SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠BCA=40°, the center of circle O is O. Find the measure of ∠BOA.
问题：如图所示，∠BCA=40°，O是圆O的圆心。求∠BOA的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(OA,OAB,BO)
Shape(CA,AO,OB,OBC)
Shape(BC,CA)
Cocircular(O,CAB)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 80
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,C)
arc_property_center_angle(1,OAB,O)
Problem: As shown in the diagram, ∠BCA=15°, the center of ⊙O is O. Find the measure of ∠BOA.
问题：如图所示，∠BCA=15°，⊙O的圆心为O。求∠BOA的大小。
Construction (CDL):
Shape(OA,AD,DO)
Shape(DA,AB,BD)
Shape(OAB,BA)
Shape(CD,DB,BC)
Shape(OD,DC,OCA,AO)
Shape(OBC,CB)
Collinear(ODB)
Collinear(ADC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),15)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 30
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, AB=6, ∠BCA=45°, O is the center of ⊙O, the diameter of circle O is AC, M bisects segment AB, N bisects segment CB. Find the length of line MN.
问题：如图所示，AB=6，∠BCA=45°，O是圆O的圆心，AC是圆O的直径，M是线段AB的中点，N平分线段CB。求直线MN的长度。
Construction (CDL):
Shape(AO,OC,OCA)
Shape(OA,AM,MN,NC,CO)
Shape(MB,BN,NM)
Shape(CN,NB,OBC)
Shape(BM,MA,OAB)
Collinear(COA)
Collinear(CNB)
Collinear(AMB)
Cocircular(O,CAB)
Conditions (CDL):
Equal(LengthOfLine(AB),6)
Equal(MeasureOfAngle(BCA),45)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AC,O)
IsMidpointOfLine(M,AB)
IsMidpointOfLine(N,CB)
Goal: Value(LengthOfLine(MN))
Answer: 3*sqrt(2)
Theorem sequence:
diameter_of_circle_property_right_angle(1,ABC,O)
sine_theorem(1,ABC)
midsegment_of_triangle_judgment_midpoint(1,NM,BCA)
midsegment_of_triangle_property_length(1,NM,BCA)
Problem: As shown in the diagram, ∠COB=100°, O is the center of ⊙O. Find the measure of ∠CAB.
问题：如图所示，∠COB=100°，O是⊙O的圆心。求∠CAB的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(AB,BO,OC,CA)
Shape(OB,OBC,CO)
Shape(AC,OCA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(COB),100)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 50
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠BAD=36°, O is the center of circle O, AB is parallel to CD. Find the measure of ∠COA.
问题：如图所示，∠BAD=36°，⊙O的圆心为O，AB平行于CD。求∠COA的大小。
Construction (CDL):
Shape(AB,OBA)
Shape(OA,OAC,CO)
Shape(AO,OC,CD,DA)
Shape(DC,OCD)
Shape(BA,AD,ODB)
Cocircular(O,BACD)
Conditions (CDL):
Equal(MeasureOfAngle(BAD),36)
IsCentreOfCircle(O,O)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(COA))
Answer: 72
Theorem sequence:
parallel_property_alternate_interior_angle(1,AB,CD)
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,D)
Problem: As shown in the diagram, ∠BCA=50°, O is the center of ⊙O. Find the measure of ∠BOA.
问题：如图所示，∠BCA=50°，⊙O的圆心为O。求∠BOA的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AO,OB,BC)
Shape(CB,OBC)
Shape(OA,OAB,BO)
Cocircular(O,CAB)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),50)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 100
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠BAC=25°, O is the center of circle O. Find the measure of ∠BOC.
问题：如图所示，∠BAC=25°，⊙O的圆心为O。求∠BOC的大小。
Construction (CDL):
Shape(OC,CD,DO)
Shape(DC,CB,BD)
Shape(BC,OCB)
Shape(AD,DB,BA)
Shape(OBA,AB)
Shape(OD,DA,OAC,CO)
Collinear(ODB)
Collinear(ADC)
Cocircular(O,CBA)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),25)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOC))
Answer: 50
Theorem sequence:
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_external(1,OCB,A)
Problem: As shown in the diagram, ∠CDB=40°, the center of ⊙O is O, the diameter of ⊙O is AB. Find the measure of ∠AOC.
