SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, AC=4, the radius of ⊙O is 5/2, the center of circle O is O, C bisects arc ODB, DE⊥CE. Find the length of line CE.
问题：如图所示，AC=4，圆O的半径为5/2，圆O的圆心为O，C是弧ODB的中点，DE垂直于CE。求直线CE的长度。
Construction (CDL):
Shape(OAD,DA)
Shape(ODC,CA,AD)
Shape(OCB,BC)
Shape(CB,BO,OA,AC)
Shape(OB,OBA,AO)
Shape(ODC,DE,EC)
Collinear(ADE)
Collinear(AOB)
Cocircular(O,ADCB)
Conditions (CDL):
Equal(LengthOfLine(AC),4)
Equal(RadiusOfCircle(O),5/2)
IsCentreOfCircle(O,O)
IsMidpointOfArc(C,ODB)
PerpendicularBetweenLine(DE,CE)
Goal: Value(LengthOfLine(CE))
Answer: 12/5
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
line_addition(1,AO,OB)
congruent_arc_judgment_length_equal(1,ODC,OCB)
congruent_arc_property_measure_equal(1,ODC,OCB)
arc_property_circumference_angle_external(1,ODC,A)
arc_property_circumference_angle_external(1,OCB,A)
diameter_of_circle_judgment_pass_centre(1,AOB,O)
diameter_of_circle_property_right_angle(1,ACB,O)
right_triangle_judgment_angle(1,ACB)
right_triangle_property_pythagorean(1,ACB)
similar_triangle_judgment_aa(1,BAC,CAE)
similar_triangle_property_line_ratio(1,ACB,AEC)
similar_triangle_property_line_ratio(1,CBA,ECA)
Problem: As shown in the diagram, ∠ABC=130°, the center of ⊙O is O. Find the measure of ∠COA.
问题：如图所示，∠ABC=130°，圆O的圆心为O。求∠COA的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(ODA,AD)
Shape(OA,AB,BC,CO)
Shape(DO,OC,CD)
Shape(DA,AO,OD)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),130)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COA))
Answer: 100
Theorem sequence:
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_internal(1,OAC,B)
Problem: As shown in the diagram, ∠BOA=50°, the center of circle O is O. Find the measure of ∠BCA.
问题：如图所示，∠BOA=50°，⊙O的圆心为O。求∠BCA的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(OAB,BO,OA)
Shape(OBC,CB)
Shape(CA,AO,OB,BC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),50)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 25
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠CBE=60°, BE⊥CE. Find the measure of ∠DAE.
问题：如图所示，∠CBE=60°，BE⊥CE。求∠DAE的大小。
Construction (CDL):
Shape(OCB,BC)
Shape(OBD,DE,EB)
Shape(ODA,AD)
Shape(OAC,CE,EA)
Shape(CB,BE,EC)
Shape(AE,ED,DA)
Collinear(BEA)
Collinear(DEC)
Cocircular(O,BDAC)
Conditions (CDL):
Equal(MeasureOfAngle(CBE),60)
PerpendicularBetweenLine(BE,CE)
Goal: Value(MeasureOfAngle(DAE))
Answer: 30
Theorem sequence:
triangle_property_angle_sum(1,BEC)
arc_property_circumference_angle_external(1,OBD,C)
arc_property_circumference_angle_external(1,OBD,A)
Problem: As shown in the diagram, ∠CAB=25°, the center of circle O is O. Find the measure of ∠CDA.
问题：如图所示，∠CAB=25°，圆O的圆心为O。求∠CDA的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(OCD,DC)
Shape(CD,DA,AC)
Shape(OBC,CB)
Shape(BC,CA,AO,OB)
Shape(OAB,BO,OA)
Collinear(AOB)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),25)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CDA))
Answer: 115
Theorem sequence:
diameter_of_circle_judgment_pass_centre(1,AOB,O)
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_internal(1,OCA,D)
Problem: As shown in the diagram, ∠CAB=44°, ∠DBC=Mul(MeasureOfAngle(CDB),2)°, the center of circle A is A. Find the measure of ∠DAC.
