SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠COB=80°, the center of circle O is O. Find the measure of ∠CAB.
问题：如图所示，∠COB=80°，O是⊙O的圆心。求∠CAB的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(DA,AB,BD)
Shape(DB,OBC,CD)
Shape(OD,DC,CO)
Shape(OC,OCA,AD,DO)
Collinear(ADC)
Collinear(ODB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(COB),80)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 40
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠DOA=136°, O is the center of ⊙O. Find the measure of ∠BCD.
问题：如图所示，∠DOA=136°，圆O的圆心为O。求∠BCD的大小。
Construction (CDL):
Shape(OA,OAD,DO)
Shape(EO,OD,DE)
Shape(BE,ED,ODB)
Shape(EB,BC,CE)
Shape(CB,OBC)
Shape(OCA,AO,OE,EC)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,CADB)
Conditions (CDL):
Equal(MeasureOfAngle(DOA),136)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCD))
Answer: 22
Theorem sequence:
flat_angle(1,EOA)
angle_addition(1,EOD,DOA)
arc_property_center_angle(1,ODB,O)
arc_property_circumference_angle_external(1,ODB,C)
Problem: As shown in the diagram, ∠BOC=140°, the center of ⊙O is O. Find the measure of ∠CAB.
问题：如图所示，∠BOC=140°，圆O的圆心为O。求∠CAB的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OB,OBC,CO)
Shape(OCA,AC)
Shape(AB,BO,OC,CA)
Cocircular(O,CAB)
Conditions (CDL):
Equal(MeasureOfAngle(BOC),140)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 110
Theorem sequence:
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_internal(1,OCB,A)
Problem: As shown in the diagram, ∠COB=72°, O is the center of circle O. Find the measure of ∠CAB.
问题：如图所示，∠COB=72°，圆O的圆心为O。求∠CAB的大小。
Construction (CDL):
Shape(OB,BD,DO)
Shape(DB,OBC,CD)
Shape(AD,DC,CA)
Shape(OCA,AC)
Shape(OD,DA,OAB,BO)
Collinear(ODC)
Collinear(BDA)
Cocircular(O,BCA)
Conditions (CDL):
Equal(MeasureOfAngle(COB),72)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 36
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠CAB=20°, O is the center of ⊙O, the diameter of ⊙O is AB. Find the measure of ∠CDA.
问题：如图所示，∠CAB=20°，⊙O的圆心为O，AB是圆O的直径。求∠CDA的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(DA,AC,CD)
Shape(CA,AO,OB,BC)
Shape(OCD,DC)
Shape(OBC,CB)
Shape(OA,OAB,BO)
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,CAB)
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_internal(1,OCA,D)
Problem: As shown in the diagram, ∠AOC=140°, the center of circle O is O. Find the measure of ∠CDB.
问题：如图所示，∠AOC=140°，O是圆O的圆心。求∠CDB的大小。
Construction (CDL):
Shape(DB,BE,ED)
Shape(OE,EC,CO)
Shape(EB,OBC,CE)
Shape(OC,OCA,AO)
Shape(ODB,BD)
Shape(EO,OA,OAD,DE)
Collinear(BEOA)
Collinear(DEC)
Cocircular(O,DBCA)
Conditions (CDL):
Equal(MeasureOfAngle(AOC),140)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CDB))
Answer: 20
Theorem sequence:
flat_angle(1,AOB)
angle_addition(1,AOC,COE)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,D)
Problem: As shown in the diagram, ∠BOC=50°, O is the center of ⊙O. Find the measure of ∠BAC.
问题：如图所示，∠BOC=50°，⊙O的圆心为O。求∠BAC的大小。
Construction (CDL):
Shape(OC,CD,DO)
Shape(DC,OCB,BD)
Shape(AD,DB,BA)
Shape(OBA,AB)
Shape(OD,DA,OAC,CO)
Collinear(CDA)
Collinear(ODB)
Cocircular(O,CBA)
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, ∠AOC=50°, the center of ⊙O is O, AE is perpendicular to CE. Find the measure of ∠BDA.