问题：如图所示，∠CDB=40°，⊙O的圆心为O，AB是⊙O的直径。求∠AOC的大小。
Construction (CDL):
Shape(ODB,BD)
Shape(ED,DB,BE)
Shape(EB,OBC,CE)
Shape(OE,EC,CO)
Shape(DE,EO,OA,OAD)
Shape(OC,OCA,AO)
Collinear(BEOA)
Collinear(DEC)
Cocircular(O,DBCA)
Conditions (CDL):
Equal(MeasureOfAngle(CDB),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(AOC))
Answer: 100
Theorem sequence:
arc_property_center_angle(1,OBA,O)
arc_property_center_angle(1,OCA,O)
arc_property_circumference_angle_external(1,OBC,D)
arc_addition_measure(1,OBC,OCA)
flat_angle(1,AOB)
Problem: As shown in the diagram, ∠CDB=33°, the center of circle O is O, the diameter of circle O is AB. Find the measure of ∠AOC.
问题：如图所示，∠CDB=33°，O是圆O的圆心，AB是⊙O的直径。求∠AOC的大小。
Construction (CDL):
Shape(DB,BE,ED)
Shape(ODB,BD)
Shape(EB,OBC,CE)
Shape(OE,EC,CO)
Shape(OC,OCA,AO)
Shape(DE,EO,OA,OAD)
Collinear(BEOA)
Collinear(DEC)
Cocircular(O,DBCA)
Conditions (CDL):
Equal(MeasureOfAngle(CDB),33)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(AOC))
Answer: 114
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,D)
flat_angle(1,AOB)
angle_addition(1,AOC,COE)
Problem: As shown in the diagram, ∠AOB=50°, O is the center of ⊙O, B bisects ⌒OCA. Find the measure of ∠BDC.
问题：如图所示，∠AOB=50°，圆O的圆心为O，B是⌒OCA的中点。求∠BDC的大小。
Construction (CDL):
Shape(AO,OB,OBA)
Shape(OA,OAD,DB,BO)
Shape(BD,DC,OCB)
Shape(ODC,CD)
Cocircular(O,BADC)
Conditions (CDL):
Equal(MeasureOfAngle(AOB),50)
IsCentreOfCircle(O,O)
IsMidpointOfArc(B,OCA)
Goal: Value(MeasureOfAngle(BDC))
Answer: 25
Theorem sequence:
arc_property_center_angle(1,OBA,O)
congruent_arc_judgment_length_equal(1,OBA,OCB)
congruent_arc_property_measure_equal(1,OBA,OCB)
arc_property_circumference_angle_external(1,OCB,D)
Problem: As shown in the diagram, the length of arc OAB is equal to the length of ⌒OCA, ∠BCA=75°, the center of circle O is O. Find the measure of ∠CAB.
问题：如图所示，弧OAB的长度与⌒OCA的长度相等，∠BCA=75°，圆O的圆心为O。求∠CAB的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCA,AC)
Shape(AB,BC,CA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(LengthOfArc(OAB),LengthOfArc(OCA))
Equal(MeasureOfAngle(BCA),75)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 30
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, ∠AOB=48°, O is the center of ⊙O, AE⊥BE, BE is perpendicular to OE, CE is perpendicular to AE. Find the measure of ∠CDA.
问题：如图所示，∠AOB=48°，圆O的圆心为O，AE垂直于BE，BE⊥OE，CE垂直于AE。求∠CDA的大小。
Construction (CDL):
Shape(AE,EB,OBA)
Shape(BE,EO,OB)
Shape(OAC,CE,EA)
Shape(CD,DA)
Shape(OE,EC,OCB,BO)
Collinear(CEB)
Collinear(AEO)
Cocircular(O,ACDB)
Conditions (CDL):
Equal(MeasureOfAngle(AOB),48)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AE,BE)
PerpendicularBetweenLine(BE,OE)
PerpendicularBetweenLine(CE,AE)
Goal: Value(MeasureOfAngle(CDA))
Answer: 24
Theorem sequence:
circle_property_chord_perpendicular_bisect_arc(1,OBC,OEA)
congruent_arc_judgment_length_equal(1,OAC,OBA)
congruent_arc_property_measure_equal(1,OAC,OBA)
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_external(1,OAC,D)
Problem: As shown in the diagram, DC=OE, ∠DCO=20°, O is the center of circle O, AB is the diameter of circle O. Find the measure of ∠EOB.