问题：如图所示，∠CAB=44°，∠DBC=Mul(MeasureOfAngle(CDB),2)°，A是⊙A的圆心。求∠DAC的大小。
Construction (CDL):
Shape(ABC,CB)
Shape(ACD,DC)
Shape(ADB,BA,AD)
Shape(AB,BE,EA)
Shape(EB,BC,CE)
Shape(EC,CD,DE)
Shape(AE,ED,DA)
Collinear(AEC)
Collinear(BED)
Cocircular(A,BCD)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),44)
Equal(MeasureOfAngle(DBC),Mul(MeasureOfAngle(CDB),2))
IsCentreOfCircle(A,A)
Goal: Value(MeasureOfAngle(DAC))
Answer: 88
Theorem sequence:
arc_property_circumference_angle_external(1,ACD,B)
arc_property_center_angle(1,ACD,A)
arc_property_circumference_angle_external(1,ABC,D)
arc_property_center_angle(1,ABC,A)
Problem: As shown in the diagram, ∠CDA=32°, O is the center of circle O, OE is perpendicular to BE. Find the measure of ∠BOC.
问题：如图所示，∠CDA=32°，O是圆O的圆心，OE垂直于BE。求∠BOC的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(OCB,BE,EC)
Shape(OAC,CE,EA)
Shape(OAC,CD,DA)
Shape(OE,EB,BO)
Shape(OBA,AO,OB)
Shape(OA,AE,EO)
Collinear(AEB)
Collinear(OEC)
Cocircular(O,DACB)
Conditions (CDL):
Equal(MeasureOfAngle(CDA),32)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(OE,BE)
Goal: Value(MeasureOfAngle(BOC))
Answer: 64
Theorem sequence:
arc_property_circumference_angle_external(1,OAC,D)
adjacent_complementary_angle(1,AEO,OEB)
circle_property_chord_perpendicular_bisect_arc(1,OAB,OEC)
congruent_arc_judgment_length_equal(1,OAC,OCB)
congruent_arc_property_measure_equal(1,OAC,OCB)
arc_property_center_angle(1,OCB,O)
Problem: As shown in the diagram, ∠ABO=50°, ∠ACO=20°, O is the center of ⊙O. Find the measure of ∠CAB.
问题：如图所示，∠ABO=50°，∠ACO=20°，O是⊙O的圆心。求∠CAB的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CE,EB)
Shape(OCA,AE,EO,OC)
Shape(AB,BE,EA)
Shape(EC,CO,OE)
Collinear(BEO)
Collinear(AEC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(ABO),50)
Equal(MeasureOfAngle(ACO),20)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 30
Theorem sequence:
triangle_property_angle_sum(1,ABE)
triangle_property_angle_sum(1,ECO)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
vertical_angle(1,BEA,OEC)
Problem: As shown in the diagram, Add(MeasureOfAngle(BOA)=MeasureOfAngle(DOC)), CD=8, the radius of ⊙O is 5, O is the center of circle O. Find the length of line AB.
问题：如图所示，Add(MeasureOfAngle(BOA)=MeasureOfAngle(DOC))，CD=8，圆O的半径为5，⊙O的圆心为O。求直线AB的长度。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CO,OB)
Shape(OCD,DC)
Shape(ODA,AO,OD)
Shape(OA,AB,BO)
Shape(OC,CD,DO)
Shape(OBE,EB)
Shape(OB,BE,EO)
Collinear(AOE)
Cocircular(O,ABCED)
Conditions (CDL):
Equal(Add(MeasureOfAngle(BOA),MeasureOfAngle(DOC)),180)
Equal(LengthOfLine(CD),8)
Equal(RadiusOfCircle(O),5)
IsCentreOfCircle(O,O)
Goal: Value(LengthOfLine(AB))
Answer: 6
Theorem sequence:
diameter_of_circle_judgment_pass_centre(1,AOE,O)
diameter_of_circle_property_right_angle(1,ABE,O)
adjacent_complementary_angle(1,EOB,BOA)
arc_property_center_angle(1,OCD,O)
arc_property_center_angle(1,OBE,O)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OE,O)
line_addition(1,AO,OE)
congruent_arc_judgment_measure_equal(1,OCD,OBE)
congruent_arc_property_chord_equal(1,OCD,OBE)
right_triangle_judgment_angle(1,ABE)
right_triangle_property_pythagorean(1,ABE)
Problem: As shown in the diagram, ∠BAD=27°, the center of ⊙O is O. Find the measure of ∠DCA.
