SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, Add(MeasureOfAngle(BAC)=MeasureOfAngle(DAE)), DE=6, the radius of ⊙A is 5, the center of ⊙A is A. Find the length of line BC.
问题：如图所示，Add(MeasureOfAngle(BAC)=MeasureOfAngle(DAE))，DE=6，⊙A的半径为5，⊙A的圆心为A。求直线BC的长度。
Construction (CDL):
Shape(ADC,CA,AD)
Shape(ACB,BC)
Shape(ABE,EA,AB)
Shape(AED,DE)
Shape(AC,CB,BA)
Shape(AE,ED,DA)
Cocircular(A,BEDC)
Conditions (CDL):
Equal(Add(MeasureOfAngle(BAC),MeasureOfAngle(DAE)),180)
Equal(LengthOfLine(DE),6)
Equal(RadiusOfCircle(A),5)
IsCentreOfCircle(A,A)
Goal: Value(LengthOfLine(BC))
Answer: 24/(5*sin(acos(7/25)/2))
Theorem sequence:
radius_of_circle_property_length_equal(1,AB,A)
radius_of_circle_property_length_equal(1,AE,A)
radius_of_circle_property_length_equal(1,AD,A)
radius_of_circle_property_length_equal(1,AC,A)
cosine_theorem(1,AED)
isosceles_triangle_judgment_line_equal(1,ACB)
isosceles_triangle_property_angle_equal(1,ACB)
triangle_property_angle_sum(1,ACB)
sine_theorem(1,CBA)
Problem: As shown in the diagram, ∠DAF=32°, ∠ECD=38°, ∠FBE=45°. Find the measure of ∠DFE.
问题：如图所示，∠DAF=32°，∠ECD=38°，∠FBE=45°。求∠DFE的大小。
Construction (CDL):
Shape(AF,FD,DA)
Shape(FB,BE,EF)
Shape(DF,FE,EC,CD)
Collinear(ADC)
Collinear(BEC)
Collinear(AFE)
Collinear(BFD)
Conditions (CDL):
Equal(MeasureOfAngle(DAF),32)
Equal(MeasureOfAngle(ECD),38)
Equal(MeasureOfAngle(FBE),45)
Goal: Value(MeasureOfAngle(DFE))
Answer: 115
Theorem sequence:
triangle_property_angle_sum(1,FBE)
triangle_property_angle_sum(1,AEC)
adjacent_complementary_angle(1,BEF,FEC)
adjacent_complementary_angle(1,DFE,EFB)
Problem: As shown in the diagram, ∠DAB=64°, O is the center of circle O. Find the measure of ∠DOB.
问题：如图所示，∠DAB=64°，O是⊙O的圆心。求∠DOB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(DAB),64)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DOB))
Answer: 128
Theorem sequence:
arc_property_circumference_angle_external(1,OBD,A)
arc_property_center_angle(1,OBD,O)
Problem: As shown in the diagram, DA=DC, ∠CBE=52°, O is the center of ⊙O. Find the measure of ∠DAC.
问题：如图所示，DA=DC，∠CBE=52°，O是⊙O的圆心。求∠DAC的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(ODA,AD)
Shape(DA,AC,CD)
Shape(AB,BC,CA)
Shape(CB,BE)
Collinear(ABE)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(LengthOfLine(DA),LengthOfLine(DC))
Equal(MeasureOfAngle(CBE),52)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DAC))
Answer: 64
Theorem sequence:
adjacent_complementary_angle(1,ABC,CBE)
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_internal(1,OCA,D)
isosceles_triangle_judgment_line_equal(1,DAC)
isosceles_triangle_property_angle_equal(1,DAC)
triangle_property_angle_sum(1,DAC)
Problem: As shown in the diagram, ∠BCD=120°, O is the center of ⊙O. Find the measure of ∠OBD.