问题：如图所示，DC=OE，∠DCO=20°，O是圆O的圆心，圆O的直径为AB。求∠EOB的大小。
Construction (CDL):
Shape(DC,CA,ODA)
Shape(ED,DO,OE)
Shape(OD,ODA,AO)
Shape(OED,DE)
Shape(EO,OB,OBE)
Shape(OA,OAB,BO)
Collinear(CAOB)
Collinear(CDE)
Cocircular(O,EDAB)
Conditions (CDL):
Equal(LengthOfLine(DC),LengthOfLine(OE))
Equal(MeasureOfAngle(DCO),20)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(EOB))
Answer: 60
Theorem sequence:
radius_of_circle_property_length_equal(1,OE,O)
radius_of_circle_property_length_equal(1,OD,O)
isosceles_triangle_judgment_line_equal(1,DCO)
isosceles_triangle_property_angle_equal(1,DCO)
triangle_property_angle_sum(1,DCO)
flat_angle(1,COB)
flat_angle(1,EDC)
angle_addition(1,EDO,ODC)
angle_addition(1,DOE,EOB)
angle_addition(1,AOD,DOB)
isosceles_triangle_judgment_line_equal(1,OED)
isosceles_triangle_property_angle_equal(1,OED)
triangle_property_angle_sum(1,EDO)
Problem: As shown in the diagram, ∠CAB=24°, the center of circle O is O, the diameter of circle O is AB. Find the measure of ∠ADC.
问题：如图所示，∠CAB=24°，圆O的圆心为O，圆O的直径为AB。求∠ADC的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(CA,AO,OE,EC)
Shape(CE,EB,OBC)
Shape(OA,AD,DE,EO)
Shape(OAD,DA)
Shape(BE,ED,ODB)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,CADB)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),24)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ADC))
Answer: 66
Theorem sequence:
arc_property_center_angle(1,OBA,O)
flat_angle(1,AOE)
arc_property_circumference_angle_external(1,OCA,D)
arc_property_circumference_angle_external(1,OBC,A)
arc_addition_measure(1,OBC,OCA)
Problem: As shown in the diagram, ∠BOC=112°, the center of ⊙O is O, AB is the diameter of ⊙O, AD∥CO. Find the measure of ∠AOD.
问题：如图所示，∠BOC=112°，O是⊙O的圆心，圆O的直径为AB，AD∥CO。求∠AOD的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(OD,DA,AO)
Shape(OB,OBD,DO)
Shape(OA,OAC,CO)
Shape(OC,OCB,BO)
Collinear(AOB)
Cocircular(O,DACB)
Conditions (CDL):
Equal(MeasureOfAngle(BOC),112)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
ParallelBetweenLine(AD,CO)
Goal: Value(MeasureOfAngle(AOD))
Answer: 44
Theorem sequence:
flat_angle(1,BOA)
angle_addition(1,BOC,COA)
parallel_property_alternate_interior_angle(1,AD,CO)
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)
Problem: As shown in the diagram, ∠ABO=50°, O is the center of circle O. Find the measure of ∠BCA.
问题：如图所示，∠ABO=50°，⊙O的圆心为O。求∠BCA的大小。
Construction (CDL):
Shape(OA,AD,DO)
Shape(DA,AB,BD)
Shape(CD,DB,BC)
Shape(OAB,BA)
Shape(OBC,CB)
Shape(AO,OD,DC,OCA)
Collinear(ADC)
Collinear(ODB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(ABO),50)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
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,OAB)
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠OBC=40°, the center of circle O is O, the diameter of ⊙O is AC. Find the measure of ∠BOA.
问题：如图所示，∠OBC=40°，⊙O的圆心为O，AC是⊙O的直径。求∠BOA的大小。
Construction (CDL):
Shape(OC,OCA,AO)
Shape(OA,OAB,BO)
Shape(OB,BC,CO)
Shape(OBC,CB)
Collinear(AOC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(OBC),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AC,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 80
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)
flat_angle(1,COA)
angle_addition(1,COB,BOA)
Problem: As shown in the diagram, Add(MeasureOfAngle(BAC)=MeasureOfAngle(DAE)), BC=8, DE=6, the center of ⊙A is A, CF is the diameter of circle A. Find the length of line AC.
