SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, AB=6, AD=5, OF=2, quadrilateral ABCD is a parallelogram. Find the perimeter of BCEF.
问题：如图所示，AB=6，AD=5，OF=2，四边形ABCD是▱。求四边形BCEF的周长。
Construction (CDL):
Shape(DA,AO,OD)
Shape(DO,OE,ED)
Shape(EO,OC,CE)
Shape(CO,OB,BC)
Shape(OA,AF,FO)
Shape(OF,FB,BO)
Collinear(DOB)
Collinear(EOF)
Collinear(AOC)
Collinear(DEC)
Collinear(AFB)
Conditions (CDL):
Equal(LengthOfLine(AB),6)
Equal(LengthOfLine(AD),5)
Equal(LengthOfLine(OF),2)
Parallelogram(ABCD)
Goal: Value(PerimeterOfQuadrilateral(BCEF))
Answer: 15
Theorem sequence:
parallelogram_property_diagonal_bisection(1,ABCD,O)
parallel_property_collinear_extend(3,BA,CD,F)
parallel_property_collinear_extend(3,DC,AF,E)
parallel_property_alternate_interior_angle(2,BA,CD)
parallel_property_alternate_interior_angle(1,EC,AF)
similar_triangle_judgment_aa(1,OCE,OAF)
similar_triangle_property_line_ratio(1,OCE,OAF)
similar_triangle_property_line_ratio(1,CEO,AFO)
similar_triangle_property_line_ratio(1,EOC,FOA)
parallelogram_property_opposite_line_equal(1,BCDA)
line_addition(1,EO,OF)
line_addition(1,AF,FB)
quadrilateral_perimeter_formula(1,BCEF)
Problem: As shown in the diagram, ∠CEB=90°, ∠DAE=130°, AD and BC are opposite sides of the ▱ ABCD. Find the measure of ∠BCE.
问题：如图所示，∠CEB=90°，∠DAE=130°，AD和BC是▱ABCD的一组对边。求∠BCE的大小。
Construction (CDL):
Shape(AE,EC,CD,DA)
Shape(EB,BC,CE)
Collinear(AEB)
Conditions (CDL):
Equal(MeasureOfAngle(CEB),90)
Equal(MeasureOfAngle(DAE),130)
Parallelogram(ABCD)
Goal: Value(MeasureOfAngle(BCE))
Answer: 40
Theorem sequence:
parallel_property_ipsilateral_internal_angle(1,AD,BC)
triangle_property_angle_sum(1,EBC)
Problem: As shown in the diagram, ∠AEM=90°, ∠FAE=40°, ∠MFC=90°, AD and BC are opposite sides of the ▱ ABCD. Find the measure of ∠ABC.
问题：如图所示，∠AEM=90°，∠FAE=40°，∠MFC=90°，四边形ABCD是平行四边形。求∠ABC的大小。
Construction (CDL):
Shape(AB,BE,EA)
Shape(AE,EM,MA)
Shape(MF,FC,CM)
Shape(AM,MC,CD,DA)
Collinear(BEMC)
Collinear(DCF)
Collinear(AMF)
Conditions (CDL):
Equal(MeasureOfAngle(AEM),90)
Equal(MeasureOfAngle(FAE),40)
Equal(MeasureOfAngle(MFC),90)
Parallelogram(ABCD)
Goal: Value(MeasureOfAngle(ABC))
Answer: 40
Theorem sequence:
vertical_angle(1,BMA,CMF)
mirror_similar_triangle_judgment_aa(1,AEM,CMF)
mirror_similar_triangle_property_angle_equal(1,AEM,CMF)
parallel_property_corresponding_angle(1,CB,DA,F)
parallelogram_property_opposite_angle_equal(1,BCDA)
Problem: As shown in the diagram, ∠ABC=70°, the measure of ⌒OCD is equal to the measure of ⌒ODA, the diameter of circle O is AB. Find the measure of ∠DAB.