问题：如图所示，∠BAD=27°，⊙O的圆心为O。求∠DCA的大小。
Construction (CDL):
Shape(OAD,DA)
Shape(ODB,BD)
Shape(BE,ED,DB)
Shape(OBC,CE,EB)
Shape(OCA,AC)
Shape(AO,OE,EC,CA)
Shape(AD,DE,EO,OA)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(BAD),27)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DCA))
Answer: 63
Theorem sequence:
diameter_of_circle_judgment_pass_centre(1,AOB,O)
diameter_of_circle_property_right_angle(1,ADB,O)
triangle_property_angle_sum(1,ADB)
arc_property_circumference_angle_external(1,OAD,B)
arc_property_circumference_angle_external(1,OAD,C)
Problem: As shown in the diagram, AB=Mul(LengthOfLine(DE),2), ∠DEB=16°, the center of circle O is O. Find the measure of ∠CBA.
问题：如图所示，AB=Mul(LengthOfLine(DE),2)，∠DEB=16°，O是⊙O的圆心。求∠CBA的大小。
Construction (CDL):
Shape(OCD,DC)
Shape(ODB,BC,CD)
Shape(ODB,BO,OD)
Shape(OBA,AO,OB)
Shape(OAC,CB,BO,OA)
Shape(ODB,DE,EB)
Shape(BD,DE,EB)
Collinear(AOBE)
Collinear(CDE)
Cocircular(O,ACDB)
Conditions (CDL):
Equal(LengthOfLine(AB),Mul(LengthOfLine(DE),2))
Equal(MeasureOfAngle(DEB),16)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CBA))
Answer: 24
Theorem sequence:
line_addition(1,AO,OB)
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,OD,O)
isosceles_triangle_judgment_line_equal(1,DEO)
isosceles_triangle_property_angle_equal(1,DEO)
arc_property_center_angle(1,ODB,O)
arc_property_circumference_angle_external(1,ODB,C)
triangle_property_angle_sum(1,BCE)
adjacent_complementary_angle(1,EBC,CBA)
Problem: As shown in the diagram, ∠BCA=22°, O is the center of circle O. Find the measure of ∠BOA.
问题：如图所示，∠BCA=22°，O是圆O的圆心。求∠BOA的大小。
Construction (CDL):
Shape(OAB,BE,EA)
Shape(OBC,CB)
Shape(OCA,AO,OE,EC)
Shape(OA,AE,EO)
Shape(EB,BC,CE)
Collinear(OEB)
Collinear(AEC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),22)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 44
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠DCA=20°, O is the center of circle O. Find the measure of ∠BAD.
问题：如图所示，∠DCA=20°，O是⊙O的圆心。求∠BAD的大小。
Construction (CDL):
Shape(OAD,DA)
Shape(EA,AD,DE)
Shape(ODB,BO,OE,ED)
Shape(CE,EO,OB,BC)
Shape(OBC,CB)
Shape(OCA,AC)
Shape(CA,AE,EC)
Collinear(AEOB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(DCA),20)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BAD))
Answer: 70
Theorem sequence:
diameter_of_circle_judgment_pass_centre(1,AOB,O)
diameter_of_circle_property_right_angle(1,BCA,O)
angle_addition(1,BCD,DCA)
arc_property_circumference_angle_external(1,ODB,C)
arc_property_circumference_angle_external(1,ODB,A)
Problem: As shown in the diagram, ∠BCA=35°, O is the center of ⊙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(OA,AE,EO)
Shape(EB,BC,CE)
Shape(OAB,BE,EA)
Shape(OBC,CB)
Shape(OCA,AO,OE,EC)
Collinear(AEC)
Collinear(OEB)
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, ∠AOB=130°, the center of ⊙O is O. Find the measure of ∠BCA.
问题：如图所示，∠AOB=130°，圆O的圆心为O。求∠BCA的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(OBC,CB)
Shape(OAD,DA)
Shape(ODB,BD)
Shape(OB,BC,CA,AO)
Shape(DO,OA,AD)
Shape(DB,BO,OD)
Cocircular(O,BCAD)
Conditions (CDL):
Equal(MeasureOfAngle(AOB),130)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 115
Theorem sequence:
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_internal(1,OBA,C)
Problem: As shown in the diagram, ∠ADC=54°, the center of circle O is O. Find the measure of ∠CAB.
