Theorem sequence:
arc_property_circumference_angle_external(1,OBE,A)
diameter_of_circle_property_right_angle(1,BEA,O)
sine_theorem(1,BEA)
cosine_theorem(1,ABE)
Problem: As shown in the diagram, the diameter of ⊙O is equal to the length of line AB, ∠BOD=60°, the measure of ⌒OCD is equal to the measure of ⌒ODB, O is the center of circle O. Find the measure of ∠COA.
问题：如图所示，圆O的直径与直线AB的长度相等，∠BOD=60°，⌒OCD的角度与⌒ODB的角度相等，⊙O的圆心为O。求∠COA的大小。
Construction (CDL):
Shape(CO,OA,OAC)
Shape(DO,OC,OCD)
Shape(BO,OD,ODB)
Shape(AO,OB,OBA)
Collinear(AOB)
Cocircular(O,ACDB)
Conditions (CDL):
Equal(DiameterOfCircle(O),LengthOfLine(AB))
Equal(MeasureOfAngle(BOD),60)
Equal(MeasureOfArc(OCD),MeasureOfArc(ODB))
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COA))
Answer: 60
Theorem sequence:
arc_addition_measure(1,OCD,ODB)
arc_property_center_angle(1,OCB,O)
arc_property_center_angle(1,ODB,O)
adjacent_complementary_angle(1,BOC,COA)
Problem: As shown in the diagram, ∠DCB=40°, the center of ⊙O is O, AB is the diameter of ⊙O. Find the measure of ∠ABD.
问题：如图所示，∠DCB=40°，圆O的圆心为O，AB是⊙O的直径。求∠ABD的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(DA,AO,OE,ED)
Shape(DE,EB,BD)
Shape(DB,OBD)
Shape(CE,EO,OA,OAC)
Shape(BE,EC,CB)
Shape(BC,OCB)
Collinear(AOEB)
Collinear(DEC)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(DCB),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ABD))
Answer: 50
Theorem sequence:
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)
triangle_property_angle_sum(1,DAB)
Problem: As shown in the diagram, ∠ABE=50°, the measure of arc OCF is equal to the measure of ⌒OEA, O is the center of ⊙O. Find the measure of ∠FDC.
问题：如图所示，∠ABE=50°，弧OCF的角度与⌒OEA的角度相等，O是圆O的圆心。求∠FDC的大小。
Construction (CDL):
Shape(DF,OFD)
Shape(FD,DO,OA,OAF)
Shape(BO,OD,ODB)
Shape(AO,OC,OCA)
Shape(CO,OB,BE,OEC)
Shape(EB,OBE)
Collinear(AOB)
Collinear(COD)
Cocircular(O,AFDBEC)
Conditions (CDL):
Equal(MeasureOfAngle(ABE),50)
Equal(MeasureOfArc(OCF),MeasureOfArc(OEA))
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(FDC))
Answer: 50
Theorem sequence:
arc_property_circumference_angle_external(1,OEA,B)
arc_property_circumference_angle_external(1,OCF,D)
Problem: As shown in the diagram, ∠COA=40°, O is the center of ⊙O, CD is the diameter of circle O, AE∥OD. Find the measure of arc ODE.
问题：如图所示，∠COA=40°，圆O的圆心为O，⊙O的直径为CD，AE∥OD。求⌒ODE的角度。
Construction (CDL):
Shape(CO,OA,OAC)
Shape(AE,OEA)
Shape(EA,AO,OE)
Shape(EO,OD,ODE)
Shape(DO,OC,OCD)
Collinear(COD)
Cocircular(O,ACDE)
Conditions (CDL):
Equal(MeasureOfAngle(COA),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(CD,O)
ParallelBetweenLine(AE,OD)
Goal: Value(MeasureOfArc(ODE))
Answer: 40
Theorem sequence:
parallel_property_collinear_extend(2,DO,EA,C)
parallel_property_alternate_interior_angle(1,OC,EA)
parallel_property_alternate_interior_angle(2,AE,OD)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OE,O)
isosceles_triangle_judgment_line_equal(1,OEA)
isosceles_triangle_property_angle_equal(1,OEA)
arc_property_center_angle(1,ODE,O)
Problem: As shown in the diagram, ∠CBA=25°, C is the center of ⊙C, △ACB is a right △. Find the measure of arc CDA.