问题：如图所示，∠ABC=70°，弧OCD的角度与⌒ODA的角度相等，⊙O的直径为AB。求∠DAB的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(DC,OCD)
Shape(CB,OBC)
Shape(DA,AE,ED)
Shape(EC,CD,DE)
Shape(EB,BC,CE)
Shape(EA,AB,BE)
Shape(BA,OAB)
Collinear(AOB)
Collinear(DEB)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),70)
Equal(MeasureOfArc(OCD),MeasureOfArc(ODA))
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(DAB))
Answer: 55
Theorem sequence:
arc_addition_measure(1,OCD,ODA)
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_external(1,ODA,B)
diameter_of_circle_property_right_angle(1,BDA,O)
triangle_property_angle_sum(1,DAB)
Problem: As shown in the diagram, ∠BOA=80°, O is the center of ⊙O. Find the measure of ∠BCA.
问题：如图所示，∠BOA=80°，⊙O的圆心为O。求∠BCA的大小。
Construction (CDL):
Shape(AO,OE,EC,OCA)
Shape(CB,OBC)
Shape(OA,AE,EO)
Shape(CE,EB,BC)
Shape(BE,EA,OAB)
Collinear(OEB)
Collinear(AEC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),80)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 40
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠CBA=28°, ∠OEB=90°, O is the center of circle O, AB is the diameter of circle O. Find the measure of ∠BCD.
问题：如图所示，∠CBA=28°，∠OEB=90°，O是圆O的圆心，圆O的直径为AB。求∠BCD的大小。
Construction (CDL):
Shape(DC,OCD)
Shape(BE,ED,ODB)
Shape(CA,OAC)
Shape(AB,OBA)
Shape(DE,EC,CD)
Shape(EB,BO,OE)
Shape(EO,OA,AC,CE)
Collinear(DEO)
Collinear(BEC)
Collinear(BOA)
Cocircular(O,BACD)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),28)
Equal(MeasureOfAngle(OEB),90)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BCD))
Answer: 31
Theorem sequence:
triangle_property_angle_sum(1,EBO)
arc_property_center_angle(1,ODB,O)
arc_property_circumference_angle_external(1,ODB,C)
Problem: As shown in the diagram, BC=6, ∠BAC=30°, the center of circle O is O. Find the radius of circle O.
问题：如图所示，BC=6，∠BAC=30°，O是圆O的圆心。求圆O的半径。
Construction (CDL):
Shape(BC,OCB)
Shape(CA,OAC)
Shape(AE,EO,OB,OBA)
Shape(EC,CB,BE)
Shape(CE,EA,AC)
Shape(EB,BO,OE)
Collinear(CEO)
Collinear(ABE)
Cocircular(O,ACB)
Conditions (CDL):
Equal(LengthOfLine(BC),6)
Equal(MeasureOfAngle(BAC),30)
IsCentreOfCircle(O,O)
Goal: Value(RadiusOfCircle(O))
Answer: 6
Theorem sequence:
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_external(1,OCB,A)
radius_of_circle_property_length_equal(1,OC,O)
radius_of_circle_property_length_equal(1,OB,O)
triangle_property_angle_sum(1,CBO)
isosceles_triangle_judgment_line_equal(1,OCB)
isosceles_triangle_property_angle_equal(1,OCB)
sine_theorem(1,CBO)
Problem: As shown in the diagram, ∠ACD=15°, the diameter of ⊙O is AB. Find the measure of ∠ABD.
问题：如图所示，∠ACD=15°，AB是圆O的直径。求∠ABD的大小。
Construction (CDL):
Shape(BE,EC,OCB)
Shape(DO,OB,BD)
Shape(DB,OBD)
Shape(AD,ODA)
Shape(CA,OAC)
Shape(CE,EA,AC)
Shape(EO,OD,DE)
Shape(AE,ED,DA)
Collinear(CED)
Collinear(AEOB)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(ACD),15)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ABD))
Answer: 15
Theorem sequence:
arc_property_circumference_angle_external(1,ODA,C)
arc_property_circumference_angle_external(1,ODA,B)
Problem: As shown in the diagram, ∠BAC=80°, the center of ⊙O is O. Find the measure of ∠BOC.
问题：如图所示，∠BAC=80°，⊙O的圆心为O。求∠BOC的大小。
Construction (CDL):
Shape(BO,OC,OCB)
Shape(AB,OBA)
Shape(CA,OAC)
Shape(OB,BA,AC,CO)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),80)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOC))
Answer: 160
Theorem sequence:
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_external(1,OCB,A)
Problem: As shown in the diagram, ∠BOA=110°, the center of ⊙O is O. Find the measure of ∠ACB.
