SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, AB=16, OA=10, ∠ABH=60°, ∠OAB=60°, O is the center of circle O, OH⊥DH. Find the length of line BC.
问题：如图所示，AB=16，OA=10，∠ABH=60°，∠OAB=60°，O是圆O的圆心，OH垂直于DH。求直线BC的长度。
Construction (CDL):
Shape(BA,AO,OD,DC,OCB)
Shape(DO,OH,HD)
Shape(HO,OA,AB,BH)
Shape(CD,DH,HB,OBC)
Collinear(CDHB)
Collinear(AOD)
Cocircular(O,BC)
Conditions (CDL):
Equal(LengthOfLine(AB),16)
Equal(LengthOfLine(OA),10)
Equal(MeasureOfAngle(ABH),60)
Equal(MeasureOfAngle(OAB),60)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(OH,DH)
Goal: Value(LengthOfLine(BC))
Answer: 26
Theorem sequence:
triangle_property_angle_sum(1,DAB)
isosceles_triangle_judgment_angle_equal(1,BDA)
isosceles_triangle_judgment_angle_equal(1,ABD)
line_addition(1,AO,OD)
triangle_property_angle_sum(1,DOH)
sine_theorem(1,DOH)
line_addition(1,DH,HB)
adjacent_complementary_angle(1,BHO,OHD)
circle_property_chord_perpendicular_bisect_chord(1,O,OH,BC)
line_addition(1,BH,HC)
Problem: As shown in the diagram, AB=24, ∠ACO=90°, the radius of circle O is 13, O is the center of circle O. Find the length of line OC.
问题：如图所示，AB=24，∠ACO=90°，圆O的半径为13，⊙O的圆心为O。求直线OC的长度。
Construction (CDL):
Shape(OA,AC,CO)
Shape(AO,OC,CB,OBA)
Shape(BC,CA,OAB)
Collinear(ACB)
Cocircular(O,AB)
Conditions (CDL):
Equal(LengthOfLine(AB),24)
Equal(MeasureOfAngle(ACO),90)
Equal(RadiusOfCircle(O),13)
IsCentreOfCircle(O,O)
Goal: Value(LengthOfLine(OC))
Answer: 5
Theorem sequence:
circle_property_chord_perpendicular_bisect_chord(1,O,OC,AB)
line_addition(1,AC,CB)
radius_of_circle_property_length_equal(1,OA,O)
right_triangle_judgment_angle(1,ACO)
right_triangle_property_pythagorean(1,ACO)
Problem: As shown in the diagram, AB=8, M bisects segment AC, N is the midpoint of segment CB. Find the length of line MN.
问题：如图所示，AB=8，M是线段AC的中点，N是线段CB的中点。求直线MN的长度。
Construction (CDL):
Collinear(AMCNB)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
IsMidpointOfLine(M,AC)
IsMidpointOfLine(N,CB)
Goal: Value(LengthOfLine(MN))
Answer: 4
Theorem sequence:
line_addition(1,AM,MC)
line_addition(1,CN,NB)
line_addition(1,AC,CB)
line_addition(1,MC,CN)
Problem: As shown in the diagram, ∠DAB=40°, ∠FBC=20°, AD is parallel to GB, BF is parallel to AE, DA is perpendicular to EA, FB is perpendicular to GB. Find the measure of ∠CBA.
问题：如图所示，∠DAB=40°，∠FBC=20°，AD∥GB，BF平行于AE，DA⊥EA，FB垂直于GB。求∠CBA的大小。
Construction (CDL):
Shape(DA,AB)
Shape(BA,AE)
Shape(FB,BC)
Shape(CB,BG)
Shape(GB,BA)
Shape(DA,AE)
Shape(FB,BG)
Shape(CB,BA)
Shape(FB,BA)
Conditions (CDL):
Equal(MeasureOfAngle(DAB),40)
Equal(MeasureOfAngle(FBC),20)
ParallelBetweenLine(AD,GB)
ParallelBetweenLine(BF,AE)
PerpendicularBetweenLine(DA,EA)
PerpendicularBetweenLine(FB,GB)
Goal: Value(MeasureOfAngle(CBA))
Answer: 110
Theorem sequence:
parallel_property_alternate_interior_angle(1,AD,GB)
angle_addition(1,FBC,CBG)
angle_addition(1,CBG,GBA)
Problem: As shown in the diagram, BC=3, BD=5, D bisects segment AC. Find the length of line AC.
问题：如图所示，BC=3，BD=5，D是线段AC的中点。求直线AC的长度。
Construction (CDL):
Collinear(ADCB)
Conditions (CDL):
Equal(LengthOfLine(BC),3)
Equal(LengthOfLine(BD),5)
IsMidpointOfLine(D,AC)
Goal: Value(LengthOfLine(AC))
Answer: 4
Theorem sequence:
line_addition(1,AD,DC)
line_addition(1,DC,CB)
Problem: As shown in the diagram, the measure of arc OBD is 60, O is the center of ⊙O, OE⊥DE. Find the measure of ∠EOC.