问题：如图所示，∠CBA=25°，圆C的圆心为C，AC和CB是直角△ACB的两个直角边。求⌒CDA的角度。
Construction (CDL):
Shape(EC,CA,CAE)
Shape(AC,CD,DA)
Shape(AD,CDA)
Shape(DC,CE,CED)
Shape(EB,BD,CED)
Collinear(ADB)
Collinear(CEB)
Cocircular(C,AED)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),25)
IsCentreOfCircle(C,C)
RightTriangle(ACB)
Goal: Value(MeasureOfArc(CDA))
Answer: 50
Theorem sequence:
triangle_property_angle_sum(1,ACD)
triangle_property_angle_sum(1,ACB)
radius_of_circle_property_length_equal(1,CA,C)
radius_of_circle_property_length_equal(1,CD,C)
isosceles_triangle_judgment_line_equal(1,CDA)
isosceles_triangle_property_angle_equal(1,CDA)
arc_property_center_angle(1,CDA,C)
Problem: As shown in the diagram, ∠COA=50°, O is the center of circle O, AB is the diameter of ⊙O, CD is the diameter of ⊙O, AE∥OD. Find the measure of ∠AOE.
问题：如图所示，∠COA=50°，圆O的圆心为O，圆O的直径为AB，CD是圆O的直径，AE平行于OD。求∠AOE的大小。
Construction (CDL):
Shape(CO,OA,OAC)
Shape(AO,OE,EA)
Shape(AE,OEA)
Shape(EO,OD,ODE)
Shape(BO,OC,OCB)
Shape(DO,OB,OBD)
Collinear(AOB)
Collinear(COD)
Cocircular(O,CBDEA)
Conditions (CDL):
Equal(MeasureOfAngle(COA),50)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsDiameterOfCircle(CD,O)
ParallelBetweenLine(AE,OD)
Goal: Value(MeasureOfAngle(AOE))
Answer: 80
Theorem sequence:
parallel_property_collinear_extend(2,DO,EA,C)
parallel_property_alternate_interior_angle(1,OC,EA)
radius_of_circle_property_length_equal(1,OE,O)
radius_of_circle_property_length_equal(1,OA,O)
isosceles_triangle_judgment_line_equal(1,OEA)
isosceles_triangle_property_angle_equal(1,OEA)
triangle_property_angle_sum(1,AOE)
Problem: As shown in the diagram, the diameter of circle O is 4, ∠FPC=30°, O is the center of ⊙O, the diameter of ⊙O is AD. Find the length of line AB.
问题：如图所示，⊙O的直径为4，∠FPC=30°，圆O的圆心为O，圆O的直径为AD。求直线AB的长度。
Construction (CDL):
Shape(AC,CF,FB,OAB)
Shape(BA,OAB)
Shape(AB,BE,EA)
Shape(AE,EP,PA)
Shape(AP,OPA)
Shape(PE,EO,OD,ODP)
Shape(EB,BD,DO,OE)
Shape(DB,OBD)
Collinear(CAP)
Collinear(AEOD)
Collinear(FBEP)
Cocircular(O,ABDP)
Conditions (CDL):
Equal(DiameterOfCircle(O),4)
Equal(MeasureOfAngle(FPC),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AD,O)
Goal: Value(LengthOfLine(AB))
Answer: 2
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,D)
arc_property_circumference_angle_external(1,OAB,P)
diameter_of_circle_property_length_equal(1,AD,O)
diameter_of_circle_property_right_angle(1,ABD,O)
sine_theorem(1,ABD)
Problem: As shown in the diagram, ∠AOD=30°, ∠COB=50°, ∠DOC=40°, O is the center of ⊙O. Find the measure of arc OAB.
