SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, CB=CD, ∠BCA=90°, ∠CAB=26°, the center of circle C is C. Find the measure of ∠BCD.
问题：如图所示，CB=CD，∠BCA=90°，∠CAB=26°，⊙C的圆心为C。求∠BCD的大小。
Construction (CDL):
Shape(BC,CD,DB)
Shape(CDB,BD)
Shape(DC,CE,CED)
Shape(EA,AD,CED)
Shape(CB,CBE,EC)
Collinear(BDA)
Collinear(CEA)
Cocircular(C,EDB)
Conditions (CDL):
Equal(LengthOfLine(CB),LengthOfLine(CD))
Equal(MeasureOfAngle(BCA),90)
Equal(MeasureOfAngle(CAB),26)
IsCentreOfCircle(C,C)
Goal: Value(MeasureOfAngle(BCD))
Answer: 52
Theorem sequence:
triangle_property_angle_sum(1,BCD)
triangle_property_angle_sum(1,BCA)
isosceles_triangle_judgment_line_equal(1,CDB)
isosceles_triangle_property_angle_equal(1,CDB)
Problem: As shown in the diagram, ∠OAB=50°, O is the center of circle O, C is the midpoint of ⌒OAB, AD⊥OD. Find the measure of ∠BOC.
问题：如图所示，∠OAB=50°，O是⊙O的圆心，C是⌒OAB的中点，AD⊥OD。求∠BOC的大小。
Construction (CDL):
Shape(OBA,AO,OB)
Shape(OA,AD,DO)
Shape(OD,DB,BO)
Shape(OAC,CD,DA)
Shape(OCB,BD,DC)
Collinear(ADB)
Collinear(ODC)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(OAB),50)
IsCentreOfCircle(O,O)
IsMidpointOfArc(C,OAB)
PerpendicularBetweenLine(AD,OD)
Goal: Value(MeasureOfAngle(BOC))
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,BOA)
circle_property_chord_perpendicular_bisect_chord(1,O,OD,AB)
median_of_triangle_judgment(1,OD,OAB)
isosceles_triangle_property_line_coincidence(2,OAB,D)
angle_addition(1,BOD,DOA)
Problem: As shown in the diagram, ∠CBA=30°, O is the center of ⊙O, the diameter of circle O is AB, OE⊥BE. Find the measure of ∠BCD.
问题：如图所示，∠CBA=30°，圆O的圆心为O，圆O的直径为AB，OE⊥BE。求∠BCD的大小。
Construction (CDL):
Shape(OCD,DC)
Shape(ODB,BE,ED)
Shape(EB,BO,OE)
Shape(DE,EC,CD)
Shape(EO,OA,AC,CE)
Shape(OAC,CA)
Shape(OBA,AO,OB)
Collinear(DEO)
Collinear(CEB)
Collinear(BOA)
Cocircular(O,ACDB)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
PerpendicularBetweenLine(OE,BE)
Goal: Value(MeasureOfAngle(BCD))
Answer: 30
Theorem sequence:
triangle_property_angle_sum(1,OEB)
arc_property_center_angle(1,ODB,O)
arc_property_circumference_angle_external(1,ODB,C)
Problem: As shown in the diagram, AB=12, the radius of circle O is 10, the center of ⊙O is O, BD is perpendicular to OD. Find the length of line CD.
问题：如图所示，AB=12，圆O的半径为10，O是圆O的圆心，BD垂直于OD。求直线CD的长度。
Construction (CDL):
Shape(OCA,AD,DC)
Shape(OBC,CD,DB)
Shape(OD,DA,AO)
Shape(OB,BD,DO)
Collinear(ADB)
Collinear(CDO)
Cocircular(O,BCA)
Conditions (CDL):
Equal(LengthOfLine(AB),12)
Equal(RadiusOfCircle(O),10)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(BD,OD)
Goal: Value(LengthOfLine(CD))
Answer: 2
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
circle_property_chord_perpendicular_bisect_chord(1,O,OD,BA)
line_addition(1,AD,DB)
right_triangle_judgment_angle(1,ODA)
right_triangle_property_pythagorean(1,ODA)
radius_of_circle_property_length_equal(1,OC,O)
line_addition(1,CD,DO)
Problem: As shown in the diagram, ∠CAB=25°, the center of circle O is O, the diameter of ⊙O is AB. Find the measure of ∠ADC.
