SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠BCA=30°, the radius of circle O is 1, the center of ⊙O is O. Find the length of line AB.
问题：如图所示，∠BCA=30°，圆O的半径为1，O是⊙O的圆心。求直线AB的长度。
Construction (CDL):
Shape(AD,DO,OA)
Shape(DA,AB,BD)
Shape(CD,DB,BC)
Shape(BA,OAB)
Shape(CB,OBC)
Shape(AO,OD,DC,OCA)
Collinear(ODB)
Collinear(ADC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),30)
Equal(RadiusOfCircle(O),1)
IsCentreOfCircle(O,O)
Goal: Value(LengthOfLine(AB))
Answer: 1
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,C)
arc_property_center_angle(1,OAB,O)
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)
isosceles_triangle_judgment_angle_equal(1,ABO)
Problem: As shown in the diagram, ∠BCA=30°, O is the center of circle O. Find the measure of ∠BOA.
问题：如图所示，∠BCA=30°，O是圆O的圆心。求∠BOA的大小。
Construction (CDL):
Shape(AO,OC,OCA)
Shape(BA,OAB)
Shape(CB,OBC)
Shape(OA,AD,DO)
Shape(OD,DC,CO)
Shape(CD,DB,BC)
Shape(DA,AB,BD)
Collinear(ODB)
Collinear(ADC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),30)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 60
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,C)
arc_property_center_angle(1,OAB,O)
Problem: As shown in the diagram, ∠ABC=40°, AD is the angle bisector of ∠CAB, the center of ⊙O is O, the diameter of circle O is AB. Find the measure of ∠CBD.
问题：如图所示，∠ABC=40°，AD是∠CAB的角平分线，圆O的圆心为O，AB是⊙O的直径。求∠CBD的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AB,BE,EA)
Shape(AE,EC,CA)
Shape(EB,BD,DE)
Shape(CE,ED,ODC)
Shape(AC,OCA)
Shape(DB,OBD)
Collinear(BEC)
Collinear(AED)
Collinear(AOB)
Cocircular(O,BDCA)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),40)
IsBisectorOfAngle(AD,CAB)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CBD))
Answer: 25
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,ABC)
angle_addition(1,CAD,DAB)
arc_property_circumference_angle_external(1,ODC,B)
arc_property_circumference_angle_external(1,ODC,A)
Problem: As shown in the diagram, ∠BAD=30°, 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=30°，O是圆O的圆心，AB是圆O的直径。求∠DCA的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AE,EC)
Shape(CE,EB,OBC)
Shape(EA,AD,DE)
Shape(BE,ED,DB)
Shape(BD,ODB)
Shape(DA,OAD)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,CADB)
Conditions (CDL):
Equal(MeasureOfAngle(BAD),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(DCA))
Answer: 60
Theorem sequence:
diameter_of_circle_property_right_angle(1,ADB,O)
triangle_property_angle_sum(1,ADB)
arc_property_circumference_angle_external(1,OAD,C)
arc_property_circumference_angle_external(1,OAD,B)
Problem: As shown in the diagram, ∠AOC=80°, O is the center of circle O, DA∥BC. Find the measure of ∠BAD.
问题：如图所示，∠AOC=80°，O是圆O的圆心，DA平行于BC。求∠BAD的大小。
Construction (CDL):
Shape(OAD,DA)
Shape(ODB,BA,AD)
Shape(OBC,CB)
Shape(OCA,AO,OC)
Shape(AB,BC,CO,OA)
Cocircular(O,BCAD)
Conditions (CDL):
Equal(MeasureOfAngle(AOC),80)
IsCentreOfCircle(O,O)
ParallelBetweenLine(DA,BC)
Goal: Value(MeasureOfAngle(BAD))
Answer: 40
Theorem sequence:
parallel_property_alternate_interior_angle(2,DA,BC)
arc_property_circumference_angle_external(1,OCA,B)
arc_property_center_angle(1,OCA,O)
Problem: As shown in the diagram, ∠CAB=40°, O is the center of ⊙O. Find the measure of ∠ABC.
