SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠COB=60°, O is the center of circle O. Find the measure of ∠CAB.
问题：如图所示，∠COB=60°，⊙O的圆心为O。求∠CAB的大小。
Construction (CDL):
Shape(OB,BD,DO)
Shape(DC,CA,AD)
Shape(BO,OD,DA,OAB)
Shape(CD,DB,OBC)
Shape(AC,OCA)
Collinear(ODC)
Collinear(BDA)
Cocircular(O,BCA)
Conditions (CDL):
Equal(MeasureOfAngle(COB),60)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 30
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠BAC=60°, O is the center of ⊙O, BD is the diameter of ⊙O. Find the measure of ∠CBD.
问题：如图所示，∠BAC=60°，O是⊙O的圆心，⊙O的直径为BD。求∠CBD的大小。
Construction (CDL):
Shape(BA,AE,EO,OB)
Shape(BO,OE,EC,CB)
Shape(ED,DC,CE)
Shape(DE,EA,OAD)
Shape(CD,ODC)
Shape(BC,OCB)
Shape(AB,OBA)
Collinear(AEC)
Collinear(BOED)
Cocircular(O,ADCB)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),60)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BD,O)
Goal: Value(MeasureOfAngle(CBD))
Answer: 30
Theorem sequence:
arc_property_circumference_angle_external(1,OCB,A)
arc_property_circumference_angle_external(1,OCB,D)
diameter_of_circle_property_right_angle(1,DCB,O)
triangle_property_angle_sum(1,DCB)
Problem: As shown in the diagram, ∠CBA=36°, O is the center of circle O, AB∥CD. Find the measure of ∠BOD.
问题：如图所示，∠CBA=36°，⊙O的圆心为O，AB∥CD。求∠BOD的大小。
Construction (CDL):
Shape(BC,CD,DO,OB)
Shape(CB,BA,OAC)
Shape(AB,OBA)
Shape(BO,OD,ODB)
Shape(DC,OCD)
Cocircular(O,ACDB)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),36)
IsCentreOfCircle(O,O)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(BOD))
Answer: 72
Theorem sequence:
parallel_property_alternate_interior_angle(2,AB,CD)
arc_property_circumference_angle_external(1,ODB,C)
arc_property_center_angle(1,ODB,O)
Problem: As shown in the diagram, ∠BAD=48°, the center of circle O is O, the diameter of circle O is AB. Find the measure of ∠DCA.
问题：如图所示，∠BAD=48°，⊙O的圆心为O，圆O的直径为AB。求∠DCA的大小。
Construction (CDL):
Shape(CE,EB,BC)
Shape(CA,AO,OE,EC)
Shape(EO,OA,AD,DE)
Shape(CB,OCB)
Shape(AC,OCA)
Shape(DA,OAD)
Shape(BE,ED,DB)
Shape(BD,ODB)
Collinear(CED)
Collinear(BEOA)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(BAD),48)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(DCA))
Answer: 42
Theorem sequence:
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, ∠BCD=40°, O is the center of ⊙O, AB is the diameter of circle O. Find the measure of ∠DBA.
问题：如图所示，∠BCD=40°，O是⊙O的圆心，AB是⊙O的直径。求∠DBA的大小。
Construction (CDL):
Shape(AO,OE,EC,OCA)
Shape(EO,OA,AD,DE)
Shape(DA,OAD)
Shape(BD,ODB)
Shape(CB,OBC)
Shape(CE,EB,BC)
Shape(ED,DB,BE)
Collinear(AOEB)
Collinear(DEC)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(DBA))
Answer: 50
Theorem sequence:
arc_property_circumference_angle_external(1,ODB,C)
arc_property_circumference_angle_external(1,ODB,A)
diameter_of_circle_property_right_angle(1,ADB,O)
triangle_property_angle_sum(1,ADB)
Problem: As shown in the diagram, ∠CAB=20°, O is the center of ⊙O, the diameter of circle O is AB. Find the measure of ∠CDA.
问题：如图所示，∠CAB=20°，圆O的圆心为O，AB是⊙O的直径。求∠CDA的大小。
Construction (CDL):
Shape(AC,CD,DA)
Shape(CA,AO,OB,BC)
Shape(DC,OCD)
Shape(CB,OBC)
Shape(AD,ODA)
Collinear(AOB)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),20)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CDA))
Answer: 110
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,BCA)
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_internal(1,OCA,D)
Problem: As shown in the diagram, ∠AED=90°, ∠BOC=70°, the center of circle O is O, the diameter of circle O is AB. Find the measure of ∠OAC.