问题：如图所示，∠AOD=30°，∠COB=50°，∠DOC=40°，O是圆O的圆心。求⌒OAB的角度。
Construction (CDL):
Shape(BO,OA,OAB)
Shape(AO,OD,ODA)
Shape(DO,OC,OCD)
Shape(CO,OB,OBC)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(AOD),30)
Equal(MeasureOfAngle(COB),50)
Equal(MeasureOfAngle(DOC),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfArc(OAB))
Answer: 240
Theorem sequence:
angle_addition(1,AOD,DOC)
angle_addition(1,AOC,COB)
arc_property_center_angle(1,OBA,O)
round_arc(1,OAB,OBA)
Problem: As shown in the diagram, ∠CDA=53°, O is the center of ⊙O, the diameter of circle O is BE, AD and BC are opposite sides of the parallelogram ABCD. Find the measure of ∠BEA.
问题：如图所示，∠CDA=53°，O是圆O的圆心，BE是⊙O的直径，BA和CD是平行四边形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,OED)
Shape(FC,CE,EF)
Shape(EC,CO,OB,OBE)
Collinear(AFE)
Collinear(DFC)
Collinear(BOCE)
Cocircular(O,ABED)
Conditions (CDL):
Equal(MeasureOfAngle(CDA),53)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BE,O)
Parallelogram(ABCD)
Goal: Value(MeasureOfAngle(BEA))
Answer: 37
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, AO=OB, CO=7/2, the radius of circle C is 2, the center of ⊙C is C, ∠BPA is the right angle of the right triangle BPA. Find the length of line AB.
问题：如图所示，AO=OB，CO=7/2，圆C的半径为2，C是圆C的圆心，三角形BPA为直角三角形。求直线AB的长度。
Construction (CDL):
Shape(AO,OP,PA)
Shape(PO,OB,BP)
Shape(PC,CD,CDP)
Shape(DC,CP,CPD)
Collinear(AOB)
Collinear(DCPO)
Cocircular(C,PD)
Conditions (CDL):
Equal(LengthOfLine(AO),LengthOfLine(OB))
Equal(LengthOfLine(CO),7/2)
Equal(RadiusOfCircle(C),2)
IsCentreOfCircle(C,C)
RightTriangle(BPA)
Goal: Value(LengthOfLine(AB))
Answer: 3
Theorem sequence:
radius_of_circle_property_length_equal(1,CP,C)
line_addition(1,DC,CP)
line_addition(1,DC,CO)
line_addition(1,DP,PO)
median_of_triangle_judgment(1,PO,PAB)
right_triangle_property_length_of_median(1,BPA,O)
Problem: As shown in the diagram, ∠BAC=30°, O is the center of circle O, DE is perpendicular to BE. Find the measure of ∠ABD.
问题：如图所示，∠BAC=30°，⊙O的圆心为O，DE⊥BE。求∠ABD的大小。
Construction (CDL):
Shape(CA,OAC)
Shape(AE,ED,ODA)
Shape(AC,CE,EA)
Shape(DE,EB,BD)
Shape(DB,OBD)
Shape(BE,EC,OCB)
Collinear(AEB)
Collinear(DEC)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),30)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(DE,BE)
Goal: Value(MeasureOfAngle(ABD))
Answer: 60
Theorem sequence:
arc_property_circumference_angle_external(1,OCB,A)
arc_property_circumference_angle_external(1,OCB,D)
triangle_property_angle_sum(1,DEB)
Problem: As shown in the diagram, ∠BOA=60°, the center of circle O is O. Find the measure of ∠BCA.
问题：如图所示，∠BOA=60°，O是圆O的圆心。求∠BCA的大小。
Construction (CDL):
Shape(OA,AD,DO)
Shape(AO,OD,DC,OCA)
Shape(CD,DB,BC)
Shape(BD,DA,OAB)
Shape(CB,OBC)
Collinear(ADC)
Collinear(ODB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),60)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 30
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠AOC=72°, O is the center of circle O. Find the measure of ∠ABC.