问题：如图所示，∠CAB=40°，O是⊙O的圆心。求∠ABC的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(OAB,BO,OA)
Shape(OBC,CB)
Shape(AO,OB,BC,CA)
Collinear(AOB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ABC))
Answer: 50
Theorem sequence:
triangle_property_angle_sum(1,ABC)
diameter_of_circle_judgment_pass_centre(1,AOB,O)
diameter_of_circle_property_right_angle(1,BCA,O)
Problem: As shown in the diagram, ∠ABD=55°, O is the center of ⊙O. Find the measure of ∠DCB.
问题：如图所示，∠ABD=55°，O是⊙O的圆心。求∠DCB的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(AO,OE,ED,DA)
Shape(OAC,CE,EO,OA)
Shape(OCB,BC)
Shape(EC,CB,BE)
Shape(OBD,DB)
Shape(DE,EB,BD)
Collinear(AOEB)
Collinear(DEC)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),55)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DCB))
Answer: 35
Theorem sequence:
diameter_of_circle_judgment_pass_centre(1,AOB,O)
diameter_of_circle_property_right_angle(1,BDA,O)
arc_property_circumference_angle_external(1,OBD,A)
arc_property_circumference_angle_external(1,OBD,C)
triangle_property_angle_sum(1,DAB)
Problem: As shown in the diagram, ∠CBP=30°, ∠CPA=80°. Find the measure of ∠DAB.
问题：如图所示，∠CBP=30°，∠CPA=80°。求∠DAB的大小。
Construction (CDL):
Shape(OCB,BC)
Shape(OBD,DP,PB)
Shape(ODA,AD)
Shape(OAC,CP,PA)
Shape(CB,BP,PC)
Shape(PD,DA,AP)
Collinear(BPA)
Collinear(CPD)
Cocircular(O,BDAC)
Conditions (CDL):
Equal(MeasureOfAngle(CBP),30)
Equal(MeasureOfAngle(CPA),80)
Goal: Value(MeasureOfAngle(DAB))
Answer: 50
Theorem sequence:
adjacent_complementary_angle(1,CPA,APD)
triangle_property_angle_sum(1,PDA)
arc_property_circumference_angle_external(1,OAC,B)
arc_property_circumference_angle_external(1,OAC,D)
Problem: As shown in the diagram, ∠ABC=30°, the center of circle O is O. Find the measure of ∠BAD.
问题：如图所示，∠ABC=30°，⊙O的圆心为O。求∠BAD的大小。
Construction (CDL):
Shape(OAD,DA)
Shape(ODB,BO,OD)
Shape(OBC,CB)
Shape(OCA,AO,OC)
Shape(OA,AD,DO)
Shape(OB,BC,CO)
Collinear(BOA)
Collinear(DOC)
Cocircular(O,DBCA)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),30)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BAD))
Answer: 30
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,D)
arc_property_circumference_angle_external(1,OCA,B)
radius_of_circle_property_length_equal(1,OD,O)
radius_of_circle_property_length_equal(1,OA,O)
isosceles_triangle_judgment_line_equal(1,OAD)
isosceles_triangle_property_angle_equal(1,OAD)
Problem: As shown in the diagram, ∠CBA=50°, the center of circle O is O, D bisects arc OCB. Find the measure of ∠BAD.
问题：如图所示，∠CBA=50°，O是⊙O的圆心，D是弧OCB的中点。求∠BAD的大小。
Construction (CDL):
Shape(OAC,CA)
Shape(OCD,DM,MC)
Shape(AC,CM,MA)
Shape(ODB,BM,MD)
Shape(AM,MB,BO,OA)
Shape(OBA,AO,OB)
Collinear(AOB)
Collinear(AMD)
Collinear(CMB)
Cocircular(O,ACDB)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),50)
IsCentreOfCircle(O,O)
IsMidpointOfArc(D,OCB)
Goal: Value(MeasureOfAngle(BAD))
Answer: 20
Theorem sequence:
diameter_of_circle_judgment_pass_centre(1,AOB,O)
diameter_of_circle_property_right_angle(1,ACB,O)
triangle_property_angle_sum(1,ACB)
arc_addition_measure(1,OCD,ODB)
congruent_arc_judgment_length_equal(1,OCD,ODB)
congruent_arc_property_measure_equal(1,OCD,ODB)
arc_property_circumference_angle_external(1,OCB,A)
arc_property_circumference_angle_external(1,ODB,A)
Problem: As shown in the diagram, ∠AOC=116°, the center of circle O is O. Find the measure of ∠CDB.