问题：如图所示，∠BCD=120°，O是圆O的圆心。求∠OBD的大小。
Construction (CDL):
Shape(OAB,BO,OA)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(ODA,AD)
Shape(AO,OB,BD,DA)
Shape(BC,CD,DB)
Collinear(AOB)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),120)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(OBD))
Answer: 30
Theorem sequence:
arc_property_circumference_angle_external(1,ODB,C)
arc_property_circumference_angle_internal(1,ODB,A)
diameter_of_circle_judgment_pass_centre(1,AOB,O)
diameter_of_circle_property_right_angle(1,BDA,O)
triangle_property_angle_sum(1,ABD)
Problem: As shown in the diagram, CD=3, ∠ABC=120°, the diameter of circle O is AD. Find the length of line AC.
问题：如图所示，CD=3，∠ABC=120°，圆O的直径为AD。求直线AC的长度。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(ODA,AO,OD)
Shape(AC,CD,DO,OA)
Shape(AB,BC,CA)
Collinear(AOD)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(LengthOfLine(CD),3)
Equal(MeasureOfAngle(ABC),120)
IsDiameterOfCircle(AD,O)
Goal: Value(LengthOfLine(AC))
Answer: 3*sqrt(3)
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_internal(1,OCA,D)
diameter_of_circle_property_right_angle(1,ACD,O)
triangle_property_angle_sum(1,ACD)
sine_theorem(1,CDA)
Problem: As shown in the diagram, ∠DCE=80°, ∠DFA=25°, the center of ⊙O is O. Find the measure of ∠CED.
问题：如图所示，∠DCE=80°，∠DFA=25°，O是⊙O的圆心。求∠CED的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(ODA,AD)
Shape(AB,BC,CD,DA)
Shape(OCD,CE,ED)
Shape(ODA,DF,FA)
Collinear(FAB)
Collinear(ECB)
Collinear(FDC)
Collinear(EDA)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(DCE),80)
Equal(MeasureOfAngle(DFA),25)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CED))
Answer: 45
Theorem sequence:
adjacent_complementary_angle(1,BCD,DCE)
triangle_property_angle_sum(1,BCF)
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_internal(1,OCA,D)
adjacent_complementary_angle(1,EDC,CDA)
triangle_property_angle_sum(1,EDC)
Problem: As shown in the diagram, ∠DCE=110°, the center of ⊙O is O. Find the measure of ∠DAB.
问题：如图所示，∠DCE=110°，圆O的圆心为O。求∠DAB的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(ODA,AD)
Shape(AB,BC,CD,DA)
Shape(DC,CE)
Collinear(BCE)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(DCE),110)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DAB))
Answer: 110
Theorem sequence:
adjacent_complementary_angle(1,BCD,DCE)
arc_property_circumference_angle_external(1,ODB,C)
arc_property_circumference_angle_internal(1,ODB,A)
Problem: As shown in the diagram, OB=2, ∠DOB=60°, ∠DOC=∠COB, O is the center of circle O. Find the length of line AC.
问题：如图所示，OB=2，∠DOB=60°，∠DOC=∠COB，圆O的圆心为O。求直线AC的长度。
Construction (CDL):
Shape(OBA,AC,CB)
Shape(OAD,DC,CA)
Shape(OB,BC,CO)
Shape(OC,CD,DO)
Collinear(BCD)
Collinear(OCA)
Cocircular(O,BAD)
Conditions (CDL):
Equal(LengthOfLine(OB),2)
Equal(MeasureOfAngle(DOB),60)
Equal(MeasureOfAngle(DOC),MeasureOfAngle(COB))
IsCentreOfCircle(O,O)
Goal: Value(LengthOfLine(AC))
Answer: 2-sqrt(3)
Theorem sequence:
angle_addition(1,DOC,COB)
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OD,O)
mirror_congruent_triangle_judgment_sas(1,OBC,OCD)
mirror_congruent_triangle_property_angle_equal(1,COB,CDO)
adjacent_complementary_angle(1,BCO,OCD)
triangle_property_angle_sum(1,OBC)
sine_theorem(1,OBC)
line_addition(1,OC,CA)
Problem: As shown in the diagram, ∠GBC=50°, the center of circle O is O, OE is perpendicular to CE. Find the measure of ∠CBD.