问题：如图所示，∠ADC=54°，O是圆O的圆心。求∠CAB的大小。
Construction (CDL):
Shape(OBC,CB)
Shape(CE,EB,BC)
Shape(CA,AO,OE,EC)
Shape(OCA,AC)
Shape(AD,DE,EO,OA)
Shape(OAD,DA)
Shape(ODB,BE,ED)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(ADC),54)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 36
Theorem sequence:
diameter_of_circle_judgment_pass_centre(1,AOB,O)
diameter_of_circle_property_right_angle(1,BCA,O)
arc_property_circumference_angle_external(1,OCA,D)
arc_property_circumference_angle_external(1,OCA,B)
triangle_property_angle_sum(1,ABC)
Problem: As shown in the diagram, ∠DAC=40°, O is the center of ⊙O, D is the midpoint of ⌒OCA. Find the measure of ∠CAB.
问题：如图所示，∠DAC=40°，圆O的圆心为O，D平分弧OCA。求∠CAB的大小。
Construction (CDL):
Shape(OBC,CB)
Shape(OCD,DC)
Shape(ODA,AD)
Shape(DA,AC,CD)
Shape(DO,OC,CD)
Shape(AO,OB,BC,CA)
Shape(DA,AO,OD)
Collinear(AOB)
Cocircular(O,BCDA)
Conditions (CDL):
Equal(MeasureOfAngle(DAC),40)
IsCentreOfCircle(O,O)
IsMidpointOfArc(D,OCA)
Goal: Value(MeasureOfAngle(CAB))
Answer: 10
Theorem sequence:
congruent_arc_judgment_length_equal(1,OCD,ODA)
congruent_arc_property_chord_equal(1,OCD,ODA)
arc_property_circumference_angle_external(1,ODA,C)
isosceles_triangle_judgment_line_equal(1,DAC)
isosceles_triangle_property_angle_equal(1,DAC)
arc_property_center_angle(1,ODA,O)
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,AOD)
angle_addition(1,DAC,CAB)
Problem: As shown in the diagram, ∠ABC=45°, ∠EAD=75°, AD bisects ∠EAC. Find the measure of ∠ACD.
问题：如图所示，∠ABC=45°，∠EAD=75°，AD平分∠EAC。求∠ACD的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(AC,CD,DA)
Shape(EA,AD)
Collinear(BAE)
Collinear(BCD)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),45)
Equal(MeasureOfAngle(EAD),75)
IsBisectorOfAngle(AD,EAC)
Goal: Value(MeasureOfAngle(ACD))
Answer: 75
Theorem sequence:
triangle_property_angle_sum(1,ABC)
adjacent_complementary_angle(1,BCA,ACD)
angle_addition(1,EAD,DAC)
adjacent_complementary_angle(1,EAC,CAB)
Problem: As shown in the diagram, ∠BOA=72°, O is the center of ⊙O. Find the measure of ∠BCA.
问题：如图所示，∠BOA=72°，O是圆O的圆心。求∠BCA的大小。
Construction (CDL):
Shape(OAB,BE,EA)
Shape(OBC,CB)
Shape(OCA,AO,OE,EC)
Shape(OA,AE,EO)
Shape(EB,BC,CE)
Collinear(OEB)
Collinear(AEC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),72)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 36
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠BOA=50°, ∠OBC=40°, O is the center of circle O. Find the measure of ∠OAC.
问题：如图所示，∠BOA=50°，∠OBC=40°，O是圆O的圆心。求∠OAC的大小。
Construction (CDL):
Shape(OBC,CB)
Shape(MB,BC,CM)
Shape(AM,MO,OA)
Shape(OCA,AO,OM,MC)
Shape(OAB,BM,MA)
Collinear(AMC)
Collinear(BMO)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),50)
Equal(MeasureOfAngle(OBC),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(OAC))
Answer: 15
Theorem sequence:
triangle_property_angle_sum(1,MOA)
vertical_angle(1,AMO,CMB)
triangle_property_angle_sum(1,BCM)
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, DC=DO, ∠DCA=18°, O is the center of circle O. Find the measure of ∠EAB.