问题：如图所示，∠AOC=116°，⊙O的圆心为O。求∠CDB的大小。
Construction (CDL):
Shape(OCA,AO,OC)
Shape(OAD,DM,MO,OA)
Shape(ODB,BD)
Shape(OBC,CM,MB)
Shape(OM,MC,CO)
Shape(MD,DB,BM)
Collinear(AOMB)
Collinear(CMD)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(AOC),116)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CDB))
Answer: 32
Theorem sequence:
adjacent_complementary_angle(1,AOC,COB)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,D)
Problem: As shown in the diagram, ∠DAC=35°, AD bisects ∠BAC, O is the center of ⊙O. Find the measure of ∠BOC.
问题：如图所示，∠DAC=35°，AD是∠BAC的角平分线，⊙O的圆心为O。求∠BOC的大小。
Construction (CDL):
Shape(OAC,CA)
Shape(OCD,DM,MC)
Shape(ODB,BO,OM,MD)
Shape(OBA,AB)
Shape(AM,MO,OB,BA)
Shape(AC,CM,MA)
Collinear(AMD)
Collinear(CMO)
Cocircular(O,ACDB)
Conditions (CDL):
Equal(MeasureOfAngle(DAC),35)
IsBisectorOfAngle(AD,BAC)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOC))
Answer: 140
Theorem sequence:
angle_addition(1,BAM,MAC)
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_external(1,OCB,A)
Problem: As shown in the diagram, ∠OAB=25°, the center of circle O is O. Find the measure of ∠BCA.
问题：如图所示，∠OAB=25°，圆O的圆心为O。求∠BCA的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OA,AB,BO)
Shape(CA,AO,OB,BC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(OAB),25)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 65
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, ∠CAB=40°, the center of ⊙O is O, OD⊥CD. Find the measure of ∠ACO.
问题：如图所示，∠CAB=40°，⊙O的圆心为O，OD垂直于CD。求∠ACO的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CD,DB)
Shape(DA,AB,BD)
Shape(OD,DC,CO)
Shape(OCA,AD,DO,OC)
Collinear(ADC)
Collinear(ODB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),40)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(OD,CD)
Goal: Value(MeasureOfAngle(ACO))
Answer: 10
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
triangle_property_angle_sum(1,ODC)
Problem: As shown in the diagram, ∠CAB=35°, ∠DEC=40°, the center of ⊙O is O. Find the measure of ∠DOB.
问题：如图所示，∠CAB=35°，∠DEC=40°，O是⊙O的圆心。求∠DOB的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(AB,BM,MA)
Shape(OBC,CM,MB)
Shape(OCD,DN,NC)
Shape(ODE,ED)
Shape(OM,MC,CO)
Shape(OC,CN,NO)
Shape(OEA,AM,MO,ON,NE)
Shape(ND,DE,EN)
Collinear(AMC)
Collinear(BMO)
Collinear(OND)
Collinear(CNE)
Cocircular(O,ABCDE)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),35)
Equal(MeasureOfAngle(DEC),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DOB))
Answer: 150
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_center_angle(1,OCD,O)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_external(1,OCD,E)
angle_addition(1,DOC,COB)
Problem: As shown in the diagram, OD=1, ∠DBO=30°, the center of ⊙O is O, CA∥DO. Find the length of line BC.
问题：如图所示，OD=1，∠DBO=30°，圆O的圆心为O，CA平行于DO。求直线BC的长度。
Construction (CDL):
Shape(OCB,BD,DC)
Shape(OD,DB,BO)
Shape(OBA,AO,OB)
Shape(OAC,CA)
Shape(OA,AC,CD,DO)
Collinear(BOA)
Collinear(BDC)
Cocircular(O,BAC)
Conditions (CDL):
Equal(LengthOfLine(OD),1)
Equal(MeasureOfAngle(DBO),30)
IsCentreOfCircle(O,O)
ParallelBetweenLine(CA,DO)
Goal: Value(LengthOfLine(BC))
Answer: 2*sqrt(3)
Theorem sequence:
parallel_property_corresponding_angle(2,CA,DO,B)
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OA,O)
line_addition(1,BO,OA)
diameter_of_circle_judgment_pass_centre(1,BOA,O)
diameter_of_circle_property_right_angle(1,ACB,O)
triangle_property_angle_sum(1,BAC)
sine_theorem(1,ODB)
sine_theorem(1,BAC)
Problem: As shown in the diagram, ∠CAB=50°, O is the center of ⊙O. Find the measure of ∠ADC.