问题：如图所示，∠AED=90°，∠BOC=70°，O是圆O的圆心，AB是圆O的直径。求∠OAC的大小。
Construction (CDL):
Shape(AC,CO,OA)
Shape(OC,CE,EO)
Shape(DE,EB,OBD)
Shape(AO,OE,ED,EDA)
Shape(CA,OAC)
Shape(BE,EC,OCB)
Collinear(CED)
Collinear(AOEB)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(AED),90)
Equal(MeasureOfAngle(BOC),70)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(OAC))
Answer: 35
Theorem sequence:
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_external(1,OCB,A)
Problem: As shown in the diagram, ∠AOD=70°, ∠BOD=40°, the center of circle O is O, CD is the diameter of ⊙O. Find the measure of ∠ACE.
问题：如图所示，∠AOD=70°，∠BOD=40°，⊙O的圆心为O，CD是圆O的直径。求∠ACE的大小。
Construction (CDL):
Shape(AC,CE,EA)
Shape(EO,OB,BE)
Shape(CA,OAC)
Shape(OE,EC,CO)
Shape(BO,OD,ODB)
Shape(AE,EB,OBA)
Collinear(CEB)
Collinear(AEO)
Collinear(COD)
Cocircular(O,CDBA)
Conditions (CDL):
Equal(MeasureOfAngle(AOD),70)
Equal(MeasureOfAngle(BOD),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(CD,O)
Goal: Value(MeasureOfAngle(ACE))
Answer: 15
Theorem sequence:
adjacent_complementary_angle(1,COA,AOD)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OC,O)
isosceles_triangle_judgment_line_equal(1,OAC)
isosceles_triangle_property_angle_equal(1,OAC)
triangle_property_angle_sum(1,OAC)
adjacent_complementary_angle(1,COB,BOD)
radius_of_circle_property_length_equal(1,OB,O)
isosceles_triangle_judgment_line_equal(1,OBC)
isosceles_triangle_property_angle_equal(1,OBC)
triangle_property_angle_sum(1,OBC)
angle_addition(1,ACE,ECO)
Problem: As shown in the diagram, ∠CAB=50°, O is the center of circle O. Find the measure of ∠OBC.
问题：如图所示，∠CAB=50°，O是圆O的圆心。求∠OBC的大小。
Construction (CDL):
Shape(AB,BO,OC,CA)
Shape(OB,BC,CO)
Shape(BA,OAB)
Shape(CB,OBC)
Shape(AC,OCA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),50)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(OBC))
Answer: 40
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
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)
Problem: As shown in the diagram, ∠CBA=110°, O is the center of circle O. Find the measure of ∠AOC.
问题：如图所示，∠CBA=110°，O是⊙O的圆心。求∠AOC的大小。
Construction (CDL):
Shape(CB,BA,AO,OC)
Shape(CO,OA,AD,DC)
Shape(BC,OCB)
Shape(AB,OBA)
Shape(DA,OAD)
Shape(CD,ODC)
Cocircular(O,BADC)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),110)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AOC))
Answer: 140
Theorem sequence:
arc_property_circumference_angle_internal(1,OCA,B)
arc_property_center_angle(1,OCA,O)
Problem: As shown in the diagram, the diameter of circle O is 8, ∠BCA=30°, the center of circle O is O, AD is the diameter of ⊙O. Find the length of line AB.
问题：如图所示，圆O的直径为8，∠BCA=30°，O是⊙O的圆心，⊙O的直径为AD。求直线AB的长度。
Construction (CDL):
Shape(CA,AE,EC)
Shape(AB,BE,EA)
Shape(EB,BD,DO,OE)
Shape(BA,OAB)
Shape(AC,OCA)
Shape(DB,OBD)
Shape(CE,EO,OD,ODC)
Collinear(AEOD)
Collinear(CEB)
Cocircular(O,ABDC)
Conditions (CDL):
Equal(DiameterOfCircle(O),8)
Equal(MeasureOfAngle(BCA),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AD,O)
Goal: Value(LengthOfLine(AB))
Answer: 4
Theorem sequence:
diameter_of_circle_property_length_equal(1,AD,O)
diameter_of_circle_property_right_angle(1,ABD,O)
arc_property_circumference_angle_external(1,OAB,C)
arc_property_circumference_angle_external(1,OAB,D)
sine_theorem(1,ABD)
Problem: As shown in the diagram, ∠BGO=90°, ∠OCA=35°, O is the center of ⊙O, the diameter of ⊙O is AB. Find the measure of ∠DOC.