问题：如图所示，∠BOA=110°，O是⊙O的圆心。求∠ACB的大小。
Construction (CDL):
Shape(AO,OB,OBA)
Shape(CA,OAC)
Shape(BC,OCB)
Shape(OA,AC,CB,BO)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),110)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(ACB))
Answer: 125
Theorem sequence:
round_angle(1,AOB,BOA)
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_external(1,OBA,C)
Problem: As shown in the diagram, ∠BCD=28°, AB is the diameter of ⊙O. Find the measure of ∠DBA.
问题：如图所示，∠BCD=28°，⊙O的直径为AB。求∠DBA的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(DE,EA,OAD)
Shape(BD,ODB)
Shape(CB,OBC)
Shape(CA,AE,EC)
Shape(CE,EB,BC)
Shape(ED,DB,BE)
Collinear(CED)
Collinear(AOEB)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),28)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(DBA))
Answer: 62
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
angle_addition(1,BCE,ECA)
arc_property_circumference_angle_external(1,OAD,B)
arc_property_circumference_angle_external(1,OAD,C)
Problem: As shown in the diagram, ∠AOC=70°, O is the center of circle O, AB is the diameter of circle O, ABCD is a ▱. Find the measure of ∠DAB.
问题：如图所示，∠AOC=70°，O是⊙O的圆心，圆O的直径为AB，BA和CD是平行四边形ABCD的一组对边。求∠DAB的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(CB,OBC)
Shape(AO,OC,OCA)
Shape(CD,DA,OCA)
Shape(OB,BC,CO)
Collinear(AOB)
Cocircular(O,BCA)
Conditions (CDL):
Equal(MeasureOfAngle(AOC),70)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Parallelogram(ABCD)
Goal: Value(MeasureOfAngle(DAB))
Answer: 145
Theorem sequence:
arc_property_center_angle(1,OCA,O)
arc_property_circumference_angle_external(1,OCA,B)
parallel_property_ipsilateral_internal_angle(1,AD,BC)
Problem: As shown in the diagram, ∠AOB=44°, the measure of arc OBA is equal to the measure of ⌒OAC, O is the center of circle O. Find the measure of ∠CDA.
问题：如图所示，∠AOB=44°，⌒OBA的角度与弧OAC的角度相等，O是圆O的圆心。求∠CDA的大小。
Construction (CDL):
Shape(DC,OCD)
Shape(CD,DA,OAC)
Shape(AO,OB,OBA)
Shape(BO,OA,AD,ODB)
Cocircular(O,BACD)
Conditions (CDL):
Equal(MeasureOfAngle(AOB),44)
Equal(MeasureOfArc(OBA),MeasureOfArc(OAC))
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CDA))
Answer: 22
Theorem sequence:
arc_property_center_angle(1,OBA,O)
arc_property_circumference_angle_external(1,OAC,D)
Problem: As shown in the diagram, ∠CDE=50°, the center of ⊙O is O, the diameter of circle O is AB, AO is parallel to DE. Find the measure of ∠DCA.
问题：如图所示，∠CDE=50°，O是圆O的圆心，⊙O的直径为AB，AO∥DE。求∠DCA的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(DO,OA,OAD)
Shape(ED,ODE)
Shape(CD,DE,OEC)
Shape(OC,CA,AO)
Collinear(COD)
Cocircular(O,ADEC)
Conditions (CDL):
Equal(MeasureOfAngle(CDE),50)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
ParallelBetweenLine(AO,DE)
Goal: Value(MeasureOfAngle(DCA))
Answer: 25
Theorem sequence:
parallel_property_alternate_interior_angle(2,AO,DE)
arc_property_center_angle(1,OAD,O)
arc_property_circumference_angle_external(1,OAD,C)
Problem: As shown in the diagram, AD=12, CD=5, ∠ABC=90°. Find the diameter of circle O.
问题：如图所示，AD=12，CD=5，∠ABC=90°。求⊙O的直径。
Construction (CDL):
Shape(BA,OAB)
Shape(CB,OBC)
Shape(DC,OCD)
Shape(AD,ODA)
Shape(AB,BC,CA)
Shape(AC,CD,DA)
Collinear(AOC)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(LengthOfLine(AD),12)
Equal(LengthOfLine(CD),5)
Equal(MeasureOfAngle(ABC),90)
Goal: Value(DiameterOfCircle(O))
Answer: 13
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_external(1,OAC,D)
round_arc(1,OAC,OCA)
right_triangle_judgment_angle(1,CDA)
right_triangle_property_pythagorean(1,CDA)
diameter_of_circle_judgment_right_angle(1,ABC,O)
diameter_of_circle_property_length_equal(1,CA,O)
Problem: As shown in the diagram, ∠CBA=52°, the measure of arc OAD is equal to the measure of arc ODC, the diameter of circle O is AB. Find the measure of ∠BAD.