问题：如图所示，弧OBD的角度为60，圆O的圆心为O，OE垂直于DE。求∠EOC的大小。
Construction (CDL):
Shape(OAC,CO,OA)
Shape(OCB,BE,EC)
Shape(OC,CE,EO)
Shape(OBD,DE,EB)
Shape(AO,OE,ED,DA)
Shape(ODA,AD)
Collinear(CED)
Collinear(AOEB)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfArc(OBD),60)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(OE,DE)
Goal: Value(MeasureOfAngle(EOC))
Answer: 60
Theorem sequence:
adjacent_complementary_angle(1,CEO,OED)
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_center_angle(1,OCB,O)
Problem: As shown in the diagram, CE=2, the radius of ⊙O is 5, O is the center of ⊙O, OE is perpendicular to AE. Find the length of line AB.
问题：如图所示，CE=2，⊙O的半径为5，⊙O的圆心为O，OE垂直于AE。求直线AB的长度。
Construction (CDL):
Shape(OAB,BE,EO,OA)
Shape(OBC,CE,EB)
Shape(OCA,AE,EC)
Shape(OE,EA,AO)
Collinear(AEB)
Collinear(CEO)
Cocircular(O,ABC)
Conditions (CDL):
Equal(LengthOfLine(CE),2)
Equal(RadiusOfCircle(O),5)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(OE,AE)
Goal: Value(LengthOfLine(AB))
Answer: 8
Theorem sequence:
adjacent_complementary_angle(1,BEO,OEA)
circle_property_chord_perpendicular_bisect_chord(1,O,OE,BA)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OC,O)
line_addition(1,CE,EO)
right_triangle_judgment_angle(1,OEA)
right_triangle_property_pythagorean(1,OEA)
line_addition(1,AE,EB)
Problem: As shown in the diagram, AB=20, CD=16, 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=20，CD=16，O是圆O的圆心，CE垂直于OE。求直线BE的长度。
Construction (CDL):
Shape(OAC,CO,OA)
Shape(OCB,BE,EC)
Shape(OC,CE,EO)
Shape(OBD,DE,EB)
Shape(ODA,AO,OE,ED)
Collinear(CED)
Collinear(AOEB)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(LengthOfLine(AB),20)
Equal(LengthOfLine(CD),16)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(CE,OE)
Goal: Value(LengthOfLine(BE))
Answer: 4
Theorem sequence:
diameter_of_circle_judgment_pass_centre(1,AOB,O)
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,OC,O)
radius_of_circle_property_length_equal(1,OB,O)
circle_property_chord_perpendicular_bisect_chord(1,O,OE,CD)
line_addition(1,CE,ED)
line_addition(1,OE,EB)
right_triangle_judgment_angle(1,CEO)
right_triangle_property_pythagorean(1,CEO)
Problem: As shown in the diagram, OA=3, O is the center of ⊙O, D bisects segment OC, AD is perpendicular to OD. Find the length of line AB.
问题：如图所示，OA=3，⊙O的圆心为O，D是线段OC的中点，AD垂直于OD。求直线AB的长度。
Construction (CDL):
Shape(OAC,CD,DA)
Shape(OCB,BD,DC)
Shape(OBA,AO,OB)
Shape(OA,AD,DO)
Shape(OD,DB,BO)
Collinear(ADB)
Collinear(ODC)
Cocircular(O,ACB)
Conditions (CDL):
Equal(LengthOfLine(OA),3)
IsCentreOfCircle(O,O)
IsMidpointOfLine(D,OC)
PerpendicularBetweenLine(AD,OD)
Goal: Value(LengthOfLine(AB))
Answer: 3*sqrt(3)
Theorem sequence:
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)
adjacent_complementary_angle(1,ADO,ODB)
line_addition(1,AD,DB)
line_addition(1,OD,DC)
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, OP=4, ∠CPB=30°, the radius of ⊙O is 3, O is the center of circle O, OD⊥AD. Find the length of line AB.
问题：如图所示，OP=4，∠CPB=30°，圆O的半径为3，圆O的圆心为O，OD垂直于AD。求直线AB的长度。
Construction (CDL):
Shape(OAC,CO,OD,DA)
Shape(OCB,BO,OC)
Shape(OBA,AD,DB)
Shape(BD,DO,OB)
Shape(OCB,CP,PB)
Collinear(ADBP)
Collinear(OCP)
Cocircular(O,ACB)
Conditions (CDL):
Equal(LengthOfLine(OP),4)
Equal(MeasureOfAngle(CPB),30)
Equal(RadiusOfCircle(O),3)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(OD,AD)
Goal: Value(LengthOfLine(AB))
Answer: 2*sqrt(5)
Theorem sequence:
adjacent_complementary_angle(1,BDO,ODA)
sine_theorem(1,OPD)
radius_of_circle_property_length_equal(1,OB,O)
right_triangle_judgment_angle(1,BDO)
right_triangle_property_pythagorean(1,BDO)
circle_property_chord_perpendicular_bisect_chord(1,O,OD,BA)
line_addition(1,AD,DB)
Problem: As shown in the diagram, CD=8, OP=3, the center of circle O is O, OP is perpendicular to CP. Find the radius of ⊙O.