问题：如图所示，∠CAB=25°，O是⊙O的圆心，AB是⊙O的直径。求∠ADC的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(CA,AE,EC)
Shape(BC,CE,EB)
Shape(OBC,CB)
Shape(EA,AD,DE)
Shape(OAD,DA)
Shape(ODB,BE,ED)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),25)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ADC))
Answer: 65
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,ABC)
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_external(1,OCA,D)
Problem: As shown in the diagram, ∠BCA=90°, ∠CAB=25°, the center of circle C is C. Find the measure of arc CDB.
问题：如图所示，∠BCA=90°，∠CAB=25°，C是圆C的圆心。求弧CDB的角度。
Construction (CDL):
Shape(CDB,BD)
Shape(CBE,EC,CB)
Shape(CED,DC,CE)
Shape(CD,DB,BC)
Shape(EA,AD,CED)
Collinear(BDA)
Collinear(CEA)
Cocircular(C,EDB)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),90)
Equal(MeasureOfAngle(CAB),25)
IsCentreOfCircle(C,C)
Goal: Value(MeasureOfArc(CDB))
Answer: 50
Theorem sequence:
triangle_property_angle_sum(1,CAB)
radius_of_circle_property_length_equal(1,CB,C)
radius_of_circle_property_length_equal(1,CD,C)
isosceles_triangle_judgment_line_equal(1,CDB)
isosceles_triangle_property_angle_equal(1,CDB)
triangle_property_angle_sum(1,CDB)
arc_property_center_angle(1,CDB,C)
Problem: As shown in the diagram, ∠BOA=80°, O is the center of circle O. Find the measure of ∠BCA.
问题：如图所示，∠BOA=80°，O是圆O的圆心。求∠BCA的大小。
Construction (CDL):
Shape(OBC,CB)
Shape(BC,CD,DB)
Shape(OA,AD,DO)
Shape(OAB,BD,DA)
Shape(OD,DC,OCA,AO)
Collinear(ODB)
Collinear(ADC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),80)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 40
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠CAB=20°, The ratio of similar arcs OCA and OCD is 1/2, O is the center of ⊙O, AB is the diameter of ⊙O, D bisects arc OCA. Find the measure of ∠DAC.
问题：如图所示，∠CAB=20°，⌒OCA和⌒OCD的相似比为1/2，O是⊙O的圆心，AB是圆O的直径，D平分弧OCA。求∠DAC的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(DA,AE,ED)
Shape(OCD,DE,EC)
Shape(OBC,CB)
Shape(BC,CE,EB)
Shape(OB,BE,EA,AO)
Collinear(AOB)
Collinear(DEB)
Collinear(AEC)
Cocircular(O,BCDA)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),20)
Equal(RatioOfSimilarArc(OCA,OCD),1/2)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsMidpointOfArc(D,OCA)
Goal: Value(MeasureOfAngle(DAC))
Answer: 35
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,ABC)
arc_property_circumference_angle_external(1,OCA,B)
similar_arc_judgment_cocircular(1,OCD,OCA)
similar_arc_property_measure_ratio(1,OCD,OCA)
arc_property_circumference_angle_external(1,OCD,A)
Problem: As shown in the diagram, BC=4, CD=4, DA=4, O is the center of circle O, the diameter of circle O is AB. Find the circumference of the circle O.
问题：如图所示，BC=4，CD=4，DA=4，圆O的圆心为O，⊙O的直径为AB。求⊙O的周长。
Construction (CDL):
Shape(ODA,AD)
Shape(OCD,DC)
Shape(OCB,BC)
Shape(OD,DA,AO)
Shape(OC,CD,DO)
Shape(OB,BC,CO)
Shape(OAB,BO,OA)
Collinear(AOB)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(LengthOfLine(BC),4)
Equal(LengthOfLine(CD),4)
Equal(LengthOfLine(DA),4)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(PerimeterOfCircle(O))
Answer: 8*pi
Theorem sequence:
congruent_arc_judgment_chord_equal(1,OCD,OBC)
congruent_arc_judgment_chord_equal(1,ODA,OBC)
congruent_arc_property_measure_equal(1,OCD,OBC)
congruent_arc_property_measure_equal(1,ODA,OBC)
arc_property_center_angle(1,ODA,O)
arc_property_center_angle(1,OCD,O)
arc_property_center_angle(1,OBC,O)
angle_addition(1,AOD,DOC)
adjacent_complementary_angle(1,AOC,COB)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OD,O)
isosceles_triangle_judgment_line_equal(1,ODA)
isosceles_triangle_property_angle_equal(1,ODA)
triangle_property_angle_sum(1,ODA)
isosceles_triangle_judgment_angle_equal(1,AOD)
circle_perimeter_formula(1,O)
Problem: As shown in the diagram, the length of arc OAB is equal to the length of ⌒OCA, ∠BCA=80°, the center of circle O is O. Find the measure of ∠CAB.