问题：如图所示，∠CAB=50°，O是圆O的圆心。求∠ADC的大小。
Construction (CDL):
Shape(OBC,CB)
Shape(OCA,AC)
Shape(OAD,DA)
Shape(ODB,BO,OM,MD)
Shape(MC,CA,AM)
Shape(MA,AD,DM)
Shape(OB,BC,CM,MO)
Collinear(BOMA)
Collinear(CMD)
Cocircular(O,BCAD)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),50)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ADC))
Answer: 40
Theorem sequence:
triangle_property_angle_sum(1,BCA)
diameter_of_circle_judgment_pass_centre(1,BOA,O)
diameter_of_circle_property_right_angle(1,BCA,O)
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_external(1,OCA,D)
Problem: As shown in the diagram, ∠BAC=40°, O is the center of ⊙O. Find the measure of ∠BOC.
问题：如图所示，∠BAC=40°，O是⊙O的圆心。求∠BOC的大小。
Construction (CDL):
Shape(OCB,BC)
Shape(OBA,AB)
Shape(OAC,CO,OM,MA)
Shape(OC,CM,MO)
Shape(MC,CB,BM)
Shape(MB,BA,AM)
Collinear(OMB)
Collinear(CMA)
Cocircular(O,CBA)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOC))
Answer: 80
Theorem sequence:
arc_property_circumference_angle_external(1,OCB,A)
arc_property_center_angle(1,OCB,O)
Problem: As shown in the diagram, ∠BOA=70°, O is the center of ⊙O. Find the measure of ∠BCA.
问题：如图所示，∠BOA=70°，O是圆O的圆心。求∠BCA的大小。
Construction (CDL):
Shape(MB,BC,CM)
Shape(OA,AM,MO)
Shape(OCA,AO,OM,MC)
Shape(OBC,CB)
Shape(OAB,BM,MA)
Collinear(AMC)
Collinear(OMB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),70)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 35
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,C)
arc_property_center_angle(1,OAB,O)
Problem: As shown in the diagram, ∠OBC=20°, O is the center of ⊙O. Find the measure of ∠CAB.
问题：如图所示，∠OBC=20°，O是⊙O的圆心。求∠CAB的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCA,AC)
Shape(AB,BO,OC,CA)
Shape(OB,BC,CO)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(OBC),20)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 70
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_external(1,OBC,A)
Problem: As shown in the diagram, AB=16, OA=10, the center of ⊙O is O, AC is perpendicular to OC. Find the length of line OC.
问题：如图所示，AB=16，OA=10，⊙O的圆心为O，AC⊥OC。求直线OC的长度。
Construction (CDL):
Shape(AC,CO,OA)
Shape(OC,CB,BO)
Shape(OAD,DC,CA)
Shape(ODB,BC,CD)
Shape(OBA,AO,OB)
Collinear(ACB)
Collinear(OCD)
Cocircular(O,ADB)
Conditions (CDL):
Equal(LengthOfLine(AB),16)
Equal(LengthOfLine(OA),10)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AC,OC)
Goal: Value(LengthOfLine(OC))
Answer: 6
Theorem sequence:
circle_property_chord_perpendicular_bisect_chord(1,O,OC,AB)
right_triangle_judgment_angle(1,ACO)
right_triangle_property_pythagorean(1,ACO)
line_addition(1,AC,CB)
Problem: As shown in the diagram, OC=6, the radius of circle O is 10, the center of ⊙O is O, BC⊥OC. Find the length of line AB.