问题：如图所示，∠AOC=50°，O是⊙O的圆心，AE⊥CE。求∠BDA的大小。
Construction (CDL):
Shape(EA,OAB,BE)
Shape(FE,EB,BF)
Shape(GF,FO,OG)
Shape(HG,GD,DH)
Shape(IH,HC,CI)
Shape(AE,EI,IA)
Shape(IE,EF,FG,GH,HI)
Shape(OF,FB,OBD,DG,GO)
Shape(ODC,CH,HD)
Shape(OCA,AI,IC)
Collinear(AEFO)
Collinear(AIHD)
Collinear(BEIC)
Collinear(BFGD)
Collinear(CHGO)
Cocircular(O,ABDC)
Conditions (CDL):
Equal(MeasureOfAngle(AOC),50)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AE,CE)
Goal: Value(MeasureOfAngle(BDA))
Answer: 25
Theorem sequence:
arc_property_center_angle(1,OCA,O)
adjacent_complementary_angle(1,AEI,IEF)
circle_property_chord_perpendicular_bisect_arc(1,OCB,OEA)
congruent_arc_judgment_length_equal(1,OAB,OCA)
congruent_arc_property_measure_equal(1,OAB,OCA)
arc_property_circumference_angle_external(1,OAB,D)
Problem: As shown in the diagram, ∠DOB=130°, O is the center of circle O. Find the measure of ∠ACD.
问题：如图所示，∠DOB=130°，O是⊙O的圆心。求∠ACD的大小。
Construction (CDL):
Shape(AC,CE,EA)
Shape(OAC,CA)
Shape(OE,EC,OCB,BO)
Shape(OB,OBD,DO)
Shape(EO,OD,DE)
Shape(AE,ED,ODA)
Collinear(AEOB)
Collinear(CED)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(DOB),130)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ACD))
Answer: 25
Theorem sequence:
flat_angle(1,AOB)
angle_addition(1,EOD,DOB)
arc_property_center_angle(1,ODA,O)
arc_property_circumference_angle_external(1,ODA,C)
Problem: As shown in the diagram, CD=AD, ∠ABC=70°, O is the center of ⊙O, AB is the diameter of ⊙O. Find the measure of ∠BAD.
问题：如图所示，CD=AD，∠ABC=70°，⊙O的圆心为O，AB是圆O的直径。求∠BAD的大小。
Construction (CDL):
Shape(OE,EC,CA,AO)
Shape(CE,EB,BC)
Shape(EO,OA,AD,DE)
Shape(OCA,AC)
Shape(OAD,DA)
Shape(BE,ED,ODB)
Shape(OBC,CB)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,CADB)
Conditions (CDL):
Equal(LengthOfLine(CD),LengthOfLine(AD))
Equal(MeasureOfAngle(ABC),70)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BAD))
Answer: 35
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,CAB)
triangle_property_angle_sum(1,CAD)
arc_property_circumference_angle_external(1,OCA,D)
arc_property_circumference_angle_external(1,OCA,B)
isosceles_triangle_judgment_line_equal(1,DCA)
isosceles_triangle_property_angle_equal(1,DCA)
angle_addition(1,CAO,OAD)
Problem: As shown in the diagram, AB=10, BC=6, O is the center of circle O, CD is perpendicular to BD. Find the length of line CD.
问题：如图所示，AB=10，BC=6，O是⊙O的圆心，CD⊥BD。求直线CD的长度。
Construction (CDL):
Shape(CA,AO,OD,DC)
Shape(CD,DB,BC)
Shape(OCA,AC)
Shape(OBC,CB)
Shape(OA,OAB,BD,DO)
Collinear(AODB)
Cocircular(O,BCA)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(LengthOfLine(BC),6)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(CD,BD)
Goal: Value(LengthOfLine(CD))
Answer: 24/5
Theorem sequence:
diameter_of_circle_judgment_pass_centre(1,BOA,O)
diameter_of_circle_property_right_angle(1,BCA,O)
right_triangle_judgment_angle(1,BCA)
right_triangle_property_pythagorean(1,BCA)
mirror_similar_triangle_judgment_aa(1,CDB,ABC)
mirror_similar_triangle_property_line_ratio(1,DBC,CAB)
mirror_similar_triangle_property_line_ratio(1,BCD,BCA)
Problem: As shown in the diagram, ∠ADC=34°, O is the center of ⊙O, the diameter of circle O is BC. Find the measure of ∠CAO.