问题：如图所示，∠GBC=50°，O是圆O的圆心，OE垂直于CE。求∠CBD的大小。
Construction (CDL):
Shape(OAD,DA)
Shape(ODC,CE,ED)
Shape(OCB,BC)
Shape(OBA,AB)
Shape(AD,DH,HA)
Shape(OH,HD,DO)
Shape(OD,DE,EO)
Shape(BA,AH,HB)
Shape(OCB,CG,GB)
Shape(BH,HO,OE,EC,CB)
Collinear(AHOE)
Collinear(BHD)
Collinear(ABG)
Collinear(DECG)
Cocircular(O,ADCB)
Conditions (CDL):
Equal(MeasureOfAngle(GBC),50)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(OE,CE)
Goal: Value(MeasureOfAngle(CBD))
Answer: 80
Theorem sequence:
adjacent_complementary_angle(1,GBC,CBA)
arc_property_circumference_angle_external(1,OAD,B)
arc_property_circumference_angle_external(1,OAC,B)
arc_property_circumference_angle_internal(1,OAC,D)
adjacent_complementary_angle(1,DEO,OEC)
triangle_property_angle_sum(1,EAD)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OD,O)
isosceles_triangle_judgment_line_equal(1,OAD)
isosceles_triangle_property_angle_equal(1,OAD)
triangle_property_angle_sum(1,OAD)
arc_property_center_angle(1,OAD,O)
angle_addition(1,CBH,HBA)
Problem: As shown in the diagram, DA=DC, ∠CBE=50°. Find the measure of ∠DAC.
问题：如图所示，DA=DC，∠CBE=50°。求∠DAC的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(DA,AC,CD)
Shape(AB,BC,CA)
Shape(CB,BE)
Collinear(ABE)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(LengthOfLine(DA),LengthOfLine(DC))
Equal(MeasureOfAngle(CBE),50)
Goal: Value(MeasureOfAngle(DAC))
Answer: 65
Theorem sequence:
adjacent_complementary_angle(1,ABC,CBE)
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_internal(1,OCA,D)
isosceles_triangle_judgment_line_equal(1,DAC)
isosceles_triangle_property_angle_equal(1,DAC)
triangle_property_angle_sum(1,DAC)
Problem: As shown in the diagram, the length of ⌒OBC is equal to the length of arc OFD, ∠ABC=110°, ∠CAB=20°. Find the measure of ∠FED.
问题：如图所示，弧OBC的长度与弧OFD的长度相等，∠ABC=110°，∠CAB=20°。求∠FED的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCF,FC)
Shape(OFD,DC,CF)
Shape(ODA,AD)
Shape(AB,BC,CA)
Shape(AC,CD,DA)
Shape(OFD,FE,ED)
Collinear(ADE)
Collinear(CFE)
Cocircular(O,ABCFD)
Conditions (CDL):
Equal(LengthOfArc(OBC),LengthOfArc(OFD))
Equal(MeasureOfAngle(ABC),110)
Equal(MeasureOfAngle(CAB),20)
Goal: Value(MeasureOfAngle(FED))
Answer: 50
Theorem sequence:
congruent_arc_judgment_length_equal(1,OBC,OFD)
congruent_arc_property_measure_equal(1,OBC,OFD)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_external(1,OFD,C)
arc_property_circumference_angle_internal(1,OCA,D)
adjacent_complementary_angle(1,EDC,CDA)
triangle_property_angle_sum(1,DCE)
Problem: As shown in the diagram, the length of ⌒OBC is equal to the length of arc OFD, ∠ABC=105°, ∠CAB=30°. Find the measure of ∠FED.
问题：如图所示，弧OBC的长度与弧OFD的长度相等，∠ABC=105°，∠CAB=30°。求∠FED的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCF,FC)
Shape(OFD,DC,CF)
Shape(ODA,AD)
Shape(AB,BC,CA)
Shape(AC,CD,DA)
Shape(OFD,FE,ED)
Collinear(ADE)
Collinear(CFE)
Cocircular(O,ABCFD)
Conditions (CDL):
Equal(LengthOfArc(OBC),LengthOfArc(OFD))
Equal(MeasureOfAngle(ABC),105)
Equal(MeasureOfAngle(CAB),30)
Goal: Value(MeasureOfAngle(FED))
Answer: 45
Theorem sequence:
congruent_arc_judgment_length_equal(1,OBC,OFD)
congruent_arc_property_measure_equal(1,OBC,OFD)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_external(1,OFD,C)
arc_property_circumference_angle_internal(1,OCA,D)
adjacent_complementary_angle(1,EDC,CDA)
triangle_property_angle_sum(1,DCE)
Problem: As shown in the diagram, AB=2, AB=AD, ∠BCD=120°. Find the perimeter of triangle ABD.