问题：如图所示，DC=DO，∠DCA=18°，O是圆O的圆心。求∠EAB的大小。
Construction (CDL):
Shape(OBE,EO,OB)
Shape(OBE,EA,AO,OB)
Shape(OE,ED,DO)
Shape(OED,DE)
Shape(ODA,AO,OD)
Shape(ODA,AE,ED)
Shape(ODA,DC,CA)
Collinear(CAOB)
Collinear(CDE)
Cocircular(O,BEDA)
Conditions (CDL):
Equal(LengthOfLine(DC),LengthOfLine(DO))
Equal(MeasureOfAngle(DCA),18)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(EAB))
Answer: 27
Theorem sequence:
isosceles_triangle_judgment_line_equal(1,DCO)
isosceles_triangle_property_angle_equal(1,DCO)
triangle_property_angle_sum(1,DCO)
adjacent_complementary_angle(1,EDO,ODC)
radius_of_circle_property_length_equal(1,OD,O)
radius_of_circle_property_length_equal(1,OE,O)
isosceles_triangle_judgment_line_equal(1,OED)
isosceles_triangle_property_angle_equal(1,OED)
angle_addition(1,AOD,DOE)
triangle_property_angle_sum(1,OED)
adjacent_complementary_angle(1,AOE,EOB)
arc_property_center_angle(1,OBE,O)
arc_property_circumference_angle_external(1,OBE,A)
Problem: As shown in the diagram, ∠DOB=120°, the center of ⊙O is O. Find the measure of ∠BCD.
问题：如图所示，∠DOB=120°，O是⊙O的圆心。求∠BCD的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(ODA,AD)
Shape(AB,BO,OD,DA)
Shape(OB,BC,CD,DO)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(DOB),120)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCD))
Answer: 120
Theorem sequence:
arc_property_center_angle(1,OBD,O)
arc_property_circumference_angle_internal(1,OBD,C)
Problem: As shown in the diagram, ∠ADC=36°, O is the center of ⊙O. Find the measure of ∠BCA.
问题：如图所示，∠ADC=36°，O是圆O的圆心。求∠BCA的大小。
Construction (CDL):
Shape(OBD,DE,EO,OB)
Shape(OCD,DC)
Shape(ED,DC,CE)
Shape(OCA,AC)
Shape(EC,CA,AE)
Shape(EA,AB,BO,OE)
Shape(OAB,BA)
Collinear(BOEC)
Collinear(AED)
Cocircular(O,BDCA)
Conditions (CDL):
Equal(MeasureOfAngle(ADC),36)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 54
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_external(1,OCA,D)
diameter_of_circle_judgment_pass_centre(1,BOC,O)
diameter_of_circle_property_right_angle(1,CAB,O)
triangle_property_angle_sum(1,CAB)
Problem: As shown in the diagram, ∠DCA=20°, the center of ⊙O is O, D bisects arc OAB. Find the measure of ∠BOA.
问题：如图所示，∠DCA=20°，O是圆O的圆心，D平分⌒OAB。求∠BOA的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(OAD,DE,EA)
Shape(ODB,BO,OD)
Shape(OE,ED,DO)
Shape(OBC,CE,EO,OB)
Shape(CA,AE,EC)
Collinear(OEA)
Collinear(CED)
Cocircular(O,CADB)
Conditions (CDL):
Equal(MeasureOfAngle(DCA),20)
IsCentreOfCircle(O,O)
IsMidpointOfArc(D,OAB)
Goal: Value(MeasureOfAngle(BOA))
Answer: 80
Theorem sequence:
arc_property_circumference_angle_external(1,OAD,C)
congruent_arc_judgment_length_equal(1,OAD,ODB)
congruent_arc_property_measure_equal(1,OAD,ODB)
arc_property_center_angle(1,OAD,O)
arc_property_center_angle(1,ODB,O)
angle_addition(1,BOD,DOA)
Problem: As shown in the diagram, ∠AMD=100°, ∠DAM=30°. Find the measure of ∠CBM.
问题：如图所示，∠AMD=100°，∠DAM=30°。求∠CBM的大小。
Construction (CDL):
Shape(OCB,BC)
Shape(MC,CB,BM)
Shape(OBD,DM,MB)
Shape(ODA,AD)
Shape(MD,DA,AM)
Shape(OAC,AM,MA)
Collinear(CMD)
Collinear(AMB)
Cocircular(O,BDAC)
Conditions (CDL):
Equal(MeasureOfAngle(AMD),100)
Equal(MeasureOfAngle(DAM),30)
Goal: Value(MeasureOfAngle(CBM))
Answer: 50
Theorem sequence:
arc_property_circumference_angle_external(1,OAC,B)
arc_property_circumference_angle_external(1,OAC,D)
triangle_property_angle_sum(1,AMD)
Problem: As shown in the diagram, ∠CAB=36°, O is the center of circle O. Find the measure of ∠COB.