问题：如图所示，∠AOC=72°，⊙O的圆心为O。求∠ABC的大小。
Construction (CDL):
Shape(AB,BD,DA)
Shape(AD,DC,OCA)
Shape(DO,OC,CD)
Shape(BA,OAB)
Shape(CO,OD,DB,OBC)
Collinear(BDC)
Collinear(ADO)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(AOC),72)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ABC))
Answer: 36
Theorem sequence:
arc_property_center_angle(1,OCA,O)
arc_property_circumference_angle_external(1,OCA,B)
Problem: As shown in the diagram, ∠OCA=55°, the center of ⊙O is O, the diameter of ⊙O is BA. Find the measure of ∠COB.
问题：如图所示，∠OCA=55°，圆O的圆心为O，⊙O的直径为BA。求∠COB的大小。
Construction (CDL):
Shape(BO,OA,OAB)
Shape(CO,OB,OBC)
Shape(OC,CA,AO)
Shape(AC,OCA)
Collinear(BOA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(OCA),55)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BA,O)
Goal: Value(MeasureOfAngle(COB))
Answer: 110
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OC,O)
isosceles_triangle_judgment_line_equal(1,OCA)
isosceles_triangle_property_angle_equal(1,OCA)
triangle_property_angle_sum(1,OCA)
adjacent_complementary_angle(1,AOC,COB)
Problem: As shown in the diagram, AB=5*sqrt(3), the radius of circle O is 5, O is the center of ⊙O, AD is perpendicular to OD. Find the measure of ∠BCA.
问题：如图所示，AB=5*sqrt(3)，⊙O的半径为5，O是⊙O的圆心，AD垂直于OD。求∠BCA的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AO,OB,BC)
Shape(CB,OBC)
Shape(BO,OD,DB)
Shape(OA,AD,DO)
Shape(BD,DA,OAB)
Collinear(BDA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(LengthOfLine(AB),5*sqrt(3))
Equal(RadiusOfCircle(O),5)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AD,OD)
Goal: Value(MeasureOfAngle(BCA))
Answer: 60
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)
altitude_of_triangle_judgment(1,OD,OAB)
isosceles_triangle_property_line_coincidence(1,OAB,D)
line_addition(1,BD,DA)
sine_theorem(1,BOD)
angle_addition(1,BOD,DOA)
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠ACO=42°, ∠OBA=22°, the center of circle O is O. Find the measure of ∠BOC.
问题：如图所示，∠ACO=42°，∠OBA=22°，⊙O的圆心为O。求∠BOC的大小。
Construction (CDL):
Shape(CA,OAC)
Shape(AC,CO,OA)
Shape(AO,OB,BA)
Shape(AB,OBA)
Shape(DO,OC,OCD)
Shape(BO,OD,ODB)
Collinear(AOD)
Cocircular(O,ACDB)
Conditions (CDL):
Equal(MeasureOfAngle(ACO),42)
Equal(MeasureOfAngle(OBA),22)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOC))
Answer: 128
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OC,O)
radius_of_circle_property_length_equal(1,OB,O)
isosceles_triangle_judgment_line_equal(1,OAC)
isosceles_triangle_judgment_line_equal(1,OBA)
isosceles_triangle_property_angle_equal(1,OAC)
isosceles_triangle_property_angle_equal(1,OBA)
triangle_property_angle_sum(1,ACO)
triangle_property_angle_sum(1,AOB)
adjacent_complementary_angle(1,AOB,BOD)
adjacent_complementary_angle(1,DOC,COA)
angle_addition(1,BOD,DOC)
Problem: As shown in the diagram, ∠BCA=50°, ∠COA=140°, the center of ⊙O is O. Find the measure of ∠CAB.
问题：如图所示，∠BCA=50°，∠COA=140°，⊙O的圆心为O。求∠CAB的大小。
Construction (CDL):
Shape(AO,OC,OCA)
Shape(OA,AD,DO)
Shape(OD,DC,CO)
Shape(DA,AB,BD)
Shape(CD,DB,BC)
Shape(CB,OBC)
Shape(BA,OAB)
Collinear(ODB)
Collinear(AOC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),50)
Equal(MeasureOfAngle(COA),140)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 20
Theorem sequence:
angle_addition(1,COD,DOA)
arc_property_center_angle(1,OAB,O)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OAB,C)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠BCA=25°, the center of circle O is O. Find the measure of ∠OAB.