问题：如图所示，∠ADC=34°，圆O的圆心为O，BC是圆O的直径。求∠CAO的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(AB,BO,OA)
Shape(AO,OE,EA)
Shape(AE,EC,CA)
Shape(OCA,AC)
Shape(OB,OBD,DE,EO)
Shape(ED,DC,CE)
Shape(ODC,CD)
Collinear(BOEC)
Collinear(AED)
Cocircular(O,ABDC)
Conditions (CDL):
Equal(MeasureOfAngle(ADC),34)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BC,O)
Goal: Value(MeasureOfAngle(CAO))
Answer: 56
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_external(1,OCA,D)
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)
diameter_of_circle_property_right_angle(1,CAB,O)
angle_addition(1,CAO,OAB)
Problem: As shown in the diagram, ∠CAO=70°, O is the center of ⊙O, CE⊥BE. Find the measure of ∠BOD.
问题：如图所示，∠CAO=70°，O是⊙O的圆心，CE垂直于BE。求∠BOD的大小。
Construction (CDL):
Shape(CA,AE,EC)
Shape(OCA,AC)
Shape(CE,EO,OB,OBC)
Shape(EA,OAD,DE)
Shape(ED,DO,OE)
Shape(OD,ODB,BO)
Collinear(AEOB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(CAO),70)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(CE,BE)
Goal: Value(MeasureOfAngle(BOD))
Answer: 140
Theorem sequence:
adjacent_complementary_angle(1,AEC,CEO)
adjacent_complementary_angle(1,BOD,DOA)
triangle_property_angle_sum(1,CAE)
arc_property_circumference_angle_external(1,OAD,C)
arc_property_center_angle(1,OAD,O)
Problem: As shown in the diagram, ∠BCA=90°, ∠BDA=90°, ∠EBD=24°, AD bisects ∠CAB. Find the measure of ∠DAB.
问题：如图所示，∠BCA=90°，∠BDA=90°，∠EBD=24°，AD是∠CAB的角平分线。求∠DAB的大小。
Construction (CDL):
Shape(CA,AE,EC)
Shape(DE,EB,BD)
Shape(EA,AB,BE)
Collinear(CEB)
Collinear(AED)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),90)
Equal(MeasureOfAngle(BDA),90)
Equal(MeasureOfAngle(EBD),24)
IsBisectorOfAngle(AD,CAB)
Goal: Value(MeasureOfAngle(DAB))
Answer: 24
Theorem sequence:
triangle_property_angle_sum(1,CAE)
triangle_property_angle_sum(1,DEB)
vertical_angle(1,AEC,DEB)
Problem: As shown in the diagram, ∠ABC=60°, BD is perpendicular to AD, CE⊥BE. Find the measure of ∠EOA.
问题：如图所示，∠ABC=60°，BD⊥AD，CE⊥BE。求∠EOA的大小。
Construction (CDL):
Shape(CA,AO,OC)
Shape(OA,AE,EO)
Shape(OE,EB,BD,DO)
Shape(CO,OD,DC)
Collinear(COE)
Collinear(BDC)
Collinear(AOD)
Collinear(AEB)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),60)
PerpendicularBetweenLine(BD,AD)
PerpendicularBetweenLine(CE,BE)
Goal: Value(MeasureOfAngle(EOA))
Answer: 60
Theorem sequence:
adjacent_complementary_angle(1,DOE,EOA)
quadrilateral_property_angle_sum(1,DOEB)
Problem: As shown in the diagram, Add(LengthOfLine(AB)=LengthOfLine(CA)), AB=5, ∠ABC=90°, ∠BCA=30°. Find the value of x.