问题：如图所示，AB=2，AB=AD，∠BCD=120°。求△ABD的周长。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(ODA,AD)
Shape(AB,BD,DA)
Shape(BC,CD,DB)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(LengthOfLine(AB),2)
Equal(LengthOfLine(AB),LengthOfLine(AD))
Equal(MeasureOfAngle(BCD),120)
Goal: Value(PerimeterOfTriangle(ABD))
Answer: 6
Theorem sequence:
arc_property_circumference_angle_external(1,ODB,C)
arc_property_circumference_angle_internal(1,ODB,A)
cosine_theorem(1,ABD)
triangle_perimeter_formula(1,ABD)
Problem: As shown in the diagram, ∠ABC=60°, the center of circle O is O. Find the measure of ∠CDA.
问题：如图所示，∠ABC=60°，圆O的圆心为O。求∠CDA的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(ODA,AD)
Shape(AB,BC,CD,DA)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),60)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CDA))
Answer: 120
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_internal(1,OCA,D)
Problem: As shown in the diagram, ∠DAO=40°, O is the center of ⊙O, C bisects ⌒OBD. Find the measure of ∠OBC.
问题：如图所示，∠DAO=40°，⊙O的圆心为O，C平分⌒OBD。求∠OBC的大小。
Construction (CDL):
Shape(OAB,BO,OA)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(ODA,AD)
Shape(AO,OB,BC,CA)
Shape(AC,CD,DA)
Collinear(AOB)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(DAO),40)
IsCentreOfCircle(O,O)
IsMidpointOfArc(C,OBD)
Goal: Value(MeasureOfAngle(OBC))
Answer: 70
Theorem sequence:
congruent_arc_judgment_length_equal(1,OBC,OCD)
congruent_arc_property_measure_equal(1,OBC,OCD)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_external(1,OCD,A)
angle_addition(1,DAC,CAO)
diameter_of_circle_judgment_pass_centre(1,AOB,O)
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,ABC)
Problem: As shown in the diagram, ∠ABC=135°, the center of circle O is O. Find the measure of ∠COA.
问题：如图所示，∠ABC=135°，O是圆O的圆心。求∠COA的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(ODA,AD)
Shape(AB,BC,CO,OA)
Shape(AO,OC,CD,DA)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),135)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COA))
Answer: 90
Theorem sequence:
arc_property_circumference_angle_internal(1,OAC,B)
arc_property_center_angle(1,OAC,O)
Problem: As shown in the diagram, ∠BAD=50°, O is the center of circle O. Find the measure of ∠BCE.
问题：如图所示，∠BAD=50°，O是⊙O的圆心。求∠BCE的大小。
Construction (CDL):
Shape(OAD,DA)
Shape(ODC,CD)
Shape(OCB,BC)
Shape(OBA,AB)
Shape(AD,DC,CB,BA)
Shape(BC,CE)
Collinear(ECD)
Cocircular(O,ADCB)
Conditions (CDL):
Equal(MeasureOfAngle(BAD),50)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCE))
Answer: 50
Theorem sequence:
arc_property_circumference_angle_external(1,ODB,A)
arc_property_circumference_angle_internal(1,ODB,C)
adjacent_complementary_angle(1,DCB,BCE)
Problem: As shown in the diagram, ∠CDA=130°, O is the center of ⊙O. Find the measure of ∠AOC.