问题：如图所示，∠BCA=25°，O是⊙O的圆心。求∠OAB的大小。
Construction (CDL):
Shape(AO,OD,DC,OCA)
Shape(OA,AD,DO)
Shape(DA,AB,BD)
Shape(CD,DB,BC)
Shape(CB,OBC)
Shape(BA,OAB)
Collinear(ODB)
Collinear(CDA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),25)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(OAB))
Answer: 65
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)
Problem: As shown in the diagram, Add(MeasureOfAngle(BAC)=MeasureOfAngle(DAE)), ED=6, the radius of circle A is 5, A is the center of ⊙A, the diameter of circle A is FC, CH is perpendicular to AH. Find the length of line HA.
问题：如图所示，Add(MeasureOfAngle(BAC)=MeasureOfAngle(DAE))，ED=6，⊙A的半径为5，⊙A的圆心为A，FC是⊙A的直径，CH⊥AH。求直线HA的长度。
Construction (CDL):
Shape(BH,HC,ACB)
Shape(CH,HA,AC)
Shape(AH,HB,BA)
Shape(AB,BF,FA)
Shape(CA,AD,ADC)
Shape(DA,AE,ED)
Shape(DE,AED)
Shape(EA,AF,AFE)
Shape(FB,ABF)
Collinear(FAC)
Collinear(CHB)
Cocircular(A,CBFED)
Conditions (CDL):
Equal(Add(MeasureOfAngle(BAC),MeasureOfAngle(DAE)),180)
Equal(LengthOfLine(ED),6)
Equal(RadiusOfCircle(A),5)
IsCentreOfCircle(A,A)
IsDiameterOfCircle(FC,A)
PerpendicularBetweenLine(CH,AH)
Goal: Value(LengthOfLine(HA))
Answer: 3
Theorem sequence:
adjacent_complementary_angle(1,FAB,BAC)
arc_property_center_angle(1,ABF,A)
arc_property_center_angle(1,AED,A)
congruent_arc_judgment_measure_equal(1,ABF,AED)
congruent_arc_property_chord_equal(1,ABF,AED)
radius_of_circle_property_length_equal(1,AC,A)
radius_of_circle_property_length_equal(1,AB,A)
radius_of_circle_property_length_equal(1,AF,A)
isosceles_triangle_judgment_line_equal(1,ACB)
altitude_of_triangle_judgment(1,AH,ACB)
isosceles_triangle_property_line_coincidence(1,ACB,H)
midsegment_of_triangle_judgment_midpoint(1,HA,CBF)
midsegment_of_triangle_property_length(1,HA,CBF)
Problem: As shown in the diagram, ∠CAB=20°, O is the center of circle O, AE⊥CE. Find the measure of ∠DOA.
问题：如图所示，∠CAB=20°，⊙O的圆心为O，AE⊥CE。求∠DOA的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(AO,OE,EC,CA)
Shape(CE,EB,OBC)
Shape(DO,OA,OAD)
Shape(OD,DE,EO)
Shape(BE,ED,ODB)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),20)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AE,CE)
Goal: Value(MeasureOfAngle(DOA))
Answer: 140
Theorem sequence:
triangle_property_angle_sum(1,CAE)
arc_property_circumference_angle_external(1,OAD,C)
arc_property_center_angle(1,OAD,O)
Problem: As shown in the diagram, ∠BCA=35°, the center of circle O is O. Find the measure of ∠BOA.
问题：如图所示，∠BCA=35°，⊙O的圆心为O。求∠BOA的大小。
Construction (CDL):
Shape(AO,OD,DC,OCA)
Shape(OA,AD,DO)
Shape(DB,BC,CD)
Shape(CB,OBC)
Shape(BD,DA,OAB)
Collinear(ODB)
Collinear(CDA)
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, ∠BAC=23°, O is the center of circle O, AB is the diameter of ⊙O. Find the measure of ∠CBA.