问题：如图所示，∠BGO=90°，∠OCA=35°，O是⊙O的圆心，AB是圆O的直径。求∠DOC的大小。
Construction (CDL):
Shape(DE,EF,FD)
Shape(EA,AO,OE)
Shape(EO,OG,GF,FE)
Shape(CF,FG,GC)
Shape(GO,OB,BG)
Shape(AE,ED,ODA)
Shape(DF,FC,OCD)
Shape(CG,GB,OBC)
Shape(BO,OA,OAB)
Collinear(CFEA)
Collinear(CGO)
Collinear(DEO)
Collinear(DFGB)
Collinear(AOB)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(BGO),90)
Equal(MeasureOfAngle(OCA),35)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(DOC))
Answer: 70
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)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
altitude_of_triangle_judgment(1,OG,OBD)
radius_of_circle_property_length_equal(1,OD,O)
radius_of_circle_property_length_equal(1,OB,O)
isosceles_triangle_judgment_line_equal(1,OBD)
isosceles_triangle_property_line_coincidence(1,OBD,G)
Problem: As shown in the diagram, OA=2, ∠CDA=30°, the center of circle O is O, the diameter of ⊙O is AB. Find the length of line AC.
问题：如图所示，OA=2，∠CDA=30°，O是⊙O的圆心，AB是圆O的直径。求直线AC的长度。
Construction (CDL):
Shape(BO,OA,AE,EB)
Shape(AC,CE,EA)
Shape(EC,CD,DE)
Shape(DC,OCD)
Shape(BE,ED,ODB)
Shape(CA,OAC)
Shape(AO,OB,OBA)
Collinear(AOB)
Collinear(AED)
Collinear(CEB)
Cocircular(O,ACDB)
Conditions (CDL):
Equal(LengthOfLine(OA),2)
Equal(MeasureOfAngle(CDA),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(LengthOfLine(AC))
Answer: 2
Theorem sequence:
diameter_of_circle_property_right_angle(1,ACB,O)
arc_property_circumference_angle_external(1,OAC,B)
arc_property_circumference_angle_external(1,OAC,D)
radius_of_circle_property_length_equal(1,OA,O)
circle_property_length_of_radius_and_diameter(1,O)
diameter_of_circle_property_length_equal(1,AB,O)
sine_theorem(1,ACB)
Problem: As shown in the diagram, ∠BCA=20°, O is the center of circle O. Find the measure of ∠OAB.
问题：如图所示，∠BCA=20°，O是圆O的圆心。求∠OAB的大小。
Construction (CDL):
Shape(BC,CE,EB)
Shape(EO,OA,AE)
Shape(BE,EA,AB)
Shape(CB,OBC)
Shape(AO,OE,EC,OCA)
Shape(BA,OAB)
Collinear(CEA)
Collinear(BEO)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),20)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(OAB))
Answer: 70
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
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, the measure of ⌒OAC is 50, O is the center of circle O, the diameter of ⊙O is CB. Find the measure of ∠ACB.
问题：如图所示，弧OAC的角度为50，O是⊙O的圆心，⊙O的直径为CB。求∠ACB的大小。
Construction (CDL):
Shape(AC,CO,OA)
Shape(AO,OB,OBA)
Shape(CA,OAC)
Shape(BO,OC,OCB)
Collinear(COB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfArc(OAC),50)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(CB,O)
Goal: Value(MeasureOfAngle(ACB))
Answer: 65
Theorem sequence:
arc_property_center_angle(1,OAC,O)
radius_of_circle_property_length_equal(1,OC,O)
radius_of_circle_property_length_equal(1,OA,O)
isosceles_triangle_judgment_line_equal(1,OAC)
isosceles_triangle_property_angle_equal(1,OAC)
triangle_property_angle_sum(1,OAC)
Problem: As shown in the diagram, AB=CD, ∠AOB=70°, O is the center of ⊙O, the diameter of circle O is ED. Find the measure of ∠DEC.