问题：如图所示，∠CDA=130°，O是⊙O的圆心。求∠AOC的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(ODA,AD)
Shape(AB,BC,CO,OA)
Shape(AO,OC,CD,DA)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(CDA),130)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AOC))
Answer: 100
Theorem sequence:
arc_property_circumference_angle_internal(1,OCA,D)
arc_property_center_angle(1,OCA,O)
Problem: As shown in the diagram, ∠DOA=80°, O is the center of circle O, AO is parallel to DC. Find the measure of ∠CBA.
问题：如图所示，∠DOA=80°，圆O的圆心为O，AO∥DC。求∠CBA的大小。
Construction (CDL):
Shape(OAD,DA)
Shape(ODC,CD)
Shape(OCB,BC)
Shape(OBA,AB)
Shape(OA,AD,DO)
Shape(AO,OD,DC,CB,BA)
Cocircular(O,ADCB)
Conditions (CDL):
Equal(MeasureOfAngle(DOA),80)
IsCentreOfCircle(O,O)
ParallelBetweenLine(AO,DC)
Goal: Value(MeasureOfAngle(CBA))
Answer: 50
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OD,O)
isosceles_triangle_judgment_line_equal(1,OAD)
isosceles_triangle_property_angle_equal(1,OAD)
triangle_property_angle_sum(1,OAD)
parallel_property_alternate_interior_angle(2,AO,DC)
angle_addition(1,ADO,ODC)
arc_property_circumference_angle_external(1,OCA,D)
arc_property_circumference_angle_internal(1,OCA,B)
Problem: As shown in the diagram, ∠ADC=140°, O is the center of ⊙O. Find the measure of ∠FOA.
问题：如图所示，∠ADC=140°，O是⊙O的圆心。求∠FOA的大小。
Construction (CDL):
Shape(OAD,DA)
Shape(ODC,CD)
Shape(OCB,BC)
Shape(OBE,EO,OF,FB)
Shape(CB,BF,FC)
Shape(CF,FA,AD,DC)
Shape(FO,OA,AF)
Collinear(EOA)
Collinear(BFA)
Collinear(CFO)
Cocircular(O,ADCBE)
Conditions (CDL):
Equal(MeasureOfAngle(ADC),140)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(FOA))
Answer: 80
Theorem sequence:
arc_property_circumference_angle_internal(1,OAC,D)
arc_property_center_angle(1,OAC,O)
Problem: As shown in the diagram, BC=CD, ∠COB=130°, O is the center of circle O. Find the measure of ∠DAB.
问题：如图所示，BC=CD，∠COB=130°，圆O的圆心为O。求∠DAB的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(ODA,AD)
Shape(CD,DO,OC)
Shape(AB,BO,OD,DA)
Shape(CO,OB,BC)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(LengthOfLine(BC),LengthOfLine(CD))
Equal(MeasureOfAngle(COB),130)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DAB))
Answer: 130
Theorem sequence:
arc_property_center_angle(1,OBC,O)
congruent_arc_judgment_chord_equal(1,OBC,OCD)
congruent_arc_property_measure_equal(1,OBC,OCD)
arc_addition_measure(1,OBC,OCD)
round_arc(1,OBD,ODB)
arc_property_circumference_angle_internal(1,ODB,A)
Problem: As shown in the diagram, ∠DAB=100°, O is the center of ⊙O. Find the measure of ∠BCD.
问题：如图所示，∠DAB=100°，圆O的圆心为O。求∠BCD的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(ODA,AD)
Shape(AB,BC,CD,DA)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),100)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCD))
Answer: 80
Theorem sequence:
arc_property_circumference_angle_external(1,OBD,A)
arc_property_circumference_angle_internal(1,OBD,C)
Problem: As shown in the diagram, ∠DOB=138°, the center of ⊙O is O. Find the measure of ∠DCE.
问题：如图所示，∠DOB=138°，圆O的圆心为O。求∠DCE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)
Shape(DC,CE)
Collinear(BCE)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(DOB),138)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DCE))
Answer: 69
Theorem sequence:
arc_property_center_angle(1,OBD,O)
arc_property_circumference_angle_internal(1,OBD,C)
adjacent_complementary_angle(1,BCD,DCE)
Problem: As shown in the diagram, ∠DAB=110°, O is the center of ⊙O. Find the measure of ∠BOD.