问题：如图所示，Add(MeasureOfAngle(BAC)=MeasureOfAngle(DAE))，BC=8，DE=6，⊙A的圆心为A，⊙A的直径为CF。求直线AC的长度。
Construction (CDL):
Shape(AC,CB,BA)
Shape(ACB,BC)
Shape(AB,BF,FA)
Shape(ABF,FB)
Shape(AF,AFE,EA)
Shape(AE,ED,DA)
Shape(AED,DE)
Shape(AD,ADC,CA)
Collinear(CAF)
Cocircular(A,CBFED)
Conditions (CDL):
Equal(Add(MeasureOfAngle(BAC),MeasureOfAngle(DAE)),180)
Equal(LengthOfLine(BC),8)
Equal(LengthOfLine(DE),6)
IsCentreOfCircle(A,A)
IsDiameterOfCircle(CF,A)
Goal: Value(LengthOfLine(AC))
Answer: 5
Theorem sequence:
flat_angle(1,FAC)
angle_addition(1,FAB,BAC)
arc_property_center_angle(1,ABF,A)
arc_property_center_angle(1,AED,A)
congruent_arc_judgment_measure_equal(1,ABF,AED)
congruent_arc_property_chord_equal(1,ABF,AED)
diameter_of_circle_property_right_angle(1,CBF,A)
right_triangle_judgment_angle(1,CBF)
right_triangle_property_pythagorean(1,CBF)
diameter_of_circle_property_length_equal(1,CF,A)
circle_property_length_of_radius_and_diameter(1,A)
radius_of_circle_property_length_equal(1,AC,A)
Problem: As shown in the diagram, AB=12, AC=12, BM=ME, MD=2, O is the center of circle O, AB is the diameter of ⊙O, BM is perpendicular to OM, OM is perpendicular to EM. Find the length of line BE.
问题：如图所示，AB=12，AC=12，BM=ME，MD=2，O是圆O的圆心，⊙O的直径为AB，BM垂直于OM，OM垂直于EM。求直线BE的长度。
Construction (CDL):
Shape(OA,OAB,BO)
Shape(OB,BM,MO)
Shape(MB,BD,DM)
Shape(OBD,DB)
Shape(EM,MD,ODE)
Shape(ODE,DC,CE)
Shape(OM,ME,EA,AO)
Shape(AE,OEA)
Collinear(BME)
Collinear(BDC)
Collinear(OMD)
Collinear(AEC)
Collinear(AOB)
Cocircular(O,ABDE)
Conditions (CDL):
Equal(LengthOfLine(AB),12)
Equal(LengthOfLine(AC),12)
Equal(LengthOfLine(BM),LengthOfLine(ME))
Equal(LengthOfLine(MD),2)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
PerpendicularBetweenLine(BM,OM)
PerpendicularBetweenLine(OM,EM)
Goal: Value(LengthOfLine(BE))
Answer: 4*sqrt(5)
Theorem sequence:
diameter_of_circle_property_length_equal(1,AB,O)
circle_property_length_of_radius_and_diameter(1,O)
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OD,O)
line_addition(1,OM,MD)
right_triangle_judgment_angle(1,BMO)
right_triangle_property_pythagorean(1,BMO)
line_addition(1,BM,ME)
Problem: As shown in the diagram, ∠CBO=30°, ∠OAC=20°, the center of ⊙O is O. Find the measure of ∠AOB.
问题：如图所示，∠CBO=30°，∠OAC=20°，⊙O的圆心为O。求∠AOB的大小。
Construction (CDL):
Shape(AC,CB,BO,OA)
Shape(OAC,CA)
Shape(OCB,BC)
Shape(AO,OB,OBA)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(CBO),30)
Equal(MeasureOfAngle(OAC),20)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AOB))
Answer: 100
Theorem sequence:
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_external(1,OBA,C)
quadrilateral_property_angle_sum(1,ACBO)
round_angle(1,BOA,AOB)
Problem: As shown in the diagram, BD=2, ∠BAC=60°, the center of ⊙O is O, the diameter of ⊙O is BD. Find the length of line BC.
问题：如图所示，BD=2，∠BAC=60°，O是圆O的圆心，BD是圆O的直径。求直线BC的长度。
Construction (CDL):
Shape(OCB,BC)
Shape(CB,BO,OE,EC)
Shape(EO,OB,BA,AE)
Shape(OBA,AB)
Shape(EA,OAD,DE)
Shape(ED,DC,CE)
Shape(ODC,CD)
Collinear(BOED)
Collinear(AEC)
Cocircular(O,BADC)
Conditions (CDL):
Equal(LengthOfLine(BD),2)
Equal(MeasureOfAngle(BAC),60)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BD,O)
Goal: Value(LengthOfLine(BC))
Answer: sqrt(3)
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)
sine_theorem(1,BDC)
Problem: As shown in the diagram, AC=4, ∠BAC=30°, the center of ⊙O is O, the diameter of circle O is AC, AD is perpendicular to BD, BD⊥CD. Find the length of line BD.