问题：如图所示，⌒OAB的长度与弧OCA的长度相等，∠BCA=80°，O是圆O的圆心。求∠CAB的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(OAB,BA)
Shape(OBC,CB)
Shape(AB,BC,CA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(LengthOfArc(OAB),LengthOfArc(OCA))
Equal(MeasureOfAngle(BCA),80)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 20
Theorem sequence:
congruent_arc_judgment_length_equal(1,OAB,OCA)
congruent_arc_property_measure_equal(1,OAB,OCA)
arc_property_circumference_angle_external(1,OAB,C)
arc_property_circumference_angle_external(1,OCA,B)
triangle_property_angle_sum(1,ABC)
Problem: As shown in the diagram, the length of arc OBC is equal to the length of ⌒OCD, the length of ⌒OCD is equal to the length of ⌒ODE, ∠COB=40°, the center of circle O is O, AB is the diameter of ⊙O. Find the measure of ∠AOE.
问题：如图所示，弧OBC的长度与弧OCD的长度相等，⌒OCD的长度与⌒ODE的长度相等，∠COB=40°，圆O的圆心为O，AB是⊙O的直径。求∠AOE的大小。
Construction (CDL):
Shape(OAB,BO,OA)
Shape(OBC,CO,OB)
Shape(OCD,DO,OC)
Shape(ODE,EO,OD)
Shape(OEA,AO,OE)
Collinear(AOB)
Cocircular(O,ABCDE)
Conditions (CDL):
Equal(LengthOfArc(OBC),LengthOfArc(OCD))
Equal(LengthOfArc(OCD),LengthOfArc(ODE))
Equal(MeasureOfAngle(COB),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(AOE))
Answer: 60
Theorem sequence:
arc_property_center_angle(1,OBC,O)
congruent_arc_judgment_length_equal(1,OBC,OCD)
congruent_arc_judgment_length_equal(1,OCD,ODE)
congruent_arc_property_measure_equal(1,OBC,OCD)
congruent_arc_property_measure_equal(1,OCD,ODE)
arc_property_center_angle(1,OCD,O)
arc_property_center_angle(1,ODE,O)
angle_addition(1,EOD,DOC)
angle_addition(1,EOC,COB)
adjacent_complementary_angle(1,AOE,EOB)
Problem: As shown in the diagram, the measure of ⌒OCD is 62, the center of ⊙O is O, the diameter of ⊙O is AB, AD is parallel to OC. Find the measure of ⌒ODA.
问题：如图所示，弧OCD的角度为62，圆O的圆心为O，AB是⊙O的直径，AD∥OC。求弧ODA的角度。
Construction (CDL):
Shape(ODA,AD)
Shape(AO,OD,DA)
Shape(OCD,DE,EC)
Shape(OBC,CB)
Shape(OE,ED,DO)
Shape(BC,CE,EB)
Shape(OB,BE,EO)
Collinear(AOB)
Collinear(DEB)
Collinear(CEO)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfArc(OCD),62)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
ParallelBetweenLine(AD,OC)
Goal: Value(MeasureOfArc(ODA))
Answer: 56
Theorem sequence:
arc_property_center_angle(1,OCD,O)
parallel_property_alternate_interior_angle(2,AD,OC)
radius_of_circle_property_length_equal(1,OD,O)
radius_of_circle_property_length_equal(1,OA,O)
isosceles_triangle_judgment_line_equal(1,ODA)
isosceles_triangle_property_angle_equal(1,ODA)
triangle_property_angle_sum(1,ODA)
arc_property_center_angle(1,ODA,O)
Problem: As shown in the diagram, ∠FCD=60°, FB∥CD. Find the sum of the measure of ∠EAF and the measure of ∠FEA.