问题：如图所示，∠CAB=36°，圆O的圆心为O。求∠COB的大小。
Construction (CDL):
Shape(OBC,CO,OB)
Shape(OC,CA,AB,BO)
Shape(OCA,AC)
Shape(OAB,BA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),36)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COB))
Answer: 72
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,OBC,O)
Problem: As shown in the diagram, ∠ABC=40°, the center of circle O is O. Find the measure of ∠AOC.
问题：如图所示，∠ABC=40°，圆O的圆心为O。求∠AOC的大小。
Construction (CDL):
Shape(OC,CM,MO)
Shape(MA,AB,BM)
Shape(OCA,AM,MC)
Shape(OAB,BA)
Shape(OBC,CO,OM,MB)
Collinear(CMB)
Collinear(OMA)
Cocircular(O,CAB)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AOC))
Answer: 80
Theorem sequence:
arc_property_center_angle(1,OCA,O)
arc_property_circumference_angle_external(1,OCA,B)
Problem: As shown in the diagram, ∠BOA=60°, the center of ⊙O is O. Find the measure of ∠BPA.
问题：如图所示，∠BOA=60°，O是圆O的圆心。求∠BPA的大小。
Construction (CDL):
Shape(OAB,BO,OA)
Shape(OB,BP,PA,AO)
Shape(OBP,PB)
Shape(OPA,AP)
Cocircular(O,ABP)
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, ∠AOC=130°, O is the center of circle O. Find the measure of ∠CDB.
问题：如图所示，∠AOC=130°，圆O的圆心为O。求∠CDB的大小。
Construction (CDL):
Shape(AO,OC,OCA)
Shape(EB,OBC,CE)
Shape(OE,EC,CO)
Shape(DE,EO,OA,OAD)
Shape(DB,BE,ED)
Shape(ODB,BD)
Collinear(BEOA)
Collinear(DEC)
Cocircular(O,DBCA)
Conditions (CDL):
Equal(MeasureOfAngle(AOC),130)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CDB))
Answer: 25
Theorem sequence:
flat_angle(1,AOB)
angle_addition(1,AOC,COE)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,D)
Problem: As shown in the diagram, the length of arc ODA is equal to the length of ⌒OCB, ∠EAC=40°, O is the center of ⊙O. Find the measure of ∠BEC.
问题：如图所示，弧ODA的长度与弧OCB的长度相等，∠EAC=40°，圆O的圆心为O。求∠BEC的大小。
Construction (CDL):
Shape(OAC,CA)
Shape(AC,CE,EA)
Shape(EC,OCB,BE)
Shape(OBD,DE,EB)
Shape(AE,ED,ODA)
Collinear(AEB)
Collinear(CED)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(LengthOfArc(ODA),LengthOfArc(OCB))
Equal(MeasureOfAngle(EAC),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BEC))
Answer: 80
Theorem sequence:
congruent_arc_judgment_length_equal(1,ODA,OCB)
congruent_arc_property_measure_equal(1,ODA,OCB)
arc_property_circumference_angle_external(1,OCB,A)
arc_property_circumference_angle_external(1,ODA,C)
triangle_property_angle_sum(1,ACE)
flat_angle(1,BEA)
angle_addition(1,BEC,CEA)
Problem: As shown in the diagram, the length of arc ODA is equal to the length of ⌒OCD, ∠CAB=40°, O is the center of circle O, the diameter of circle O is AB. Find the measure of ∠DAC.
问题：如图所示，弧ODA的长度与⌒OCD的长度相等，∠CAB=40°，⊙O的圆心为O，圆O的直径为AB。求∠DAC的大小。
Construction (CDL):
Shape(DA,AE,ED)
Shape(ODA,AD)
Shape(EA,AO,OB,BE)
Shape(CE,EB,BC)
Shape(OBC,CB)
Shape(OCD,DE,EC)
Shape(OA,OAB,BO)
Collinear(AOB)
Collinear(AEC)
Collinear(BED)
Cocircular(O,CDAB)
Conditions (CDL):
Equal(LengthOfArc(ODA),LengthOfArc(OCD))
Equal(MeasureOfAngle(CAB),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(DAC))
Answer: 25
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,ABC)
congruent_arc_judgment_length_equal(1,ODA,OCD)
congruent_arc_property_measure_equal(1,ODA,OCD)
arc_property_circumference_angle_external(1,OCD,A)
arc_property_circumference_angle_external(1,OCA,B)
arc_addition_measure(1,OCD,ODA)
Problem: As shown in the diagram, ∠AOC=60°, ∠BCA=50°, the center of ⊙O is O. Find the measure of ∠CAB.