问题：如图所示，AB=CD，∠AOB=70°，O是圆O的圆心，⊙O的直径为ED。求∠DEC的大小。
Construction (CDL):
Shape(AO,OB,BA)
Shape(OE,EC,CD,DO)
Shape(AB,OBA)
Shape(EO,OA,OAE)
Shape(CE,OEC)
Shape(DC,OCD)
Shape(BO,OD,ODB)
Collinear(EOD)
Cocircular(O,ECDBA)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(CD))
Equal(MeasureOfAngle(AOB),70)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(ED,O)
Goal: Value(MeasureOfAngle(DEC))
Answer: 35
Theorem sequence:
congruent_arc_judgment_chord_equal(1,OBA,OCD)
congruent_arc_property_measure_equal(1,OBA,OCD)
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_external(1,OCD,E)
Problem: As shown in the diagram, ∠OBA=40°, O is the center of ⊙O. Find the measure of ∠ACB.
问题：如图所示，∠OBA=40°，圆O的圆心为O。求∠ACB的大小。
Construction (CDL):
Shape(AO,OD,DA)
Shape(DC,CB,BD)
Shape(AD,DB,BA)
Shape(CD,DO,OA,OAC)
Shape(BC,OCB)
Shape(AB,OBA)
Collinear(ADC)
Collinear(ODB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(OBA),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ACB))
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)
triangle_property_angle_sum(1,OBA)
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_external(1,OBA,C)
Problem: As shown in the diagram, ∠CDA=35°, the center of circle O is O, AB is the diameter of ⊙O. Find the measure of ∠BOC.
问题：如图所示，∠CDA=35°，O是⊙O的圆心，AB是圆O的直径。求∠BOC的大小。
Construction (CDL):
Shape(DA,AE,ED)
Shape(EC,CO,OE)
Shape(CE,EA,OAC)
Shape(BO,OC,OCB)
Shape(DE,EO,OB,OBD)
Shape(AD,ODA)
Collinear(AEOB)
Collinear(DEC)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(CDA),35)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BOC))
Answer: 110
Theorem sequence:
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,D)
adjacent_complementary_angle(1,BOC,COA)
Problem: As shown in the diagram, AB=16, CD=20, ∠AEO=90°, the center of ⊙O is O, CD is the diameter of ⊙O. Find the length of line EP.
问题：如图所示，AB=16，CD=20，∠AEO=90°，O是⊙O的圆心，圆O的直径为CD。求直线EP的长度。
Construction (CDL):
Shape(CO,OD,ODC)
Shape(AO,OC,OCA)
Shape(OA,AE,EO)
Shape(DO,OE,EB,OBD)
Shape(BA,OAB)
Collinear(COD)
Collinear(AEB)
Collinear(OEP)
Cocircular(O,APBDC)
Conditions (CDL):
Equal(LengthOfLine(AB),16)
Equal(LengthOfLine(CD),20)
Equal(MeasureOfAngle(AEO),90)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(CD,O)
Goal: Value(LengthOfLine(EP))
Answer: 4
Theorem sequence:
circle_property_chord_perpendicular_bisect_chord(1,O,OE,AB)
line_addition(1,AE,EB)
diameter_of_circle_property_length_equal(1,CD,O)
circle_property_length_of_radius_and_diameter(1,O)
radius_of_circle_property_length_equal(1,OA,O)
right_triangle_judgment_angle(1,AEO)
right_triangle_property_pythagorean(1,AEO)
radius_of_circle_property_length_equal(1,OP,O)
line_addition(1,OE,EP)
Problem: As shown in the diagram, the length of arc OBC is equal to the length of ⌒OCD, ∠DAB=50°, the center of ⊙O is O, the diameter of circle O is AB. Find the measure of ∠ABC.
问题：如图所示，⌒OBC的长度与弧OCD的长度相等，∠DAB=50°，⊙O的圆心为O，圆O的直径为AB。求∠ABC的大小。
Construction (CDL):
Shape(DA,AO,OC,CD)
Shape(CO,OB,BC)
Shape(AD,ODA)
Shape(DC,OCD)
Shape(CB,OBC)
Shape(BO,OA,OAB)
Collinear(AOB)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(LengthOfArc(OBC),LengthOfArc(OCD))
Equal(MeasureOfAngle(DAB),50)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ABC))
Answer: 65
Theorem sequence:
congruent_arc_judgment_length_equal(1,OBC,OCD)
congruent_arc_property_measure_equal(1,OBC,OCD)
arc_property_circumference_angle_external(1,OBD,A)
arc_addition_measure(1,OBC,OCD)
arc_property_center_angle(1,OBC,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_property_angle_equal(1,OBC)
triangle_property_angle_sum(1,OBC)
Problem: As shown in the diagram, ∠DOB=110°, O is the center of circle O. Find the measure of ∠BCD.