问题：如图所示，AC=4，∠BAC=30°，⊙O的圆心为O，AC是圆O的直径，AD垂直于BD，BD垂直于CD。求直线BD的长度。
Construction (CDL):
Shape(OBA,AB)
Shape(BA,AO,OD,DB)
Shape(BD,DC,CB)
Shape(OCB,BC)
Shape(OA,OAC,CD,DO)
Collinear(AODC)
Cocircular(O,BAC)
Conditions (CDL):
Equal(LengthOfLine(AC),4)
Equal(MeasureOfAngle(BAC),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AC,O)
PerpendicularBetweenLine(AD,BD)
PerpendicularBetweenLine(BD,CD)
Goal: Value(LengthOfLine(BD))
Answer: sqrt(3)
Theorem sequence:
diameter_of_circle_property_right_angle(1,CBA,O)
triangle_property_angle_sum(1,BAC)
sine_theorem(1,BAD)
sine_theorem(1,ACB)
Problem: As shown in the diagram, ∠BOC=50°, O is the center of circle O. Find the measure of ∠BAC.
问题：如图所示，∠BOC=50°，O是⊙O的圆心。求∠BAC的大小。
Construction (CDL):
Shape(OC,OCB,BO)
Shape(OB,OBC,OAC,CO)
Shape(BA,AC)
Cocircular(O,BAC)
Conditions (CDL):
Equal(MeasureOfAngle(BOC),50)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BAC))
Answer: 25
Theorem sequence:
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_external(1,OCB,A)
Problem: As shown in the diagram, ∠ACB=50°, O is the center of ⊙O. Find the measure of ∠OBA.
问题：如图所示，∠ACB=50°，⊙O的圆心为O。求∠OBA的大小。
Construction (CDL):
Shape(OCB,BC)
Shape(CB,BO,OA,AC)
Shape(OB,BA,AO)
Shape(OBA,AB)
Shape(OCA,AC)
Cocircular(O,CBA)
Conditions (CDL):
Equal(MeasureOfAngle(ACB),50)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(OBA))
Answer: 40
Theorem sequence:
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_external(1,OBA,C)
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OA,O)
isosceles_triangle_judgment_line_equal(1,OBA)
isosceles_triangle_property_angle_equal(1,OBA)
triangle_property_angle_sum(1,OBA)
Problem: As shown in the diagram, the length of ⌒OBA is equal to the length of ⌒OAC, ∠CDA=20°. Find the measure of arc OBA.
问题：如图所示，弧OBA的长度与弧OAC的长度相等，∠CDA=20°。求弧OBA的角度。
Construction (CDL):
Shape(OCD,DC)
Shape(OAC,CD,DA)
Shape(AO,OB,OBA)
Shape(OA,AD,ODB,BO)
Cocircular(O,CDBA)
Conditions (CDL):
Equal(LengthOfArc(OBA),LengthOfArc(OAC))
Equal(MeasureOfAngle(CDA),20)
Goal: Value(MeasureOfArc(OBA))
Answer: 40
Theorem sequence:
congruent_arc_judgment_length_equal(1,OAC,OBA)
congruent_arc_property_measure_equal(1,OAC,OBA)
arc_property_circumference_angle_external(1,OAC,D)
Problem: As shown in the diagram, ∠BOC=40°, the center of circle O is O, OB∥AC. Find the measure of ∠OAB.
问题：如图所示，∠BOC=40°，O是⊙O的圆心，OB平行于AC。求∠OAB的大小。
Construction (CDL):
Shape(OA,AD,DO)
Shape(DA,AC,CD)
Shape(OD,DB,BO)
Shape(OAC,CA)
Shape(BD,DC,OCB)
Shape(AO,OB,OBA)
Collinear(ADB)
Collinear(ODC)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(BOC),40)
IsCentreOfCircle(O,O)
ParallelBetweenLine(OB,AC)
Goal: Value(MeasureOfAngle(OAB))
Answer: 20
Theorem sequence:
parallel_property_alternate_interior_angle(1,OB,AC)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OC,O)
radius_of_circle_property_length_equal(1,OB,O)
isosceles_triangle_judgment_line_equal(1,OAC)
isosceles_triangle_judgment_line_equal(1,OAB)
isosceles_triangle_property_angle_equal(1,OAC)
isosceles_triangle_property_angle_equal(1,OAB)
triangle_property_angle_sum(1,OAC)
triangle_property_angle_sum(1,OAB)
angle_addition(1,BOD,DOA)
Problem: As shown in the diagram, ∠OCD=32°, the center of circle O is O, CE is perpendicular to OE, OE⊥DE. Find the measure of ∠DAB.