问题：如图所示，∠DAB=110°，O是圆O的圆心。求∠BOD的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(ODA,AD)
Shape(AB,BO,OD,DA)
Shape(BC,CD,DO,OB)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),110)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOD))
Answer: 140
Theorem sequence:
arc_property_circumference_angle_internal(1,ODB,A)
arc_property_center_angle(1,ODB,O)
Problem: As shown in the diagram, ∠COA=120°, O is the center of ⊙O. Find the measure of ∠ABC.
问题：如图所示，∠COA=120°，O是圆O的圆心。求∠ABC的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(ODA,AD)
Shape(AB,BC,CO,OA)
Shape(AO,OC,CD,DA)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(COA),120)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ABC))
Answer: 120
Theorem sequence:
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_internal(1,OAC,B)
Problem: As shown in the diagram, the length of ⌒OBC is equal to the length of arc OFD, ∠ABC=105°, ∠CAB=25°. Find the measure of ∠FED.
问题：如图所示，弧OBC的长度与⌒OFD的长度相等，∠ABC=105°，∠CAB=25°。求∠FED的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCF,FC)
Shape(OFD,DC,CF)
Shape(ODA,AD)
Shape(AB,BC,CA)
Shape(AC,CD,DA)
Shape(OFD,FE,ED)
Collinear(ADE)
Collinear(CFE)
Cocircular(O,ABCFD)
Conditions (CDL):
Equal(LengthOfArc(OBC),LengthOfArc(OFD))
Equal(MeasureOfAngle(ABC),105)
Equal(MeasureOfAngle(CAB),25)
Goal: Value(MeasureOfAngle(FED))
Answer: 50
Theorem sequence:
congruent_arc_judgment_length_equal(1,OBC,OFD)
congruent_arc_property_measure_equal(1,OBC,OFD)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_external(1,OFD,C)
arc_property_circumference_angle_internal(1,OCA,D)
adjacent_complementary_angle(1,EDC,CDA)
triangle_property_angle_sum(1,DCE)
Problem: As shown in the diagram, ∠COA=100°, O is the center of circle O. Find the measure of ∠ABC.
问题：如图所示，∠COA=100°，⊙O的圆心为O。求∠ABC的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCA,AO,OC)
Shape(OA,AB,BC,CO)
Cocircular(O,ABC)
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, ∠BCD=140°, O is the center of circle O. Find the measure of ∠DOB.
问题：如图所示，∠BCD=140°，O是圆O的圆心。求∠DOB的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(ODA,AD)
Shape(CD,DO,OB,BC)
Shape(DA,AB,BO,OD)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),140)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DOB))
Answer: 80
Theorem sequence:
arc_property_circumference_angle_internal(1,OBD,C)
arc_property_center_angle(1,OBD,O)
Problem: As shown in the diagram, ∠FCB=100°, O is the center of circle O, AB is parallel to DC. Find the measure of ∠FEC.
问题：如图所示，∠FCB=100°，圆O的圆心为O，AB平行于DC。求∠FEC的大小。
Construction (CDL):
Shape(OAD,DA)
Shape(ODE,EF,FD)
Shape(OEC,CE)
Shape(EC,CF,FE)
Shape(OCB,BC)
Shape(OBA,AB)
Shape(AD,DF,FA)
Shape(AF,FC,CB,BA)
Collinear(DFC)
Collinear(AFE)
Cocircular(O,ADECB)
Conditions (CDL):
Equal(MeasureOfAngle(FCB),100)
IsCentreOfCircle(O,O)
ParallelBetweenLine(AB,DC)
Goal: Value(MeasureOfAngle(FEC))
Answer: 100
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,CD,BA)
arc_property_circumference_angle_external(1,OAC,B)
arc_property_circumference_angle_internal(1,OAC,E)
Problem: As shown in the diagram, ∠DOB=90°, O is the center of ⊙O. Find the measure of ∠BCD.
问题：如图所示，∠DOB=90°，O是圆O的圆心。求∠BCD的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(ODA,AD)
Shape(BC,CD,DO,OB)
Shape(BO,OD,DA,AB)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(DOB),90)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCD))
Answer: 135
Theorem sequence:
arc_property_center_angle(1,OBD,O)
arc_property_circumference_angle_internal(1,OBD,C)
Problem: As shown in the diagram, ∠DOB=100°, the center of circle O is O. Find the measure of ∠BCD.