问题：如图所示，∠FCD=60°，FB平行于CD。求∠EAF的大小与∠FEA的大小之和。
Construction (CDL):
Shape(AF,FE,EA)
Shape(EF,FB)
Shape(FC,CD)
Collinear(EFC)
Collinear(AFB)
Conditions (CDL):
Equal(MeasureOfAngle(FCD),60)
ParallelBetweenLine(FB,CD)
Goal: Value(Add(MeasureOfAngle(EAF),MeasureOfAngle(FEA)))
Answer: 60
Theorem sequence:
parallel_property_corresponding_angle(1,FB,CD,E)
adjacent_complementary_angle(1,AFE,EFB)
triangle_property_angle_sum(1,AFE)
Problem: As shown in the diagram, the diameter of circle O is 100, AB=60, O is the center of ⊙O, AC is perpendicular to OC. Find the length of line CD.
问题：如图所示，圆O的直径为100，AB=60，O是⊙O的圆心，AC⊥OC。求直线CD的长度。
Construction (CDL):
Shape(OAB,BA)
Shape(OC,CB,BO)
Shape(OBD,DO,OB)
Shape(ODA,AC,CO,OD)
Collinear(ACB)
Collinear(COD)
Cocircular(O,ABD)
Conditions (CDL):
Equal(DiameterOfCircle(O),100)
Equal(LengthOfLine(AB),60)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AC,OC)
Goal: Value(LengthOfLine(CD))
Answer: 90
Theorem sequence:
circle_property_chord_perpendicular_bisect_chord(1,O,OC,AB)
right_triangle_judgment_angle(1,OCB)
line_addition(1,AC,CB)
circle_property_length_of_radius_and_diameter(1,O)
radius_of_circle_property_length_equal(1,OB,O)
right_triangle_property_pythagorean(1,OCB)
radius_of_circle_property_length_equal(1,OD,O)
line_addition(1,CO,OD)
Problem: As shown in the diagram, ∠CDA=56°, the center of ⊙O is O, BE is the diameter of ⊙O, quadrilateral ABCD is a ▱. Find the measure of ∠BEA.
问题：如图所示，∠CDA=56°，⊙O的圆心为O，BE是⊙O的直径，AD和BC是▱ABCD的一组对边。求∠BEA的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(ODA,AD)
Shape(DA,AF,FD)
Shape(AB,BO,OC,CF,FA)
Shape(OED,DF,FE)
Shape(CE,EF,FC)
Shape(OBE,EC,CO,OB)
Collinear(BOCE)
Collinear(AFE)
Collinear(DFC)
Cocircular(O,ABED)
Conditions (CDL):
Equal(MeasureOfAngle(CDA),56)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BE,O)
Parallelogram(ABCD)
Goal: Value(MeasureOfAngle(BEA))
Answer: 34
Theorem sequence:
parallelogram_property_opposite_angle_equal(1,DABC)
diameter_of_circle_property_right_angle(1,EAB,O)
triangle_property_angle_sum(1,ABE)
Problem: As shown in the diagram, ∠DCA=45°, ∠DMA=75°, the center of circle O is O. Find the measure of ∠CDB.
问题：如图所示，∠DCA=45°，∠DMA=75°，O是⊙O的圆心。求∠CDB的大小。
Construction (CDL):
Shape(O)
Shape(OCA,AC)
Shape(CA,AM,MC)
Shape(OBC,CM,MB)
Shape(ODB,BD)
Shape(BM,MD,DB)
Shape(OAD,DM,MA)
Collinear(AMB)
Collinear(CMD)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(DCA),45)
Equal(MeasureOfAngle(DMA),75)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CDB))
Answer: 30
Theorem sequence:
arc_property_circumference_angle_external(1,OAD,C)
arc_property_circumference_angle_external(1,OAD,B)
adjacent_complementary_angle(1,BMD,DMA)
triangle_property_angle_sum(1,BMD)
Problem: As shown in the diagram, ∠COA=100°, the center of circle O is O. Find the measure of ∠ABC.
问题：如图所示，∠COA=100°，圆O的圆心为O。求∠ABC的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(ODA,AD)
Shape(OC,CD,DA,AO)
Shape(OA,AB,BC,CO)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(COA),100)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ABC))
Answer: 130
Theorem sequence:
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_internal(1,OAC,B)
Problem: As shown in the diagram, ∠BAO=25°, O is the center of ⊙O, AC is parallel to OB. Find the measure of ∠COB.