问题：如图所示，∠AOC=60°，∠BCA=50°，O是圆O的圆心。求∠CAB的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(CA,AD,DC)
Shape(OC,CD,DO)
Shape(DA,AB,BD)
Shape(OBC,CO,OD,DB)
Collinear(CDB)
Collinear(ODA)
Cocircular(O,CAB)
Conditions (CDL):
Equal(MeasureOfAngle(AOC),60)
Equal(MeasureOfAngle(BCA),50)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 100
Theorem sequence:
arc_property_center_angle(1,OCA,O)
arc_property_circumference_angle_external(1,OAB,C)
arc_property_circumference_angle_external(1,OBC,A)
arc_addition_measure(1,OCA,OAB)
round_arc(1,OCB,OBC)
Problem: As shown in the diagram, ∠BAC=40°, the center of ⊙O is O. Find the measure of ∠CDA.
问题：如图所示，∠BAC=40°，圆O的圆心为O。求∠CDA的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(DA,AO,OE,ED)
Shape(EO,OA,AC,CE)
Shape(OAC,CA)
Shape(EC,CB,BE)
Shape(OCB,BC)
Shape(OBD,DE,EB)
Collinear(DEC)
Collinear(AOEB)
Cocircular(O,DACB)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CDA))
Answer: 50
Theorem sequence:
flat_angle(1,EOA)
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAC,D)
arc_property_circumference_angle_external(1,OCB,A)
arc_addition_measure(1,OAC,OCB)
Problem: As shown in the diagram, ∠AOB=60°, O is the center of ⊙O. Find the measure of ∠ACB.
问题：如图所示，∠AOB=60°，⊙O的圆心为O。求∠ACB的大小。
Construction (CDL):
Shape(OAC,CA)
Shape(OA,AC,CB,BO)
Shape(AO,OB,OBA)
Shape(OCB,BC)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(AOB),60)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ACB))
Answer: 30
Theorem sequence:
arc_property_circumference_angle_external(1,OBA,C)
arc_property_center_angle(1,OBA,O)
Problem: As shown in the diagram, ∠COB=40°, O is the center of ⊙O. Find the measure of ∠CDA.
问题：如图所示，∠COB=40°，⊙O的圆心为O。求∠CDA的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(DA,AO,OC,CD)
Shape(CO,OB,BC)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(OA,OAB,BO)
Collinear(AOB)
Cocircular(O,DABC)
Conditions (CDL):
Equal(MeasureOfAngle(COB),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CDA))
Answer: 110
Theorem sequence:
flat_angle(1,BOA)
angle_addition(1,COB,BOA)
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,D)
Problem: As shown in the diagram, AE=10, ∠CAE=∠ABC, the center of circle O is O, the diameter of ⊙O is AE. Find the length of line AC.
问题：如图所示，AE=10，∠CAE=∠ABC，O是圆O的圆心，AE是⊙O的直径。求直线AC的长度。
Construction (CDL):
Shape(OAB,BA)
Shape(AB,BD,DO,OA)
Shape(DB,OBE,ED)
Shape(DE,EC,CD)
Shape(OEC,CE)
Shape(AO,OC,CA)
Shape(OD,DC,CO)
Shape(OCA,AC)
Collinear(AODE)
Collinear(BDC)
Cocircular(O,ABEC)
Conditions (CDL):
Equal(LengthOfLine(AE),10)
Equal(MeasureOfAngle(CAE),MeasureOfAngle(ABC))
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AE,O)
Goal: Value(LengthOfLine(AC))
Answer: 5*sqrt(2)
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_external(1,OEC,A)
congruent_arc_judgment_measure_equal(1,OEC,OCA)
congruent_arc_property_chord_equal(1,OEC,OCA)
diameter_of_circle_property_right_angle(1,ECA,O)
right_triangle_judgment_angle(1,ECA)
right_triangle_property_pythagorean(1,ECA)