问题：如图所示，OC=6，圆O的半径为10，O是⊙O的圆心，BC⊥OC。求直线AB的长度。
Construction (CDL):
Shape(OBA,AB)
Shape(AO,OC,CA)
Shape(BC,CO,OB)
Shape(OAB,BO,OA)
Collinear(ACB)
Cocircular(O,AB)
Conditions (CDL):
Equal(LengthOfLine(OC),6)
Equal(RadiusOfCircle(O),10)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(BC,OC)
Goal: Value(LengthOfLine(AB))
Answer: 16
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
adjacent_complementary_angle(1,BCO,OCA)
right_triangle_judgment_angle(1,OCA)
right_triangle_judgment_angle(1,BCO)
right_triangle_property_pythagorean(1,OCA)
right_triangle_property_pythagorean(1,BCO)
line_addition(1,AC,CB)
Problem: As shown in the diagram, AB=16, OC=6, O is the center of circle O, OC⊥BC. Find the length of line OA.
问题：如图所示，AB=16，OC=6，O是圆O的圆心，OC⊥BC。求直线OA的长度。
Construction (CDL):
Shape(OAB,BC,CA)
Shape(OBA,AO,OB)
Shape(OA,AC,CO)
Shape(OC,CB,BO)
Collinear(ACB)
Cocircular(O,AB)
Conditions (CDL):
Equal(LengthOfLine(AB),16)
Equal(LengthOfLine(OC),6)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(OC,BC)
Goal: Value(LengthOfLine(OA))
Answer: 10
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
adjacent_complementary_angle(1,ACO,OCB)
right_triangle_judgment_angle(1,ACO)
right_triangle_judgment_angle(1,OCB)
right_triangle_property_pythagorean(1,ACO)
right_triangle_property_pythagorean(1,OCB)
line_addition(1,AC,CB)
Problem: As shown in the diagram, Mul(LengthOfLine(OD)=2), the radius of ⊙O is 4, the center of ⊙O is O, OD is perpendicular to AD. Find the length of line AB.
问题：如图所示，Mul(LengthOfLine(OD)=2)，⊙O的半径为4，⊙O的圆心为O，OD垂直于AD。求直线AB的长度。
Construction (CDL):
Shape(OAB,BO,OA)
Shape(OBA,AB)
Shape(OD,DA,AO)
Shape(OB,BD,DO)
Collinear(ADB)
Cocircular(O,AB)
Conditions (CDL):
Equal(Mul(LengthOfLine(OD),2),RadiusOfCircle(O))
Equal(RadiusOfCircle(O),4)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(OD,AD)
Goal: Value(LengthOfLine(AB))
Answer: 4*sqrt(3)
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
adjacent_complementary_angle(1,BDO,ODA)
right_triangle_judgment_angle(1,ODA)
right_triangle_judgment_angle(1,BDO)
right_triangle_property_pythagorean(1,ODA)
right_triangle_property_pythagorean(1,BDO)
line_addition(1,AD,DB)
Problem: As shown in the diagram, AB=8, AC=6, the center of ⊙O is O, AD⊥OD, CA is perpendicular to BA, CE⊥OE. Find the length of line OA.
问题：如图所示，AB=8，AC=6，O是圆O的圆心，AD⊥OD，CA⊥BA，CE⊥OE。求直线OA的长度。
Construction (CDL):
Shape(OE,EA,AO)
Shape(OA,AD,DO)
Shape(OCA,AE,EC)
Shape(OAB,BD,DA)
Shape(OBC,CE,EO,OD,DB)
Collinear(CEA)
Collinear(ADB)
Cocircular(O,CAB)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(LengthOfLine(AC),6)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AD,OD)
PerpendicularBetweenLine(CA,BA)
PerpendicularBetweenLine(CE,OE)
Goal: Value(LengthOfLine(OA))
Answer: 5
Theorem sequence:
line_addition(1,CE,EA)
line_addition(1,AD,DB)
circle_property_chord_perpendicular_bisect_chord(1,O,OE,CA)
circle_property_chord_perpendicular_bisect_chord(1,O,OD,AB)
parallel_judgment_ipsilateral_internal_angle(1,EO,AD)
parallel_judgment_ipsilateral_internal_angle(1,AE,DO)
parallelogram_judgment_parallel_and_parallel(1,OEAD)
parallelogram_property_opposite_line_equal(1,OEAD)
right_triangle_judgment_angle(1,OEA)
right_triangle_property_pythagorean(1,OEA)
Problem: As shown in the diagram, AB=8, the radius of circle O is 5, O is the center of circle O, BC⊥OC. Find the length of line OC.