问题：如图所示，∠BAO=25°，⊙O的圆心为O，AC平行于OB。求∠COB的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(CA,AD,DC)
Shape(OBC,CD,DB)
Shape(OB,BD,DO)
Shape(OD,DA,AO)
Shape(OAB,BO,OA)
Collinear(CDO)
Collinear(ADB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BAO),25)
IsCentreOfCircle(O,O)
ParallelBetweenLine(AC,OB)
Goal: Value(MeasureOfAngle(COB))
Answer: 50
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)
parallel_property_alternate_interior_angle(1,AC,OB)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,OBC,O)
Problem: As shown in the diagram, ∠CAB=15°, the radius of ⊙O is 2, O is the center of circle O, AB is the diameter of ⊙O, DE⊥OE. Find the length of line CD.
问题：如图所示，∠CAB=15°，⊙O的半径为2，圆O的圆心为O，AB是⊙O的直径，DE⊥OE。求直线CD的长度。
Construction (CDL):
Shape(OCA,AC)
Shape(OC,CA,AO)
Shape(OE,EC,CO)
Shape(OBC,CE,EB)
Shape(ODB,BE,ED)
Shape(OAD,DE,EO,OA)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),15)
Equal(RadiusOfCircle(O),2)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
PerpendicularBetweenLine(DE,OE)
Goal: Value(LengthOfLine(CD))
Answer: 2
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,COE)
circle_property_chord_perpendicular_bisect_chord(1,O,OE,DC)
sine_theorem(1,COE)
line_addition(1,CE,ED)
Problem: As shown in the diagram, ∠BAC=36°, ∠OCD=27°, the center of circle O is O. Find the measure of ∠DBA.
问题：如图所示，∠BAC=36°，∠OCD=27°，O是⊙O的圆心。求∠DBA的大小。
Construction (CDL):
Shape(OCB,BD,DC)
Shape(OC,CD,DO)
Shape(OBA,AB)
Shape(BA,AD,DB)
Shape(OAC,CO,OD,DA)
Collinear(BDO)
Collinear(CDA)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),36)
Equal(MeasureOfAngle(OCD),27)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DBA))
Answer: 63
Theorem sequence:
arc_property_circumference_angle_external(1,OCB,A)
arc_property_center_angle(1,OCB,O)
triangle_property_angle_sum(1,OCD)
adjacent_complementary_angle(1,CDO,ODA)
adjacent_complementary_angle(1,ODA,ADB)
triangle_property_angle_sum(1,ADB)
Problem: As shown in the diagram, ∠EDB=30°, the radius of ⊙O is 6, the center of circle O is O, AB is the diameter of ⊙O, OE is perpendicular to CE. Find the length of line OE.
问题：如图所示，∠EDB=30°，⊙O的半径为6，⊙O的圆心为O，AB是⊙O的直径，OE⊥CE。求直线OE的长度。
Construction (CDL):
Shape(OCA,AO,OC)
Shape(OE,EC,CO)
Shape(OBC,CB)
Shape(EB,BC,CE)
Shape(ODB,BD)
Shape(ED,DB,BE)
Shape(OAD,DE,EO,OA)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(EDB),30)
Equal(RadiusOfCircle(O),6)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
PerpendicularBetweenLine(OE,CE)
Goal: Value(LengthOfLine(OE))
Answer: 3
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,D)
arc_property_center_angle(1,OBC,O)
right_triangle_judgment_angle(1,OEC)
radius_of_circle_property_length_equal(1,OC,O)
triangle_property_angle_sum(1,COE)
sine_theorem(1,ECO)
right_triangle_property_pythagorean(1,OEC)
Problem: As shown in the diagram, ∠BOA=64°, O is the center of circle O. Find the measure of ∠BCA.
问题：如图所示，∠BOA=64°，⊙O的圆心为O。求∠BCA的大小。
Construction (CDL):
Shape(AO,OD,DC,OCA)
Shape(AD,DO,OA)
Shape(DA,AB,BD)
Shape(DB,BC,CD)
Shape(BA,OAB)
Shape(CB,OBC)
Collinear(ODB)
Collinear(ADC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),64)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 32
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠AOB=70°, O is the center of ⊙O, BE⊥OE. Find the measure of ∠CDA.