问题：如图所示，∠DOB=110°，O是圆O的圆心。求∠BCD的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(AB,BO,OD,DA)
Shape(OB,BC,CD,DO)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(DOB),110)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCD))
Answer: 125
Theorem sequence:
arc_property_center_angle(1,OBD,O)
arc_property_circumference_angle_internal(1,OBD,C)
Problem: As shown in the diagram, ∠DCE=75°, O is the center of circle O. Find the measure of ∠DAB.
问题：如图所示，∠DCE=75°，⊙O的圆心为O。求∠DAB的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(AB,BC,CD,DA)
Shape(DC,CE)
Collinear(BCE)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(DCE),75)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DAB))
Answer: 75
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, ∠CDE=80°, the center of ⊙O is O. Find the measure of ∠CBA.
问题：如图所示，∠CDE=80°，圆O的圆心为O。求∠CBA的大小。
Construction (CDL):
Shape(OBA,AB)
Shape(OAD,DA)
Shape(ODC,CD)
Shape(OCB,BC)
Shape(AD,DC,CB,BA)
Shape(CD,DE)
Collinear(ADE)
Cocircular(O,ADCB)
Conditions (CDL):
Equal(MeasureOfAngle(CDE),80)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CBA))
Answer: 80
Theorem sequence:
adjacent_complementary_angle(1,ADC,CDE)
arc_property_circumference_angle_external(1,OCA,D)
arc_property_circumference_angle_internal(1,OCA,B)
Problem: As shown in the diagram, ∠CDA=65°, P is the center of circle O. Find the measure of ∠ABC.
问题：如图所示，∠CDA=65°，P是⊙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,CD,DA)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(CDA),65)
IsCentreOfCircle(P,O)
Goal: Value(MeasureOfAngle(ABC))
Answer: 115
Theorem sequence:
arc_property_circumference_angle_external(1,OAC,D)
arc_property_circumference_angle_internal(1,OAC,B)
Problem: As shown in the diagram, ∠ECB=70°, O is the center of ⊙O. Find the measure of ∠DAB.
问题：如图所示，∠ECB=70°，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(EC,CB)
Collinear(ECD)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(ECB),70)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DAB))
Answer: 70
Theorem sequence:
adjacent_complementary_angle(1,ECB,BCD)
arc_property_circumference_angle_external(1,ODB,C)
arc_property_circumference_angle_internal(1,ODB,A)
Problem: As shown in the diagram, ∠DFE=27°, ∠EAB=53°, the center of circle O is O. Find the measure of ∠BCD.
问题：如图所示，∠DFE=27°，∠EAB=53°，O是⊙O的圆心。求∠BCD的大小。
Construction (CDL):
Shape(OEA,AE)
Shape(OAB,BA)
Shape(OBD,DB)
Shape(ODE,ED)
Shape(AB,BD,DE,EA)
Shape(OBD,BC,CD)
Shape(ODE,DF,FE)
Collinear(EDC)
Collinear(BDF)
Collinear(AEF)
Collinear(ABC)
Cocircular(O,ABDE)
Conditions (CDL):
Equal(MeasureOfAngle(DFE),27)
Equal(MeasureOfAngle(EAB),53)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCD))
Answer: 47
Theorem sequence:
triangle_property_angle_sum(1,FAB)
arc_property_circumference_angle_external(1,ODA,B)
arc_property_circumference_angle_internal(1,ODA,E)
triangle_property_angle_sum(1,CEA)
Problem: As shown in the diagram, ∠DEN=30°, ∠MAB=25°, the center of ⊙O is O. Find the measure of ∠NOM.
问题：如图所示，∠DEN=30°，∠MAB=25°，O是圆O的圆心。求∠NOM的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(AB,BM,MA)
Shape(OBC,CM,MB)
Shape(OM,MC,CN,NO)
Shape(OCD,DN,NC)
Shape(ODE,ED)
Shape(EN,ND,DE)
Shape(OEA,AM,MO,ON,NE)
Collinear(AMC)
Collinear(BMO)
Collinear(OND)
Collinear(CNE)
Cocircular(O,ABCDE)
Conditions (CDL):
Equal(MeasureOfAngle(DEN),30)
Equal(MeasureOfAngle(MAB),25)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(NOM))
Answer: 110
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_external(1,OCD,E)
arc_addition_measure(1,OBC,OCD)
arc_property_center_angle(1,OBD,O)
Problem: As shown in the diagram, ∠ABC=80°, the center of circle O is O. Find the measure of ∠CDA.