问题：如图所示，Add(LengthOfLine(AB)=LengthOfLine(CA))，AB=5，∠ABC=90°，∠BCA=30°。求x的值。
Construction (CDL):
Shape(AB,BC,CA)
Conditions (CDL):
Equal(Add(LengthOfLine(AB),LengthOfLine(CA)),x)
Equal(LengthOfLine(AB),5)
Equal(MeasureOfAngle(ABC),90)
Equal(MeasureOfAngle(BCA),30)
Goal: Value(x)
Answer: 15
Theorem sequence:
sine_theorem(1,ABC)
Problem: As shown in the diagram, ∠ABC=90°, ∠CAB=50°. Find the measure of ∠BCA.
问题：如图所示，∠ABC=90°，∠CAB=50°。求∠BCA的大小。
Construction (CDL):
Shape(CA,AB,BC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),90)
Equal(MeasureOfAngle(CAB),50)
Goal: Value(MeasureOfAngle(BCA))
Answer: 40
Theorem sequence:
triangle_property_angle_sum(1,CAB)
Problem: As shown in the diagram, ∠COB=100°, the center of ⊙O is O. Find the measure of ∠CAB.
问题：如图所示，∠COB=100°，圆O的圆心为O。求∠CAB的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(OB,OBC,CO)
Shape(OCA,AC)
Shape(AB,BO,OC,CA)
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, AB=8, OA=5, O is the center of ⊙O, OC is perpendicular to BC. Find the length of line CD.
问题：如图所示，AB=8，OA=5，O是⊙O的圆心，OC垂直于BC。求直线CD的长度。
Construction (CDL):
Shape(OA,AC,CO)
Shape(CA,OAD,DC)
Shape(BC,CD,ODB)
Shape(OC,CB,OBA,AO)
Collinear(ACB)
Collinear(OCD)
Cocircular(O,ADB)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(LengthOfLine(OA),5)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(OC,BC)
Goal: Value(LengthOfLine(CD))
Answer: 2
Theorem sequence:
adjacent_complementary_angle(1,ACO,OCB)
circle_property_chord_perpendicular_bisect_chord(1,O,OC,AB)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OD,O)
line_addition(1,AC,CB)
line_addition(1,OC,CD)
right_triangle_judgment_angle(1,ACO)
right_triangle_property_pythagorean(1,ACO)
Problem: As shown in the diagram, CD=8, OC=5, the center of circle O is O, AD⊥OD, OD is perpendicular to BD. Find the length of line AB.
问题：如图所示，CD=8，OC=5，O是圆O的圆心，AD垂直于OD，OD垂直于BD。求直线AB的长度。
Construction (CDL):
Shape(OC,OCA,AO)
Shape(OA,AD,DO)
Shape(OD,DB,BO)
Shape(OB,OBC,CO)
Shape(DA,OAB,BD)
Collinear(ADB)
Collinear(COD)
Cocircular(O,CAB)
Conditions (CDL):
Equal(LengthOfLine(CD),8)
Equal(LengthOfLine(OC),5)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AD,OD)
PerpendicularBetweenLine(OD,BD)
Goal: Value(LengthOfLine(AB))
Answer: 8
Theorem sequence:
line_addition(1,AD,DB)
line_addition(1,CO,OD)
radius_of_circle_property_length_equal(1,OC,O)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
right_triangle_judgment_angle(1,ADO)
right_triangle_judgment_angle(1,ODB)
right_triangle_property_pythagorean(1,ADO)
right_triangle_property_pythagorean(1,ODB)
Problem: As shown in the diagram, AB=600, OA=325, the center of circle O is O, OC is perpendicular to BC. Find the length of line CD.