问题：如图所示，∠AOB=70°，O是圆O的圆心，BE⊥OE。求∠CDA的大小。
Construction (CDL):
Shape(DC,OCD)
Shape(CD,DH,HC)
Shape(ODB,BO,OH,HD)
Shape(EO,OB,BE)
Shape(AE,EB,OBA)
Shape(AF,FE,EA)
Shape(CF,FA,OAC)
Shape(CH,HF,FC)
Shape(FH,HO,OE,EF)
Collinear(CFEB)
Collinear(AFHD)
Collinear(CHO)
Collinear(AEO)
Cocircular(O,ACDB)
Conditions (CDL):
Equal(MeasureOfAngle(AOB),70)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(BE,OE)
Goal: Value(MeasureOfAngle(CDA))
Answer: 35
Theorem sequence:
radius_of_circle_property_length_equal(1,OC,O)
radius_of_circle_property_length_equal(1,OB,O)
adjacent_complementary_angle(1,BEO,OEC)
mirror_congruent_triangle_judgment_hl(1,BEO,COE)
mirror_congruent_triangle_property_angle_equal(1,OBE,OEC)
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,D)
Problem: As shown in the diagram, ∠CBD=30°, the center of ⊙O is O, the diameter of ⊙O is DB. Find the measure of ∠BAC.
问题：如图所示，∠CBD=30°，圆O的圆心为O，DB是圆O的直径。求∠BAC的大小。
Construction (CDL):
Shape(AB,OBA)
Shape(BA,AH,HO,OB)
Shape(BO,OH,HC,CB)
Shape(HD,DC,CH)
Shape(BC,OCB)
Shape(DH,HA,OAD)
Shape(CD,ODC)
Collinear(BOHD)
Collinear(AHC)
Cocircular(O,ADCB)
Conditions (CDL):
Equal(MeasureOfAngle(CBD),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(DB,O)
Goal: Value(MeasureOfAngle(BAC))
Answer: 60
Theorem sequence:
diameter_of_circle_property_right_angle(1,DCB,O)
triangle_property_angle_sum(1,CBD)
arc_property_circumference_angle_external(1,OCB,A)
arc_property_circumference_angle_external(1,OCB,D)
Problem: As shown in the diagram, the length of ⌒OAB is equal to the length of arc OBC, ∠CDB=25°, O is the center of circle O. Find the measure of ∠BOA.
问题：如图所示，弧OAB的长度与⌒OBC的长度相等，∠CDB=25°，圆O的圆心为O。求∠BOA的大小。
Construction (CDL):
Shape(BO,OA,OAB)
Shape(CD,DB,OBC)
Shape(DC,OCD)
Shape(AO,OB,BD,ODA)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(LengthOfArc(OAB),LengthOfArc(OBC))
Equal(MeasureOfAngle(CDB),25)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 50
Theorem sequence:
congruent_arc_judgment_length_equal(1,OAB,OBC)
congruent_arc_property_measure_equal(1,OAB,OBC)
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OBC,D)
Problem: As shown in the diagram, ∠DAC=110°, the center of ⊙O is O, △ACD is mirror congruent to △EDC. Find the measure of ∠DBC.
问题：如图所示，∠DAC=110°，O是⊙O的圆心，△ACD镜像全等于△EDC。求∠DBC的大小。
Construction (CDL):
Shape(CB,OBC)
Shape(CA,AD,DB,BC)
Shape(BD,ODB)
Shape(CD,DA,AC)
Shape(CE,ED,DC)
Shape(EC,OCE)
Shape(DE,OED)
Cocircular(O,CEDB)
Conditions (CDL):
Equal(MeasureOfAngle(DAC),110)
IsCentreOfCircle(O,O)
MirrorCongruentBetweenTriangle(ACD,EDC)
Goal: Value(MeasureOfAngle(DBC))
Answer: 70
Theorem sequence:
mirror_congruent_triangle_property_angle_equal(1,ACD,EDC)
arc_property_circumference_angle_internal(1,OCD,E)
arc_property_circumference_angle_external(1,OCD,B)
Problem: As shown in the diagram, ∠CAB=40°, O is the center of ⊙O, the diameter of ⊙O is AB, CH is perpendicular to BH. Find the sum of the measure of ∠DOA and the measure of ∠DBA.