问题：如图所示，AB=8，圆O的半径为5，⊙O的圆心为O，BC垂直于OC。求直线OC的长度。
Construction (CDL):
Shape(AO,OC,CA)
Shape(OAB,BC,CO,OA)
Shape(OBA,AC,CB)
Collinear(ACB)
Cocircular(O,BA)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(RadiusOfCircle(O),5)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(BC,OC)
Goal: Value(LengthOfLine(OC))
Answer: 3
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
circle_property_chord_perpendicular_bisect_chord(1,O,OC,BA)
line_addition(1,AC,CB)
right_triangle_judgment_angle(1,OCA)
right_triangle_property_pythagorean(1,OCA)
Problem: As shown in the diagram, ∠BOA=40°, O is the center of ⊙O, CB∥OA. Find the measure of ∠OAC.
问题：如图所示，∠BOA=40°，⊙O的圆心为O，CB∥OA。求∠OAC的大小。
Construction (CDL):
Shape(OAB,BM,MA)
Shape(OBC,CB)
Shape(MB,BC,CM)
Shape(OA,AM,MO)
Shape(OCA,AO,OM,MC)
Collinear(AMC)
Collinear(OMB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),40)
IsCentreOfCircle(O,O)
ParallelBetweenLine(CB,OA)
Goal: Value(MeasureOfAngle(OAC))
Answer: 20
Theorem sequence:
parallel_property_alternate_interior_angle(1,CB,OA)
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, the diameter of ⊙O is 12, the center of ⊙O is O, AM perpendicular bisects OC. Find the length of line AB.
问题：如图所示，圆O的直径为12，O是圆O的圆心，AM是OC的垂直平分线。求直线AB的长度。
Construction (CDL):
Shape(AO,OM,MA)
Shape(OB,BM,MO)
Shape(OAB,BO,OA)
Shape(OBC,CM,MB)
Shape(OCA,AM,MC)
Collinear(OMC)
Collinear(AMB)
Cocircular(O,BCA)
Conditions (CDL):
Equal(DiameterOfCircle(O),12)
IsCentreOfCircle(O,O)
IsPerpendicularBisectorOfLine(AM,OC)
Goal: Value(LengthOfLine(AB))
Answer: 6*sqrt(3)
Theorem sequence:
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,OC,O)
radius_of_circle_property_length_equal(1,OA,O)
line_addition(1,OM,MC)
line_addition(1,AM,MB)
adjacent_complementary_angle(1,BMO,OMA)
right_triangle_judgment_angle(1,OMA)
right_triangle_judgment_angle(1,BMO)
right_triangle_property_pythagorean(1,OMA)
right_triangle_property_pythagorean(1,BMO)
Problem: As shown in the diagram, AB=10, CD=6, O is the center of circle O, CE is perpendicular to OE. Find the length of line BE.
问题：如图所示，AB=10，CD=6，圆O的圆心为O，CE⊥OE。求直线BE的长度。
Construction (CDL):
Shape(OAC,CE,EO,OA)
Shape(OCB,BE,EC)
Shape(OBD,DE,EB)
Shape(ODA,AO,OE,ED)
Shape(OC,CE,EO)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(LengthOfLine(CD),6)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(CE,OE)
Goal: Value(LengthOfLine(BE))
Answer: 1
Theorem sequence:
line_addition(1,AO,OE)
line_addition(1,AO,OB)
line_addition(1,AE,EB)
line_addition(1,CE,ED)
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)
circle_property_chord_perpendicular_bisect_chord(1,O,OE,CD)
right_triangle_judgment_angle(1,CEO)
right_triangle_property_pythagorean(1,CEO)
Problem: As shown in the diagram, the measure of ⌒OAB is 120, the radius of circle O is 2, O is the center of ⊙O, AC⊥OC. Find the length of line OC.