问题：如图所示，∠BAC=23°，O是⊙O的圆心，AB是⊙O的直径。求∠CBA的大小。
Construction (CDL):
Shape(CA,OAC)
Shape(AC,CB,BO,OA)
Shape(BC,OCB)
Shape(AO,OB,OBA)
Collinear(AOB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),23)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CBA))
Answer: 67
Theorem sequence:
diameter_of_circle_property_right_angle(1,ACB,O)
triangle_property_angle_sum(1,ACB)
Problem: As shown in the diagram, ∠CAB=50°, the center of circle O is O. Find the measure of ∠COB.
问题：如图所示，∠CAB=50°，⊙O的圆心为O。求∠COB的大小。
Construction (CDL):
Shape(BO,OD,DA,OAB)
Shape(OB,BD,DO)
Shape(DC,CA,AD)
Shape(AC,OCA)
Shape(CD,DB,OBC)
Collinear(ODC)
Collinear(ADB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),50)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COB))
Answer: 100
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠BOD=44°, the center of ⊙O is O. Find the measure of ∠BCD.
问题：如图所示，∠BOD=44°，圆O的圆心为O。求∠BCD的大小。
Construction (CDL):
Shape(AO,OE,EC,OCA)
Shape(CE,EB,BC)
Shape(CB,OBC)
Shape(DO,OA,OAD)
Shape(EO,OD,DE)
Shape(BE,ED,ODB)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(BOD),44)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCD))
Answer: 22
Theorem sequence:
arc_property_center_angle(1,ODB,O)
arc_property_circumference_angle_external(1,ODB,C)
Problem: As shown in the diagram, AB=8, ∠DBC=30°, O is the center of circle O, the diameter of circle O is AB. Find the length of line CD.
问题：如图所示，AB=8，∠DBC=30°，O是圆O的圆心，AB是圆O的直径。求直线CD的长度。
Construction (CDL):
Shape(AO,OD,ODA)
Shape(DO,OE,ED)
Shape(CD,DE,EC)
Shape(DC,OCD)
Shape(CE,EB,BC)
Shape(CB,OBC)
Shape(EO,OB,BE)
Shape(BO,OA,OAB)
Collinear(AOB)
Collinear(DEB)
Collinear(CEO)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(MeasureOfAngle(DBC),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(LengthOfLine(CD))
Answer: 4
Theorem sequence:
arc_property_center_angle(1,OCD,O)
arc_property_circumference_angle_external(1,OCD,B)
diameter_of_circle_property_length_equal(1,AB,O)
circle_property_length_of_radius_and_diameter(1,O)
radius_of_circle_property_length_equal(1,OC,O)
radius_of_circle_property_length_equal(1,OD,O)
cosine_theorem(1,OCD)
Problem: As shown in the diagram, ∠CAB=42°, O is the center of ⊙O, AB is the diameter of ⊙O, D bisects segment CA. Find the measure of ∠DOC.
问题：如图所示，∠CAB=42°，⊙O的圆心为O，圆O的直径为AB，D是线段CA的中点。求∠DOC的大小。
Construction (CDL):
Shape(AD,DC,OCA)
Shape(DA,AO,OD)
Shape(CD,DO,OC)
Shape(CO,OB,BC)
Shape(CB,OBC)
Shape(BO,OA,OAB)
Collinear(ADC)
Collinear(AOB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),42)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsMidpointOfLine(D,CA)
Goal: Value(MeasureOfAngle(DOC))
Answer: 48
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OC,O)
isosceles_triangle_judgment_line_equal(1,OCA)
isosceles_triangle_property_angle_equal(1,OCA)
median_of_triangle_judgment(1,OD,OCA)
isosceles_triangle_property_line_coincidence(2,OCA,D)
triangle_property_angle_sum(1,CDO)
Problem: As shown in the diagram, the diameter of circle O is 6, ∠BCA=30°, the center of circle O is O, CB is the diameter of ⊙O. Find the length of line AB.