问题：如图所示，∠CAB=40°，⊙O的圆心为O，⊙O的直径为AB，CH⊥BH。求∠DOA的大小与∠DBA的大小之和。
Construction (CDL):
Shape(AC,OCA)
Shape(AO,OH,HC,CA)
Shape(CH,HB,OBC)
Shape(DO,OA,OAD)
Shape(DH,HO,OD)
Shape(DB,BH,HD)
Shape(BD,ODB)
Collinear(AOHB)
Collinear(CHD)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
PerpendicularBetweenLine(CH,BH)
Goal: Value(Add(MeasureOfAngle(DOA),MeasureOfAngle(DBA)))
Answer: 150
Theorem sequence:
arc_property_circumference_angle_external(1,OAD,C)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_external(1,OBC,D)
adjacent_complementary_angle(1,AHC,CHB)
adjacent_complementary_angle(1,CHB,BHD)
triangle_property_angle_sum(1,AHC)
triangle_property_angle_sum(1,DBH)
arc_property_center_angle(1,OAD,O)
Problem: As shown in the diagram, ∠AED=20°, the center of circle O is O, the diameter of ⊙O is AB. Find the measure of ∠BCD.
问题：如图所示，∠AED=20°，⊙O的圆心为O，AB是⊙O的直径。求∠BCD的大小。
Construction (CDL):
Shape(EA,OAE)
Shape(AE,EF,FA)
Shape(AF,FG,GA)
Shape(AG,GD,ODA)
Shape(DG,GC,CD)
Shape(DC,OCD)
Shape(BC,CG,GF,FO,OB)
Shape(CB,OBC)
Shape(BO,OF,FE,OEB)
Collinear(AFOB)
Collinear(EFGD)
Collinear(AGC)
Cocircular(O,EBCDA)
Conditions (CDL):
Equal(MeasureOfAngle(AED),20)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BCD))
Answer: 110
Theorem sequence:
arc_property_circumference_angle_external(1,ODA,E)
arc_property_circumference_angle_external(1,ODA,C)
diameter_of_circle_property_right_angle(1,BCA,O)
angle_addition(1,BCA,ACD)
Problem: As shown in the diagram, ∠OCA=20°, the center of circle O is O, the diameter of ⊙O is AB. Find the measure of ∠COB.
问题：如图所示，∠OCA=20°，O是⊙O的圆心，⊙O的直径为AB。求∠COB的大小。
Construction (CDL):
Shape(AO,OC,CA)
Shape(AC,OCA)
Shape(CO,OB,OBC)
Shape(BO,OA,OAB)
Collinear(AOB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(OCA),20)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(COB))
Answer: 40
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,AOC)
adjacent_complementary_angle(1,AOC,COB)
Problem: As shown in the diagram, the diameter of circle O is 100, AB=80, O is the center of ⊙O, AC is perpendicular to OC. Find the length of line CD.
问题：如图所示，⊙O的直径为100，AB=80，⊙O的圆心为O，AC⊥OC。求直线CD的长度。
Construction (CDL):
Shape(AO,OB,OBA)
Shape(AC,CO,OA)
Shape(OC,CB,BO)
Shape(DC,CA,OAD)
Shape(BC,CD,ODB)
Collinear(ACB)
Collinear(OCD)
Cocircular(O,ADB)
Conditions (CDL):
Equal(DiameterOfCircle(O),100)
Equal(LengthOfLine(AB),80)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AC,OC)
Goal: Value(LengthOfLine(CD))
Answer: 20
Theorem sequence:
circle_property_length_of_radius_and_diameter(1,O)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OD,O)
radius_of_circle_property_length_equal(1,OB,O)
adjacent_complementary_angle(1,ACO,OCB)
mirror_congruent_triangle_judgment_hl(1,ACO,BOC)
mirror_congruent_triangle_property_line_equal(1,OAC,OCB)
line_addition(1,AC,CB)
right_triangle_judgment_angle(1,OCB)
right_triangle_property_pythagorean(1,OCB)
line_addition(1,OC,CD)
Problem: As shown in the diagram, ∠BOA=130°, O is the center of circle O. Find the measure of ∠ACB.
问题：如图所示，∠BOA=130°，O是圆O的圆心。求∠ACB的大小。
Construction (CDL):
Shape(AO,OB,OBA)
Shape(AB,BO,OA)
Shape(AC,CB,BA)
Shape(BC,OCB)
Shape(CA,OAC)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),130)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ACB))
Answer: 115
Theorem sequence:
arc_property_center_angle(1,OBA,O)
round_angle(1,BOA,AOB)
arc_property_circumference_angle_external(1,OBA,C)