问题：如图所示，⌒OAB的角度为120，圆O的半径为2，⊙O的圆心为O，AC⊥OC。求直线OC的长度。
Construction (CDL):
Shape(OAB,BC,CA)
Shape(OBA,AO,OB)
Shape(OC,CB,BO)
Shape(OA,AC,CO)
Collinear(ACB)
Cocircular(O,AB)
Conditions (CDL):
Equal(MeasureOfArc(OAB),120)
Equal(RadiusOfCircle(O),2)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AC,OC)
Goal: Value(LengthOfLine(OC))
Answer: 1
Theorem sequence:
arc_property_center_angle(1,OAB,O)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
isosceles_triangle_judgment_line_equal(1,OAB)
isosceles_triangle_property_angle_equal(1,OAB)
triangle_property_angle_sum(1,BOA)
sine_theorem(1,OAC)
Problem: As shown in the diagram, AB=10, CD=6, the center of ⊙O is O, CE⊥OE. Find the length of line AC.
问题：如图所示，AB=10，CD=6，O是圆O的圆心，CE垂直于OE。求直线AC的长度。
Construction (CDL):
Shape(OCD,DC)
Shape(ODC,CD)
Shape(OAB,BD,OCD,CA)
Shape(OBA,AC,ODC,DB)
Shape(OC,CE,EO)
Collinear(ACEDB)
Cocircular(O,AB)
Cocircular(O,CD)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(LengthOfLine(CD),6)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(CE,OE)
Goal: Value(LengthOfLine(AC))
Answer: 2
Theorem sequence:
line_addition(1,AC,CE)
line_addition(1,AC,CD)
line_addition(1,AE,ED)
line_addition(1,AE,EB)
circle_property_chord_perpendicular_bisect_chord(1,O,OE,AB)
circle_property_chord_perpendicular_bisect_chord(1,O,OE,CD)
Problem: As shown in the diagram, AC=7, AE=24, ∠CAO=∠AOB, the center of ⊙O is O, EK is perpendicular to OK. Find the radius of circle O.
问题：如图所示，AC=7，AE=24，∠CAO=∠AOB，O是⊙O的圆心，EK垂直于OK。求⊙O的半径。
Construction (CDL):
Shape(OCA,AC)
Shape(OAE,EK,KO,OA)
Shape(OEB,BK,KE)
Shape(OBC,CA,AK,KB)
Shape(KA,AO,OK)
Shape(CA,AO,OM,MC)
Collinear(AOM)
Collinear(BKO)
Collinear(AKE)
Cocircular(O,AMEBC)
Conditions (CDL):
Equal(LengthOfLine(AC),7)
Equal(LengthOfLine(AE),24)
Equal(MeasureOfAngle(CAO),MeasureOfAngle(AOB))
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(EK,OK)
Goal: Value(RadiusOfCircle(O))
Answer: 25/2
Theorem sequence:
diameter_of_circle_judgment_pass_centre(1,AOM,O)
diameter_of_circle_property_right_angle(1,MCA,O)
line_addition(1,AO,OM)
line_addition(1,AK,KE)
circle_property_chord_perpendicular_bisect_chord(1,O,OK,EA)
right_triangle_judgment_angle(1,OKA)
right_triangle_judgment_angle(1,MCA)
right_triangle_property_pythagorean(1,OKA)
right_triangle_property_pythagorean(1,MCA)
mirror_similar_triangle_judgment_aa(1,AOK,MCA)
mirror_similar_triangle_property_line_ratio(1,AOK,MCA)
mirror_similar_triangle_property_line_ratio(1,OKA,AMC)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OM,O)
Problem: As shown in the diagram, AB=8, OP=2, the center of ⊙O is O, BP⊥OP. Find the length of line BC.
问题：如图所示，AB=8，OP=2，⊙O的圆心为O，BP垂直于OP。求直线BC的长度。
Construction (CDL):
Shape(OCA,AO,OP,PC)
Shape(OAB,BO,OA)
Shape(OBC,CP,PB)
Shape(OB,BP,PO)
Collinear(AOB)
Collinear(CPB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(LengthOfLine(OP),2)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(BP,OP)
Goal: Value(LengthOfLine(BC))
Answer: 4*sqrt(3)
Theorem sequence:
line_addition(1,AO,OB)
line_addition(1,CP,PB)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
circle_property_chord_perpendicular_bisect_chord(1,O,OP,BC)
right_triangle_judgment_angle(1,BPO)
right_triangle_property_pythagorean(1,BPO)