问题：如图所示，∠CBA=52°，弧OAD的角度与弧ODC的角度相等，⊙O的直径为AB。求∠BAD的大小。
Construction (CDL):
Shape(BC,OCB)
Shape(CD,ODC)
Shape(DA,OAD)
Shape(AB,OBA)
Shape(CB,BD,DC)
Shape(DB,BA,AD)
Collinear(BOA)
Cocircular(O,ADCB)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),52)
Equal(MeasureOfArc(OAD),MeasureOfArc(ODC))
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BAD))
Answer: 64
Theorem sequence:
arc_property_circumference_angle_external(1,OAD,B)
arc_property_circumference_angle_external(1,OAC,B)
arc_addition_measure(1,OAD,ODC)
diameter_of_circle_property_right_angle(1,ADB,O)
triangle_property_angle_sum(1,DBA)
Problem: As shown in the diagram, AB=AC, AB=AD, ∠CAB=42°, ∠DBC=Mul(MeasureOfAngle(CDB),2)°, A is the center of circle A. Find the measure of ∠DAC.
问题：如图所示，AB=AC，AB=AD，∠CAB=42°，∠DBC=Mul(MeasureOfAngle(CDB),2)°，A是圆A的圆心。求∠DAC的大小。
Construction (CDL):
Shape(AB,BE,EA)
Shape(EB,BC,CE)
Shape(EC,CD,DE)
Shape(AE,ED,DA)
Shape(AE,ED,DA)
Shape(CB,ABC)
Shape(DC,ACD)
Shape(BA,AD,ADB)
Collinear(AEC)
Collinear(BED)
Cocircular(A,BCD)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(LengthOfLine(AB),LengthOfLine(AD))
Equal(MeasureOfAngle(CAB),42)
Equal(MeasureOfAngle(DBC),Mul(MeasureOfAngle(CDB),2))
IsCentreOfCircle(A,A)
Goal: Value(MeasureOfAngle(DAC))
Answer: 84
Theorem sequence:
arc_property_center_angle(1,ABC,A)
arc_property_center_angle(1,ACD,A)
arc_property_circumference_angle_external(1,ABC,D)
arc_property_circumference_angle_external(1,ACD,B)
Problem: As shown in the diagram, OA=2, ∠CBA=30°, the center of ⊙O is O, the diameter of ⊙O is AB. Find the length of line AC.
问题：如图所示，OA=2，∠CBA=30°，O是圆O的圆心，AB是⊙O的直径。求直线AC的长度。
Construction (CDL):
Shape(AB,OBA)
Shape(CE,EA,AC)
Shape(CA,OAC)
Shape(DC,OCD)
Shape(BE,ED,ODB)
Shape(AE,EB,BA)
Shape(EC,CD,DE)
Collinear(AOB)
Collinear(AED)
Collinear(BEC)
Cocircular(O,ACDB)
Conditions (CDL):
Equal(LengthOfLine(OA),2)
Equal(MeasureOfAngle(CBA),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(LengthOfLine(AC))
Answer: 2
Theorem sequence:
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)
diameter_of_circle_property_right_angle(1,ACB,O)
sine_theorem(1,ACB)
Problem: As shown in the diagram, AB=2, AC=AD, ∠CBA=30°, the center of ⊙O is O, AB is the diameter of ⊙O. Find the length of line CD.