问题：如图所示，CD=8，OP=3，圆O的圆心为O，OP垂直于CP。求圆O的半径。
Construction (CDL):
Shape(OAD,DP,PO,OA)
Shape(ODB,BP,PD)
Shape(OBC,CP,PB)
Shape(OCA,AO,OC)
Shape(CO,OP,PC)
Collinear(AOPB)
Collinear(CPD)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(LengthOfLine(CD),8)
Equal(LengthOfLine(OP),3)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(OP,CP)
Goal: Value(RadiusOfCircle(O))
Answer: 5
Theorem sequence:
radius_of_circle_property_length_equal(1,OC,O)
adjacent_complementary_angle(1,DPO,OPC)
circle_property_chord_perpendicular_bisect_chord(1,O,OP,DC)
line_addition(1,CP,PD)
right_triangle_judgment_angle(1,OPC)
right_triangle_property_pythagorean(1,OPC)
Problem: As shown in the diagram, AB=8, OC=3, the center of circle O is O, AC is perpendicular to OC. Find the radius of circle O.
问题：如图所示，AB=8，OC=3，O是圆O的圆心，AC垂直于OC。求圆O的半径。
Construction (CDL):
Shape(OA,AC,CO)
Shape(OAB,BC,CA)
Shape(OBA,AO,OC,CB)
Collinear(ACB)
Cocircular(O,AB)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(LengthOfLine(OC),3)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AC,OC)
Goal: Value(RadiusOfCircle(O))
Answer: 5
Theorem sequence:
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)
Problem: As shown in the diagram, AB=8, OC=5, the center of ⊙O is O, AD⊥OD. Find the length of line DC.
问题：如图所示，AB=8，OC=5，O是⊙O的圆心，AD垂直于OD。求直线DC的长度。
Construction (CDL):
Shape(OAC,CD,DA)
Shape(OCB,BD,DC)
Shape(OBA,AO,OD,DB)
Shape(OA,AD,DO)
Collinear(ADB)
Collinear(ODC)
Cocircular(O,ACB)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(LengthOfLine(OC),5)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AD,OD)
Goal: Value(LengthOfLine(DC))
Answer: 2
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OC,O)
circle_property_chord_perpendicular_bisect_chord(1,O,OD,AB)
line_addition(1,AD,DB)
line_addition(1,OD,DC)
right_triangle_judgment_angle(1,ADO)
right_triangle_property_pythagorean(1,ADO)
Problem: As shown in the diagram, AB=4, CD=3, the center of ⊙O is O, DE is perpendicular to CE. Find the radius of ⊙O.
问题：如图所示，AB=4，CD=3，O是圆O的圆心，DE⊥CE。求圆O的半径。
Construction (CDL):
Shape(OAM,MA)
Shape(AM,MF,FA)
Shape(AF,FH,HA)
Shape(FM,MG,GF)
Shape(HF,FG,GO,OI,IC,CE,EH)
Shape(EI,IC,CE)
Shape(OMB,BG,GM)
Shape(GB,BI,IO,OG)
Shape(OBC,CI,IB)
Shape(OCD,DC)
Shape(CD,DE,EC)
Shape(DH,HE,ED)
Shape(ODA,AH,HD)
Collinear(AHEC)
Collinear(MGOIC)
Collinear(MFHD)
Collinear(AFGB)
Collinear(DEIB)
Cocircular(O,AMBCD)
Conditions (CDL):
Equal(LengthOfLine(AB),4)
Equal(LengthOfLine(CD),3)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(DE,CE)
Goal: Value(RadiusOfCircle(O))
Answer: 5/2
Theorem sequence:
diameter_of_circle_judgment_pass_centre(1,MOC,O)
diameter_of_circle_property_right_angle(1,CAM,O)
diameter_of_circle_property_right_angle(1,CDM,O)
adjacent_complementary_angle(1,DEC,CEI)
parallel_judgment_corresponding_angle(1,EI,AM,C)
parallel_property_collinear_extend(1,EI,AM,D)
parallel_property_alternate_interior_angle(1,DE,AM)
arc_property_circumference_angle_external(1,OMB,D)
arc_property_circumference_angle_external(1,ODA,M)
congruent_arc_judgment_measure_equal(1,OMB,ODA)
congruent_arc_property_length_equal(1,OMB,ODA)
arc_addition_length(1,ODA,OAM)
arc_addition_length(1,OAM,OMB)
congruent_arc_judgment_length_equal(1,ODM,OAB)
congruent_arc_property_chord_equal(1,ODM,OAB)
right_triangle_judgment_angle(1,CDM)
right_triangle_property_pythagorean(1,CDM)
diameter_of_circle_property_length_equal(1,MC,O)
circle_property_length_of_radius_and_diameter(1,O)
Problem: As shown in the diagram, AB=16, OC=6, O is the center of ⊙O, OC⊥BC. Find the diameter of circle O.