问题：如图所示，AB=600，OA=325，O是圆O的圆心，OC⊥BC。求直线CD的长度。
Construction (CDL):
Shape(OA,AC,CO)
Shape(CA,OAD,DC)
Shape(BC,CD,ODB)
Shape(OC,CB,OBA,AO)
Collinear(ACB)
Collinear(OCD)
Cocircular(O,ADB)
Conditions (CDL):
Equal(LengthOfLine(AB),600)
Equal(LengthOfLine(OA),325)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(OC,BC)
Goal: Value(LengthOfLine(CD))
Answer: 200
Theorem sequence:
adjacent_complementary_angle(1,ACO,OCB)
circle_property_chord_perpendicular_bisect_chord(1,O,OC,AB)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OD,O)
line_addition(1,AC,CB)
line_addition(1,OC,CD)
right_triangle_judgment_angle(1,ACO)
right_triangle_property_pythagorean(1,ACO)
Problem: As shown in the diagram, ∠FDA=130°, EA is perpendicular to DA. Find the measure of ∠GEC.
问题：如图所示，∠FDA=130°，EA⊥DA。求∠GEC的大小。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Shape(FD,DA)
Shape(AE,EG)
Collinear(ADB)
Collinear(AEC)
Collinear(FDEG)
Conditions (CDL):
Equal(MeasureOfAngle(FDA),130)
PerpendicularBetweenLine(EA,DA)
Goal: Value(MeasureOfAngle(GEC))
Answer: 40
Theorem sequence:
flat_angle(1,FDE)
angle_addition(1,FDA,ADE)
triangle_property_angle_sum(1,ADE)
vertical_angle(1,FEA,GEC)
Problem: As shown in the diagram, AB=8, OB=5, the center of circle O is O, OC is perpendicular to BC. Find the length of line OC.
问题：如图所示，AB=8，OB=5，O是⊙O的圆心，OC垂直于BC。求直线OC的长度。
Construction (CDL):
Shape(OC,CB,BO)
Shape(CA,OAB,BC)
Shape(OB,OBA,AC,CO)
Collinear(ACB)
Cocircular(O,AB)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(LengthOfLine(OB),5)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(OC,BC)
Goal: Value(LengthOfLine(OC))
Answer: 3
Theorem sequence:
adjacent_complementary_angle(1,ACO,OCB)
circle_property_chord_perpendicular_bisect_chord(1,O,OC,AB)
line_addition(1,AC,CB)
right_triangle_judgment_angle(1,OCB)
right_triangle_property_pythagorean(1,OCB)
Problem: As shown in the diagram, ∠ABD=56°, the center of ⊙O is O, the diameter of ⊙O is AB. Find the measure of ∠DCB.
问题：如图所示，∠ABD=56°，O是⊙O的圆心，⊙O的直径为AB。求∠DCB的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(DA,AO,OE,ED)
Shape(DE,EB,BD)
Shape(OBD,DB)
Shape(EO,OA,OAC,CE)
Shape(EC,CB,BE)
Shape(ECB,BC)
Collinear(AOEB)
Collinear(DEC)
Cocircular(O,DACB)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),56)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(DCB))
Answer: 34
Theorem sequence:
diameter_of_circle_property_right_angle(1,BDA,O)
triangle_property_angle_sum(1,DAB)
arc_property_circumference_angle_external(1,OBD,A)
arc_property_circumference_angle_external(1,OBD,C)
Problem: As shown in the diagram, DO=1, OA=4, the center of circle O is O, OD is perpendicular to ED. Find the length of line EF.
问题：如图所示，DO=1，OA=4，⊙O的圆心为O，OD垂直于ED。求直线EF的长度。
Construction (CDL):
Shape(DE,EO,OD)
Shape(OE,OEA,AO)
Shape(OA,OAB,BO)
Shape(OB,OBF,FD,DO)
Shape(DF,OFE,ED)
Collinear(AOB)
Collinear(EDF)
Cocircular(O,BFEA)
Conditions (CDL):
Equal(LengthOfLine(DO),1)
Equal(LengthOfLine(OA),4)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(OD,ED)
Goal: Value(LengthOfLine(EF))
Answer: 2*sqrt(15)
Theorem sequence:
radius_of_circle_property_length_equal(1,OE,O)
radius_of_circle_property_length_equal(1,OA,O)
right_triangle_judgment_angle(1,ODE)
right_triangle_property_pythagorean(1,ODE)
adjacent_complementary_angle(1,FDO,ODE)
circle_property_chord_perpendicular_bisect_chord(1,O,OD,FE)
line_addition(1,ED,DF)
Problem: As shown in the diagram, AB=16, OA=10, O is the center of circle O, BD is perpendicular to OD. Find the length of line CD.