问题：如图所示，∠OCD=32°，圆O的圆心为O，CE⊥OE，OE垂直于DE。求∠DAB的大小。
Construction (CDL):
Shape(CO,OA,OAC)
Shape(OC,CE,EO)
Shape(EC,OCB,BE)
Shape(DE,EB,OBD)
Shape(AO,OE,ED,DA)
Shape(ODA,AD)
Collinear(CED)
Collinear(AOEB)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(OCD),32)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(CE,OE)
PerpendicularBetweenLine(OE,DE)
Goal: Value(MeasureOfAngle(DAB))
Answer: 29
Theorem sequence:
triangle_property_angle_sum(1,OCE)
arc_property_center_angle(1,OCB,O)
circle_property_chord_perpendicular_bisect_arc(1,OCD,OEB)
congruent_arc_judgment_length_equal(1,OCB,OBD)
congruent_arc_property_measure_equal(1,OCB,OBD)
arc_property_circumference_angle_external(1,OBD,A)
Problem: As shown in the diagram, ∠AOB=110°, O is the center of circle O. Find the measure of ∠BCA.
问题：如图所示，∠AOB=110°，⊙O的圆心为O。求∠BCA的大小。
Construction (CDL):
Shape(OA,OAB,BO)
Shape(AO,OB,BC,CA)
Shape(OCA,AC)
Shape(OBC,CB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(AOB),110)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 125
Theorem sequence:
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_internal(1,OBA,C)
Problem: As shown in the diagram, ∠CAD=56°, O is the center of circle O, the diameter of circle O is AD. Find the measure of ∠ABC.
问题：如图所示，∠CAD=56°，⊙O的圆心为O，圆O的直径为AD。求∠ABC的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(AB,BE,EA)
Shape(AE,EC,CA)
Shape(OCA,AC)
Shape(EB,OBD,DO,OE)
Shape(CE,EO,OD,DC)
Shape(ODC,CD)
Collinear(AEOD)
Collinear(BEC)
Cocircular(O,ABDC)
Conditions (CDL):
Equal(MeasureOfAngle(CAD),56)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AD,O)
Goal: Value(MeasureOfAngle(ABC))
Answer: 34
Theorem sequence:
diameter_of_circle_property_right_angle(1,DCA,O)
triangle_property_angle_sum(1,CAD)
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_external(1,OCA,D)
Problem: As shown in the diagram, OA=8, OB=8, ∠BCA=30°, the center of circle O is O. Find the length of line AB.
问题：如图所示，OA=8，OB=8，∠BCA=30°，⊙O的圆心为O。求直线AB的长度。
Construction (CDL):
Shape(OA,AD,DO)
Shape(DA,AB,BD)
Shape(OAB,BA)
Shape(CD,DB,BC)
Shape(OBC,CB)
Shape(OD,DC,OCA,AO)
Collinear(ADC)
Collinear(ODB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(LengthOfLine(OA),8)
Equal(LengthOfLine(OB),8)
Equal(MeasureOfAngle(BCA),30)
IsCentreOfCircle(O,O)
Goal: Value(LengthOfLine(AB))
Answer: 8
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,C)
arc_property_center_angle(1,OAB,O)
isosceles_triangle_judgment_line_equal(1,OAB)
isosceles_triangle_property_angle_equal(1,OAB)
triangle_property_angle_sum(1,OAB)
isosceles_triangle_judgment_angle_equal(1,ABO)
Problem: As shown in the diagram, AB=3, BE=3/2, DC=3, OB=7/2, the center of ⊙O is O, BE⊥OE. Find the length of line OE.