问题：如图所示，∠DOB=100°，⊙O的圆心为O。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)
Shape(DC,CE)
Collinear(BCE)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(DOB),100)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCD))
Answer: 130
Theorem sequence:
arc_property_center_angle(1,OBD,O)
arc_property_circumference_angle_internal(1,OBD,C)
Problem: As shown in the diagram, ∠AOB=130°, O is the center of circle O. Find the measure of ∠BCA.
问题：如图所示，∠AOB=130°，⊙O的圆心为O。求∠BCA的大小。
Construction (CDL):
Shape(OAB,BO,OA)
Shape(OBC,CB)
Shape(OCA,AC)
Shape(AO,OB,BC,CA)
Cocircular(O,ABC)
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, ∠DAB=115°, O is the center of ⊙O. Find the measure of ∠BOD.
问题：如图所示，∠DAB=115°，⊙O的圆心为O。求∠BOD的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(ODA,AD)
Shape(AB,BO,OD,DA)
Shape(DO,OB,BC,CD)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),115)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOD))
Answer: 130
Theorem sequence:
arc_property_circumference_angle_internal(1,ODB,A)
arc_property_center_angle(1,ODB,O)
Problem: As shown in the diagram, ∠BFC=35°, ∠CED=35°, the center of ⊙O is O. Find the measure of ∠DAB.
问题：如图所示，∠BFC=35°，∠CED=35°，O是圆O的圆心。求∠DAB的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(ODA,AD)
Shape(AB,BC,CD,DA)
Shape(OBC,BF,FC)
Shape(OCD,CE,ED)
Collinear(ADE)
Collinear(ABF)
Collinear(DCF)
Collinear(ECB)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(BFC),35)
Equal(MeasureOfAngle(CED),35)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DAB))
Answer: 55
Theorem sequence:
triangle_property_angle_sum(1,EDC)
adjacent_complementary_angle(1,EDC,CDA)
triangle_property_angle_sum(1,FCB)
adjacent_complementary_angle(1,ABC,CBF)
vertical_angle(1,DCE,FCB)
arc_property_circumference_angle_external(1,OAC,D)
arc_property_circumference_angle_internal(1,OAC,B)
triangle_property_angle_sum(1,EAB)
Problem: As shown in the diagram, ∠ABC=35°, O is the center of circle O. Find the measure of ∠AOC.
问题：如图所示，∠ABC=35°，O是⊙O的圆心。求∠AOC的大小。
Construction (CDL):
Shape(OCA,AO,OC)
Shape(OAB,BA)
Shape(OA,AB,BC,CO)
Shape(OBC,CB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),35)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AOC))
Answer: 70
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,B)
arc_property_center_angle(1,OCA,O)
Problem: As shown in the diagram, ∠OBC=42°, the center of ⊙O is O. Find the measure of ∠BAC.
问题：如图所示，∠OBC=42°，O是圆O的圆心。求∠BAC的大小。
Construction (CDL):
Shape(OCB,BO,OC)
Shape(OB,BC,CO)
Shape(BA,AC,CB)
Shape(OBA,AB)
Shape(OAC,CA)
Cocircular(O,BAC)
Conditions (CDL):
Equal(MeasureOfAngle(OBC),42)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BAC))
Answer: 132
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_internal(1,OBC,A)
Problem: As shown in the diagram, ∠CBA=40°. Find the measure of ∠ADC.
问题：如图所示，∠CBA=40°。求∠ADC的大小。
Construction (CDL):
Shape(OBA,AB)
Shape(OCB,BC)
Shape(BA,AD,DC,CB)
Shape(OAD,DA)
Shape(ODC,CD)
Cocircular(O,ADCB)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),40)
Goal: Value(MeasureOfAngle(ADC))
Answer: 140
Theorem sequence:
arc_property_circumference_angle_external(1,OAC,B)
arc_property_circumference_angle_internal(1,OAC,D)