问题：如图所示，AB=16，OA=10，⊙O的圆心为O，BD垂直于OD。求直线CD的长度。
Construction (CDL):
Shape(AD,DC,OCA)
Shape(CD,DB,OBC)
Shape(DA,AO,OD)
Shape(OA,OAB,BD,DO)
Collinear(CDO)
Collinear(ADB)
Cocircular(O,CAB)
Conditions (CDL):
Equal(LengthOfLine(AB),16)
Equal(LengthOfLine(OA),10)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(BD,OD)
Goal: Value(LengthOfLine(CD))
Answer: 4
Theorem sequence:
circle_property_chord_perpendicular_bisect_chord(1,O,OD,BA)
line_addition(1,CD,DO)
line_addition(1,AD,DB)
radius_of_circle_property_length_equal(1,OC,O)
radius_of_circle_property_length_equal(1,OA,O)
right_triangle_judgment_angle(1,ODA)
right_triangle_property_pythagorean(1,ODA)
Problem: As shown in the diagram, OB=5, OC=3, O is the center of circle O, OC⊥BC. Find the length of line AB.
问题：如图所示，OB=5，OC=3，O是圆O的圆心，OC⊥BC。求直线AB的长度。
Construction (CDL):
Shape(OC,CB,BO)
Shape(CA,OAB,BC)
Shape(OB,OBA,AC,CO)
Collinear(ACB)
Cocircular(O,AB)
Conditions (CDL):
Equal(LengthOfLine(OB),5)
Equal(LengthOfLine(OC),3)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(OC,BC)
Goal: Value(LengthOfLine(AB))
Answer: 8
Theorem sequence:
adjacent_complementary_angle(1,ACO,OCB)
circle_property_chord_perpendicular_bisect_chord(1,O,OC,AB)
line_addition(1,AC,CB)
right_triangle_judgment_angle(1,OCB)
right_triangle_property_pythagorean(1,OCB)
Problem: As shown in the diagram, AO=50, BO=50, CO=30, DE=40, DO=30, DE is perpendicular to AE. Find the measure of ∠AOB.
问题：如图所示，AO=50，BO=50，CO=30，DE=40，DO=30，DE垂直于AE。求∠AOB的大小。
Construction (CDL):
Shape(AO,OF,FE,EA)
Shape(EF,FB,BE)
Shape(OD,DF,FO)
Shape(OC,CD,DO)
Collinear(AEB)
Collinear(AOD)
Collinear(EFD)
Collinear(COFB)
Conditions (CDL):
Equal(LengthOfLine(AO),50)
Equal(LengthOfLine(BO),50)
Equal(LengthOfLine(CO),30)
Equal(LengthOfLine(DE),40)
Equal(LengthOfLine(DO),30)
PerpendicularBetweenLine(DE,AE)
Goal: Value(MeasureOfAngle(AOB))
Answer: 120
Theorem sequence:
line_addition(1,Ao,OD)
sine_theorem(1,OBA)
sine_theorem(1,DEA)
triangle_property_angle_sum(1,AOB)
Problem: As shown in the diagram, ∠ABC=50°, ∠BCA=60°, CF is perpendicular to DF, DE⊥BE. Find the measure of ∠EDF.