问题：如图所示，AB=2，AC=AD，∠CBA=30°，圆O的圆心为O，⊙O的直径为AB。求直线CD的长度。
Construction (CDL):
Shape(BC,OCB)
Shape(AB,OBA)
Shape(CA,OAC)
Shape(BO,OC,CB)
Shape(CO,OA,AC)
Shape(AD,DC,OAC)
Collinear(BOAD)
Cocircular(O,ACB)
Conditions (CDL):
Equal(LengthOfLine(AB),2)
Equal(LengthOfLine(AC),LengthOfLine(AD))
Equal(MeasureOfAngle(CBA),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(LengthOfLine(CD))
Answer: sqrt(3)
Theorem sequence:
arc_property_center_angle(1,OAC,O)
arc_property_circumference_angle_external(1,OAC,B)
diameter_of_circle_property_right_angle(1,ACB,O)
triangle_property_angle_sum(1,CBA)
adjacent_complementary_angle(1,BAC,CAD)
isosceles_triangle_judgment_line_equal(1,ADC)
isosceles_triangle_property_angle_equal(1,ADC)
triangle_property_angle_sum(1,CAD)
radius_of_circle_property_length_equal(1,OC,O)
circle_property_length_of_radius_and_diameter(1,O)
diameter_of_circle_property_length_equal(1,AB,O)
sine_theorem(1,COD)
Problem: As shown in the diagram, ∠CAB=35°, the measure of ⌒OCD is 80, O is the center of circle O. Find the measure of ∠DOB.
问题：如图所示，∠CAB=35°，弧OCD的角度为80，O是⊙O的圆心。求∠DOB的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(CE,EB,OBC)
Shape(DO,OC,OCD)
Shape(AE,EO,OD,ODA)
Shape(BE,EA,AB)
Shape(OE,EC,CO)
Collinear(AEC)
Collinear(BEO)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),35)
Equal(MeasureOfArc(OCD),80)
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)
angle_addition(1,DOC,COE)
Problem: As shown in the diagram, AD=3, ∠DCB=30°, the diameter of circle O is AB. Find the diameter of circle O.
问题：如图所示，AD=3，∠DCB=30°，圆O的直径为AB。求圆O的直径。
Construction (CDL):
Shape(AD,ODA)
Shape(DB,OBD)
Shape(BC,OCB)
Shape(CE,EA,OAC)
Shape(DA,AE,ED)
Shape(DE,EB,BD)
Shape(BE,EC,CB)
Collinear(AOEB)
Collinear(DEC)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(LengthOfLine(AD),3)
Equal(MeasureOfAngle(DCB),30)
IsDiameterOfCircle(AB,O)
Goal: Value(DiameterOfCircle(O))
Answer: 2*sqrt(3)
Theorem sequence:
arc_property_circumference_angle_external(1,OBD,A)
arc_property_circumference_angle_external(1,OBD,C)
diameter_of_circle_property_right_angle(1,BDA,O)
triangle_property_angle_sum(1,DAB)
sine_theorem(1,ABD)
diameter_of_circle_property_length_equal(1,AB,O)
Problem: As shown in the diagram, ∠EOD=50°, the center of ⊙O is O, AB is the diameter of ⊙O, the diameter of circle O is CD, ED∥AO. Find the measure of ∠DOB.
问题：如图所示，∠EOD=50°，O是圆O的圆心，圆O的直径为AB，圆O的直径为CD，ED平行于AO。求∠DOB的大小。
Construction (CDL):
Shape(AO,OE,OEA)
Shape(CO,OA,OAC)
Shape(BO,OC,OCB)
Shape(DO,OB,OBD)
Shape(OD,DE,EO)
Shape(ED,ODE)
Collinear(AOB)
Collinear(COD)
Cocircular(O,ACBDE)
Conditions (CDL):
Equal(MeasureOfAngle(EOD),50)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
IsDiameterOfCircle(CD,O)
ParallelBetweenLine(ED,AO)
Goal: Value(MeasureOfAngle(DOB))
Answer: 65
Theorem sequence:
radius_of_circle_property_length_equal(1,OD,O)
radius_of_circle_property_length_equal(1,OE,O)
triangle_property_angle_sum(1,DEO)
isosceles_triangle_judgment_line_equal(1,ODE)
isosceles_triangle_property_angle_equal(1,ODE)
parallel_property_alternate_interior_angle(1,ED,AO)
angle_addition(1,AOE,EOD)
adjacent_complementary_angle(1,AOD,DOB)
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(AO,OE,EC,OCA)
Shape(BE,EA,OAB)
Shape(CB,OBC)
Shape(AE,EO,OA)
Shape(EB,BC,CE)
Collinear(OEB)
Collinear(AEC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),70)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 35
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, OM=8, ON=6, ∠MON=90°. Find the radius of circle E.