问题：如图所示，圆O的直径为6，∠BCA=30°，O是圆O的圆心，⊙O的直径为CB。求直线AB的长度。
Construction (CDL):
Shape(CB,OBC)
Shape(CA,AO,OC)
Shape(OA,AB,BO)
Shape(AC,OCA)
Shape(BA,OAB)
Collinear(COB)
Cocircular(O,CAB)
Conditions (CDL):
Equal(DiameterOfCircle(O),6)
Equal(MeasureOfAngle(BCA),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(CB,O)
Goal: Value(LengthOfLine(AB))
Answer: 3
Theorem sequence:
diameter_of_circle_property_length_equal(1,CB,O)
diameter_of_circle_property_right_angle(1,CAB,O)
sine_theorem(1,BCA)
Problem: As shown in the diagram, ∠BCA=112°, O is the center of ⊙O. Find the measure of ∠AOB.
问题：如图所示，∠BCA=112°，O是⊙O的圆心。求∠AOB的大小。
Construction (CDL):
Shape(CB,OBC)
Shape(AC,OCA)
Shape(CA,AO,OB,BC)
Shape(BO,OA,OAB)
Cocircular(O,BCA)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),112)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AOB))
Answer: 136
Theorem sequence:
arc_property_circumference_angle_internal(1,OBA,C)
arc_property_center_angle(1,OBA,O)
Problem: As shown in the diagram, ∠BCO=50°, ∠BOA=40°, the center of circle O is O. Find the measure of ∠OAC.
问题：如图所示，∠BCO=50°，∠BOA=40°，⊙O的圆心为O。求∠OAC的大小。
Construction (CDL):
Shape(OA,AD,DO)
Shape(OD,DC,CO)
Shape(CD,DB,BC)
Shape(BD,DA,OAB)
Shape(AO,OC,OCA)
Collinear(ADC)
Collinear(ODB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCO),50)
Equal(MeasureOfAngle(BOA),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(OAC))
Answer: 30
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,OBC)
isosceles_triangle_judgment_line_equal(1,OAC)
isosceles_triangle_property_angle_equal(1,OBC)
triangle_property_angle_sum(1,OBC)
angle_addition(1,COD,DOA)
isosceles_triangle_property_angle_equal(1,OAC)
triangle_property_angle_sum(1,OAC)
Problem: As shown in the diagram, ∠ACD=40°, O is the center of circle O. Find the measure of ∠DAB.
问题：如图所示，∠ACD=40°，圆O的圆心为O。求∠DAB的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(AE,ED,DA)
Shape(DE,EB,BD)
Shape(DB,OBD)
Shape(CA,OAC)
Shape(AC,CE,EA)
Shape(BE,EC,OCB)
Collinear(AEOB)
Collinear(DEC)
Cocircular(O,DACB)
Conditions (CDL):
Equal(MeasureOfAngle(ACD),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DAB))
Answer: 50
Theorem sequence:
diameter_of_circle_judgment_pass_centre(1,AOB,O)
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)
triangle_property_angle_sum(1,DAB)
Problem: As shown in the diagram, ∠BCA=32°, O is the center of ⊙O. Find the measure of ∠BOA.
问题：如图所示，∠BCA=32°，O是⊙O的圆心。求∠BOA的大小。
Construction (CDL):
Shape(OA,AD,DO)
Shape(CD,DB,BC)
Shape(BD,DA,OAB)
Shape(OD,DC,CO)
Shape(CB,OBC)
Shape(AO,OC,OCA)
Collinear(ADC)
Collinear(ODB)
Cocircular(O,CAB)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),32)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 64
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,C)
arc_property_center_angle(1,OAB,O)
Problem: As shown in the diagram, ∠DAB=32°, the center of ⊙O is O. Find the measure of ∠COB.
问题：如图所示，∠DAB=32°，⊙O的圆心为O。求∠COB的大小。
Construction (CDL):
Shape(OA,AD,DO)
Shape(DA,AB,BD)
Shape(BA,OAB)
Shape(CD,DB,DBC)
Shape(CO,OD,DC)
Shape(AO,OC,OCA)
Collinear(ADC)
Collinear(ODB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),32)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COB))
Answer: 64
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,OBC,O)