问题：如图所示，∠ABC=50°，∠BCA=60°，CF⊥DF，DE垂直于BE。求∠EDF的大小。
Construction (CDL):
Shape(AE,ED,DF,FA)
Shape(EB,BD,DE)
Shape(FD,DC,CF)
Collinear(BDC)
Collinear(AEB)
Collinear(AFC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),50)
Equal(MeasureOfAngle(BCA),60)
PerpendicularBetweenLine(CF,DF)
PerpendicularBetweenLine(DE,BE)
Goal: Value(MeasureOfAngle(EDF))
Answer: 110
Theorem sequence:
adjacent_complementary_angle(1,AED,DEB)
adjacent_complementary_angle(1,CFD,DFA)
triangle_property_angle_sum(1,BCA)
quadrilateral_property_angle_sum(1,AEDF)
Problem: As shown in the diagram, AB=8, OC=3, O is the center of circle O, OC⊥BC. Find the diameter of ⊙O.
问题：如图所示，AB=8，OC=3，⊙O的圆心为O，OC垂直于BC。求⊙O的直径。
Construction (CDL):
Shape(OA,AC,CO)
Shape(OAB,BA)
Shape(OC,CB,OBA,AO)
Collinear(ACB)
Cocircular(O,AB)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(LengthOfLine(OC),3)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(OC,BC)
Goal: Value(DiameterOfCircle(O))
Answer: 10
Theorem sequence:
adjacent_complementary_angle(1,ACO,OCB)
circle_property_chord_perpendicular_bisect_chord(1,O,OC,AB)
line_addition(1,AC,CB)
right_triangle_judgment_angle(1,ACO)
right_triangle_property_pythagorean(1,ACO)
radius_of_circle_property_length_equal(1,OA,O)
circle_property_length_of_radius_and_diameter(1,O)
Problem: As shown in the diagram, ∠CAB=25°, O is the center of circle O, OE⊥CE. Find the measure of ∠BOD.
问题：如图所示，∠CAB=25°，O是⊙O的圆心，OE垂直于CE。求∠BOD的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(CA,AO,OE,EC)
Shape(CE,EB,OBC)
Shape(OA,OAD,DO)
Shape(OD,DE,EO)
Shape(ED,ODB,BE)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),25)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(OE,CE)
Goal: Value(MeasureOfAngle(BOD))
Answer: 50
Theorem sequence:
adjacent_complementary_angle(1,DEO,OEC)
circle_property_chord_perpendicular_bisect_arc(1,ODC,OEB)
congruent_arc_judgment_length_equal(1,ODB,OBC)
congruent_arc_property_measure_equal(1,ODB,OBC)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,ODB,O)
Problem: As shown in the diagram, DE=8, OC=5, the center of ⊙O is O, OF is perpendicular to DF. Find the length of line OF.
问题：如图所示，DE=8，OC=5，O是圆O的圆心，OF⊥DF。求直线OF的长度。
Construction (CDL):
Shape(OC,OCA,AO)
Shape(CO,OD,ODC)
Shape(DO,OF,FD)
Shape(OED,DE)
Shape(FO,OB,OBE,EF)
Shape(OA,OAB,BO)
Collinear(DFE)
Collinear(AOB)
Cocircular(O,DCABE)
Conditions (CDL):
Equal(LengthOfLine(DE),8)
Equal(LengthOfLine(OC),5)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(OF,DF)
Goal: Value(LengthOfLine(OF))
Answer: 3
Theorem sequence:
adjacent_complementary_angle(1,EFO,OFD)
circle_property_chord_perpendicular_bisect_chord(1,O,OF,ED)
line_addition(1,DF,FE)
right_triangle_judgment_angle(1,OFD)
radius_of_circle_property_length_equal(1,OD,O)
radius_of_circle_property_length_equal(1,OC,O)
right_triangle_property_pythagorean(1,OFD)
Problem: As shown in the diagram, AB=16, OB=10, the center of circle O is O, OC is perpendicular to BC. Find the length of line OC.