问题：如图所示，∠ABC=80°，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),80)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CDA))
Answer: 100
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_internal(1,OCA,D)
Problem: As shown in the diagram, ∠ADC=130°, the center of circle O is O. Find the measure of ∠COA.
问题：如图所示，∠ADC=130°，圆O的圆心为O。求∠COA的大小。
Construction (CDL):
Shape(OBA,AB)
Shape(OAD,DA)
Shape(ODC,CD)
Shape(OCB,BC)
Shape(AO,OC,CB,BA)
Shape(OA,AD,DC,CO)
Cocircular(O,ADCB)
Conditions (CDL):
Equal(MeasureOfAngle(ADC),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,D)
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°，O是⊙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(OC,CD,DA,AO)
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, ∠DAB=100°, O is the center of ⊙O. Find the measure of ∠DCE.
问题：如图所示，∠DAB=100°，O是⊙O的圆心。求∠DCE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(DAB),100)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DCE))
Answer: 100
Theorem sequence:
arc_property_circumference_angle_external(1,OBD,A)
arc_property_circumference_angle_internal(1,OBD,C)
adjacent_complementary_angle(1,BCD,DCE)
Problem: As shown in the diagram, ∠GBC=48°, O is the center of circle O, OE is perpendicular to CE. Find the measure of ∠CBD.
问题：如图所示，∠GBC=48°，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),48)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(OE,CE)
Goal: Value(MeasureOfAngle(CBD))
Answer: 84
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, ∠BDE=25°, the center of circle O is O, C is the midpoint of arc OAB. Find the measure of ∠EOA.
问题：如图所示，∠BDE=25°，O是圆O的圆心，C平分⌒OAB。求∠EOA的大小。
Construction (CDL):
Shape(OAC,CE,EO,OA)
Shape(OCB,BE,EC)
Shape(OBD,DB)
Shape(BD,DE,EB)
Shape(ODA,AO,OE,ED)
Collinear(OEB)
Collinear(CED)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(BDE),25)
IsCentreOfCircle(O,O)
IsMidpointOfArc(C,OAB)
Goal: Value(MeasureOfAngle(EOA))
Answer: 100
Theorem sequence:
arc_property_circumference_angle_external(1,OCB,D)
congruent_arc_judgment_length_equal(1,OAC,OCB)
congruent_arc_property_measure_equal(1,OAC,OCB)
arc_addition_measure(1,OAC,OCB)
arc_property_center_angle(1,OAB,O)
Problem: As shown in the diagram, AB=2, ∠ADE=30°, O is the center of circle O. Find the length of line BC.
问题：如图所示，AB=2，∠ADE=30°，圆O的圆心为O。求直线BC的长度。
Construction (CDL):
Shape(OAD,DA)
Shape(ODB,BE,ED)
Shape(OBC,CB)
Shape(OCA,AC)
Shape(AD,DE,EO,OA)
Shape(AO,OE,EC,CA)
Shape(CE,EB,BC)
Collinear(DEC)
Collinear(AOEB)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(LengthOfLine(AB),2)
Equal(MeasureOfAngle(ADE),30)
IsCentreOfCircle(O,O)
Goal: Value(LengthOfLine(BC))
Answer: sqrt(3)
Theorem sequence:
diameter_of_circle_judgment_pass_centre(1,AOB,O)
arc_property_circumference_angle_external(1,OCA,D)
arc_property_circumference_angle_external(1,OCA,B)
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,ABC)
sine_theorem(1,BCA)
Problem: As shown in the diagram, ∠BCA=35°, the center of ⊙O is O. Find the measure of ∠BOA.
问题：如图所示，∠BCA=35°，圆O的圆心为O。求∠BOA的大小。
Construction (CDL):
Shape(OBC,CB)
Shape(OCA,AC)
Shape(OAB,BO,OA)
Shape(CA,AO,OB,BC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),35)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 70
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,C)
arc_property_center_angle(1,OAB,O)