问题：如图所示，OM=8，ON=6，∠MON=90°。求⊙E的半径。
Construction (CDL):
Shape(OM,EMO)
Shape(NO,EON)
Shape(MN,ONM)
Shape(ON,NM,MO)
Cocircular(E,ONM)
Conditions (CDL):
Equal(LengthOfLine(OM),8)
Equal(LengthOfLine(ON),6)
Equal(MeasureOfAngle(MON),90)
Goal: Value(RadiusOfCircle(E))
Answer: 5
Theorem sequence:
right_triangle_judgment_angle(1,MON)
right_triangle_property_pythagorean(1,MON)
diameter_of_circle_judgment_right_angle(1,MON,E)
diameter_of_circle_property_length_equal(1,NM,E)
circle_property_length_of_radius_and_diameter(1,E)
Problem: As shown in the diagram, ∠AEC=90°, ∠CAB=40°. Find the measure of ∠DBA.
问题：如图所示，∠AEC=90°，∠CAB=40°。求∠DBA的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(DO,OA,OAD)
Shape(BD,ODB)
Shape(CE,EB,OBC)
Shape(AO,OE,EC,CA)
Shape(EO,OD,DE)
Shape(ED,DB,BE)
Collinear(CED)
Collinear(AOEB)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(AEC),90)
Equal(MeasureOfAngle(CAB),40)
Goal: Value(MeasureOfAngle(DBA))
Answer: 50
Theorem sequence:
triangle_property_angle_sum(1,AEC)
arc_property_circumference_angle_external(1,OAD,B)
arc_property_circumference_angle_external(1,OAD,C)
Problem: As shown in the diagram, ∠CAB=40°, the measure of ⌒OCD is equal to the measure of arc ODA, the center of circle O is O, AB is the diameter of ⊙O. Find the measure of ∠DAC.
问题：如图所示，∠CAB=40°，弧OCD的角度与⌒ODA的角度相等，O是⊙O的圆心，AB是⊙O的直径。求∠DAC的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(DC,OCD)
Shape(CB,OBC)
Shape(AC,CD,DA)
Shape(CA,AB,BC)
Shape(BA,OAB)
Collinear(AOB)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),40)
Equal(MeasureOfArc(OCD),MeasureOfArc(ODA))
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(DAC))
Answer: 25
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,ABC)
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_external(1,OCD,A)
arc_property_circumference_angle_external(1,ODA,C)
arc_property_circumference_angle_internal(1,OCA,D)
triangle_property_angle_sum(1,DAC)
Problem: As shown in the diagram, ∠AOC=130°, O is the center of ⊙O, the diameter of ⊙O is AB. Find the measure of ∠CDB.
问题：如图所示，∠AOC=130°，O是⊙O的圆心，AB是圆O的直径。求∠CDB的大小。
Construction (CDL):
Shape(BD,ODB)
Shape(CE,EB,OBC)
Shape(AO,OC,OCA)
Shape(DE,DA,OAD)
Shape(BE,ED,DB)
Shape(EC,CO,OE)
Collinear(DEC)
Collinear(BEOA)
Cocircular(O,BCAD)
Conditions (CDL):
Equal(MeasureOfAngle(AOC),130)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CDB))
Answer: 25
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, the diameter of circle O is 6, ∠BAC=60°, the center of circle O is O, BD is the diameter of ⊙O. Find the length of line BC.
问题：如图所示，⊙O的直径为6，∠BAC=60°，⊙O的圆心为O，⊙O的直径为BD。求直线BC的长度。
Construction (CDL):
Shape(AB,OBA)
Shape(DE,EA,OAD)
Shape(CD,ODC)
Shape(BC,OCB)
Shape(BA,AE,EB)
Shape(BE,EC,CB)
Shape(CE,ED,DC)
Collinear(BOED)
Collinear(AEC)
Cocircular(O,ADCB)
Conditions (CDL):
Equal(DiameterOfCircle(O),6)
Equal(MeasureOfAngle(BAC),60)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BD,O)
Goal: Value(LengthOfLine(BC))
Answer: 3*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)
diameter_of_circle_property_length_equal(1,BD,O)
sine_theorem(1,BDC)
Problem: As shown in the diagram, ∠ABC=50°. Find the measure of ∠AEC.