问题：如图所示，AB=16，OB=10，⊙O的圆心为O，OC⊥BC。求直线OC的长度。
Construction (CDL):
Shape(OC,CB,BO)
Shape(CA,OAB,BC)
Shape(OB,OBA,AC,CO)
Collinear(ACB)
Cocircular(O,AB)
Conditions (CDL):
Equal(LengthOfLine(AB),16)
Equal(LengthOfLine(OB),10)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(OC,BC)
Goal: Value(LengthOfLine(OC))
Answer: 6
Theorem sequence:
adjacent_complementary_angle(1,ACO,OCB)
circle_property_chord_perpendicular_bisect_chord(1,O,OC,AB)
line_addition(1,AC,CB)
right_triangle_judgment_angle(1,OCB)
right_triangle_property_pythagorean(1,OCB)
Problem: As shown in the diagram, ∠BCA=30°, ∠DAB=20°, BD⊥AD. Find the measure of ∠CAB.
问题：如图所示，∠BCA=30°，∠DAB=20°，BD垂直于AD。求∠CAB的大小。
Construction (CDL):
Shape(AB,BD,DA)
Shape(AD,DC,CA)
Collinear(BDC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),30)
Equal(MeasureOfAngle(DAB),20)
PerpendicularBetweenLine(BD,AD)
Goal: Value(MeasureOfAngle(CAB))
Answer: 80
Theorem sequence:
triangle_property_angle_sum(1,ABD)
triangle_property_angle_sum(1,ABC)
Problem: As shown in the diagram, ∠BCD=90°, ∠DAB=30°, BD is the angle bisector of ∠ABC. Find the measure of ∠CDB.
问题：如图所示，∠BCD=90°，∠DAB=30°，BD平分∠ABC。求∠CDB的大小。
Construction (CDL):
Shape(CD,DB,BC)
Shape(DA,AB,BD)
Collinear(CDA)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),90)
Equal(MeasureOfAngle(DAB),30)
IsBisectorOfAngle(BD,ABC)
Goal: Value(MeasureOfAngle(CDB))
Answer: 60
Theorem sequence:
triangle_property_angle_sum(1,CDB)
triangle_property_angle_sum(1,BCA)
angle_addition(1,ABD,DBC)
Problem: As shown in the diagram, ∠ABC=72°, ∠CAB=90°, AD is the angle bisector of ∠CAB, AE is perpendicular to CE. Find the measure of ∠ECD.
问题：如图所示，∠ABC=72°，∠CAB=90°，AD是∠CAB的角平分线，AE⊥CE。求∠ECD的大小。
Construction (CDL):
Shape(CA,AD,DC)
Shape(CD,DE,EC)
Shape(DA,AB,BD)
Collinear(ADEF)
Collinear(CDB)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),72)
Equal(MeasureOfAngle(CAB),90)
IsBisectorOfAngle(AD,CAB)
PerpendicularBetweenLine(AE,CE)
Goal: Value(MeasureOfAngle(ECD))
Answer: 27
Theorem sequence:
angle_addition(1,CAD,DAB)
angle_addition(1,ECD,DCA)
triangle_property_angle_sum(1,CAB)
triangle_property_angle_sum(1,CAE)
Problem: As shown in the diagram, AB=10, ∠BCA=45°, O is the center of circle O, CB is the diameter of circle O, M is the midpoint of segment AB, N bisects segment AC. Find the length of line MN.
问题：如图所示，AB=10，∠BCA=45°，圆O的圆心为O，圆O的直径为CB，M平分线段AB，N是线段AC的中点。求直线MN的长度。
Construction (CDL):
Shape(NA,AM,MN)
Shape(CN,NM,MB,BC)
Shape(OCA,AC)
Shape(OAB,BA)
Shape(CO,OB,OBC)
Collinear(COB)
Collinear(AMB)
Collinear(ANC)
Cocircular(O,CAB)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(MeasureOfAngle(BCA),45)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(CB,O)
IsMidpointOfLine(M,AB)
IsMidpointOfLine(N,AC)
Goal: Value(LengthOfLine(MN))
Answer: 5*sqrt(2)
Theorem sequence:
line_addition(1,AM,MB)
line_addition(1,AN,NC)
diameter_of_circle_property_right_angle(1,CAB,O)
sine_theorem(1,BCA)
similar_triangle_judgment_sas(1,AMN,ABC)
similar_triangle_property_line_ratio(1,AMN,ABC)
similar_triangle_property_line_ratio(1,NAM,CAB)