问题：如图所示，AB=3，BE=3/2，DC=3，OB=7/2，O是⊙O的圆心，BE⊥OE。求直线OE的长度。
Construction (CDL):
Shape(DA,OAF,FD)
Shape(OD,DF,OFG,GC,CO)
Shape(EA,AD,DO,OE)
Shape(BE,EO,OB)
Shape(BO,OC,CB)
Shape(OBA,AE,EB)
Shape(BC,CG,OGB)
Collinear(FDOCG)
Collinear(AEB)
Cocircular(O,AFGB)
Conditions (CDL):
Equal(LengthOfLine(AB),3)
Equal(LengthOfLine(BE),3/2)
Equal(LengthOfLine(DC),3)
Equal(LengthOfLine(OB),7/2)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(BE,OE)
Goal: Value(LengthOfLine(OE))
Answer: sqrt(10)
Theorem sequence:
right_triangle_judgment_angle(1,BEO)
right_triangle_property_pythagorean(1,BEO)
Problem: As shown in the diagram, AB=16, AO=10, OD=10, O is the center of ⊙O, BC is perpendicular to OC, OC⊥AC. Find the length of line CD.
问题：如图所示，AB=16，AO=10，OD=10，O是⊙O的圆心，BC⊥OC，OC⊥AC。求直线CD的长度。
Construction (CDL):
Shape(CA,AO,OC)
Shape(BC,CO,OB)
Shape(AC,CB,OBA)
Shape(OA,OAD,DO)
Shape(BO,OD,ODB)
Collinear(ACB)
Collinear(COD)
Cocircular(O,ADB)
Conditions (CDL):
Equal(LengthOfLine(AB),16)
Equal(LengthOfLine(AO),10)
Equal(LengthOfLine(OD),10)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(BC,OC)
PerpendicularBetweenLine(OC,AC)
Goal: Value(LengthOfLine(CD))
Answer: 16
Theorem sequence:
circle_property_chord_perpendicular_bisect_chord(1,O,OC,BA)
line_addition(1,AC,CB)
line_addition(1,CO,OD)
right_triangle_judgment_angle(1,OCA)
right_triangle_property_pythagorean(1,OCA)
Problem: As shown in the diagram, the length of arc OAC is equal to the length of ⌒OBD, ∠BOA=40°, the center of circle O is O. Find the measure of ∠DOC.
问题：如图所示，弧OAC的长度与⌒OBD的长度相等，∠BOA=40°，O是圆O的圆心。求∠DOC的大小。
Construction (CDL):
Shape(OC,OCD,DO)
Shape(OB,OBC,CO)
Shape(OA,OAB,BO)
Shape(OD,ODA,AO)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(LengthOfArc(OAC),LengthOfArc(OBD))
Equal(MeasureOfAngle(BOA),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DOC))
Answer: 40
Theorem sequence:
arc_addition_length(1,OAB,OBC)
arc_addition_length(1,OBC,OCD)
congruent_arc_judgment_length_equal(1,OAB,OCD)
congruent_arc_property_measure_equal(1,OAB,OCD)
arc_property_center_angle(1,OAB,O)
arc_property_center_angle(1,OCD,O)
Problem: As shown in the diagram, ∠EOD=40°, O is the center of circle O, AB is the diameter of ⊙O, CD is the diameter of ⊙O, BA is parallel to DE. Find the measure of ∠BOC.
问题：如图所示，∠EOD=40°，⊙O的圆心为O，AB是圆O的直径，圆O的直径为CD，BA平行于DE。求∠BOC的大小。
Construction (CDL):
Shape(OA,OAC,CO)
Shape(OC,OCB,BO)
Shape(AO,OE,OEA)
Shape(EO,OD,DE)
Shape(ODE,ED)
Shape(OB,OBD,DO)
Collinear(AOB)
Collinear(COD)
Cocircular(O,ACBDE)
Conditions (CDL):
Equal(MeasureOfAngle(EOD),40)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsDiameterOfCircle(CD,O)
ParallelBetweenLine(BA,DE)
Goal: Value(MeasureOfAngle(BOC))
Answer: 110
Theorem sequence:
radius_of_circle_property_length_equal(1,OD,O)
radius_of_circle_property_length_equal(1,OE,O)
isosceles_triangle_judgment_line_equal(1,ODE)
isosceles_triangle_property_angle_equal(1,ODE)
triangle_property_angle_sum(1,EOD)
parallel_property_collinear_extend(3,BA,DE,O)
parallel_property_corresponding_angle(1,OA,DE,C)
flat_angle(1,BOA)
angle_addition(1,BOC,COA)