问题：如图所示，∠ABC=50°。求∠AEC的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(EM,MB,OBE)
Shape(CE,OEC)
Shape(AC,OCA)
Shape(AB,BM,MA)
Shape(ME,EC,CM)
Shape(AM,MC,CA)
Collinear(AME)
Collinear(BMC)
Cocircular(O,ABEC)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),50)
Goal: Value(MeasureOfAngle(AEC))
Answer: 50
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_external(1,OCA,E)
Problem: As shown in the diagram, the measure of ⌒OAB is 60, the center of circle O is O, the diameter of ⊙O is BD, AE is perpendicular to OE. Find the measure of ∠CDB.
问题：如图所示，⌒OAB的角度为60，O是⊙O的圆心，BD是圆O的直径，AE⊥OE。求∠CDB的大小。
Construction (CDL):
Shape(AO,OD,ODA)
Shape(BE,EA,OAB)
Shape(CE,EB,OBC)
Shape(DC,OCD)
Shape(AE,EO,OA)
Shape(DO,OC,CD)
Shape(OE,EC,CO)
Collinear(DOEB)
Collinear(AEC)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfArc(OAB),60)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(BD,O)
PerpendicularBetweenLine(AE,OE)
Goal: Value(MeasureOfAngle(CDB))
Answer: 30
Theorem sequence:
circle_property_chord_perpendicular_bisect_arc(1,OAC,OEB)
congruent_arc_judgment_length_equal(1,OAB,OBC)
congruent_arc_property_measure_equal(1,OAB,OBC)
arc_property_circumference_angle_external(1,OBC,D)
Problem: As shown in the diagram, ∠ACD=20°, O is the center of circle O, AB is the diameter of ⊙O. Find the measure of ∠DAB.
问题：如图所示，∠ACD=20°，O是⊙O的圆心，圆O的直径为AB。求∠DAB的大小。
Construction (CDL):
Shape(BC,OCB)
Shape(DE,EB,OBD)
Shape(AD,ODA)
Shape(CA,OAC)
Shape(CB,BE,EC)
Shape(CE,EA,AC)
Shape(AE,ED,DA)
Collinear(CED)
Collinear(BOEA)
Cocircular(O,BDAC)
Conditions (CDL):
Equal(MeasureOfAngle(ACD),20)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(DAB))
Answer: 70
Theorem sequence:
diameter_of_circle_property_right_angle(1,ACB,O)
angle_addition(1,ACE,ECB)
arc_property_circumference_angle_external(1,OBD,A)
arc_property_circumference_angle_external(1,OBD,C)
Problem: As shown in the diagram, ∠ABO=50°, O is the center of ⊙O. Find the measure of ∠BCA.
问题：如图所示，∠ABO=50°，圆O的圆心为O。求∠BCA的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(CB,OBC)
Shape(AO,OE,EC,OCA)
Shape(AB,BE,EA)
Shape(EB,BC,CE)
Shape(AE,EO,OA)
Collinear(AEC)
Collinear(OEB)
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,ABO)
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, ∠OBC=60°, the center of ⊙O is O, the diameter of circle O is AB. Find the measure of ∠CAB.
问题：如图所示，∠OBC=60°，O是⊙O的圆心，⊙O的直径为AB。求∠CAB的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(BA,OAB)
Shape(CB,OBC)
Shape(CA,AO,OC)
Shape(CO,OB,BC)
Collinear(AOB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(OBC),60)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 30
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,COB)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
Problem: As shown in the diagram, ∠CBA=50°, the measure of arc OAD is equal to the measure of arc ODC, the center of ⊙O is O, AB is the diameter of circle O. Find the measure of ∠BAD.
问题：如图所示，∠CBA=50°，⌒OAD的角度与⌒ODC的角度相等，圆O的圆心为O，圆O的直径为AB。求∠BAD的大小。
Construction (CDL):
Shape(BC,OCB)
Shape(CD,ODC)
Shape(DA,OAD)
Shape(AB,OBA)
Shape(BD,DC,CB)
Shape(DB,BA,AD)
Collinear(BOA)
Cocircular(O,BADC)
Conditions (CDL):
Equal(MeasureOfAngle(CBA),50)
Equal(MeasureOfArc(OAD),MeasureOfArc(ODC))
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(BAD))
Answer: 65
Theorem sequence:
arc_property_circumference_angle_external(1,OAD,B)
arc_property_circumference_angle_external(1,ODC,B)
angle_addition(1,CBD,DBA)
diameter_of_circle_property_right_angle(1,ADB,O)
triangle_property_angle_sum(1,DBA)