问题：如图所示，AB=16，OC=6，O是圆O的圆心，OC垂直于BC。求⊙O的直径。
Construction (CDL):
Shape(OAB,BC,CA)
Shape(OA,AC,CO)
Shape(OBA,AO,OC,CB)
Collinear(ACB)
Cocircular(O,AB)
Conditions (CDL):
Equal(LengthOfLine(AB),16)
Equal(LengthOfLine(OC),6)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(OC,BC)
Goal: Value(DiameterOfCircle(O))
Answer: 20
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, AB=10, OE=3, the center of circle O is O, OE⊥CE. Find the length of line CD.
问题：如图所示，AB=10，OE=3，圆O的圆心为O，OE垂直于CE。求直线CD的长度。
Construction (CDL):
Shape(OAD,DE,EO,OA)
Shape(ODB,BE,ED)
Shape(OBC,CE,EB)
Shape(OE,EC,CO)
Shape(OCA,AO,OC)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(LengthOfLine(OE),3)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(OE,CE)
Goal: Value(LengthOfLine(CD))
Answer: 8
Theorem sequence:
diameter_of_circle_judgment_pass_centre(1,AOB,O)
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,OC,O)
right_triangle_judgment_angle(1,OEC)
right_triangle_property_pythagorean(1,OEC)
adjacent_complementary_angle(1,DEO,OEC)
circle_property_chord_perpendicular_bisect_chord(1,O,OE,DC)
line_addition(1,CE,ED)
Problem: As shown in the diagram, AB=16, the radius of ⊙O is 10, the center of ⊙O is O, OD is perpendicular to AD. Find the length of line CD.
问题：如图所示，AB=16，圆O的半径为10，⊙O的圆心为O，OD垂直于AD。求直线CD的长度。
Construction (CDL):
Shape(OAB,BD,DO,OA)
Shape(OBC,CD,DB)
Shape(OCA,AD,DC)
Shape(AO,OD,DA)
Collinear(ADB)
Collinear(CDO)
Cocircular(O,ABC)
Conditions (CDL):
Equal(LengthOfLine(AB),16)
Equal(RadiusOfCircle(O),10)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(OD,AD)
Goal: Value(LengthOfLine(CD))
Answer: 4
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OC,O)
adjacent_complementary_angle(1,BDO,ODA)
circle_property_chord_perpendicular_bisect_chord(1,O,OD,BA)
line_addition(1,AD,DB)
right_triangle_judgment_angle(1,ODA)
right_triangle_property_pythagorean(1,ODA)
line_addition(1,CD,DO)
Problem: As shown in the diagram, ∠EOC=70°, the center of circle O is O, OE⊥DE. Find the measure of ∠EBD.
问题：如图所示，∠EOC=70°，⊙O的圆心为O，OE⊥DE。求∠EBD的大小。
Construction (CDL):
Shape(OAC,CO,OA)
Shape(OCB,BE,EC)
Shape(OC,CE,EO)
Shape(OBD,DB)
Shape(EB,BD,DE)
Shape(OE,ED,DO)
Shape(ODA,AO,OD)
Collinear(CED)
Collinear(AOEB)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(EOC),70)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(OE,DE)
Goal: Value(MeasureOfAngle(EBD))
Answer: 55
Theorem sequence:
adjacent_complementary_angle(1,CEO,OED)
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_center_angle(1,OCB,O)
arc_property_center_angle(1,OBD,O)
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OD,O)
isosceles_triangle_judgment_line_equal(1,OBD)
isosceles_triangle_property_angle_equal(1,OBD)
triangle_property_angle_sum(1,BDO)
Problem: As shown in the diagram, OC=3, the radius of ⊙O is 5, the center of circle O is O, AC is perpendicular to OC. Find the length of line AB.
问题：如图所示，OC=3，圆O的半径为5，圆O的圆心为O，AC垂直于OC。求直线AB的长度。
Construction (CDL):
Shape(OAB,BC,CA)
Shape(OBA,AO,OC,CB)
Shape(OA,AC,CO)
Collinear(ACB)
Cocircular(O,AB)
Conditions (CDL):
Equal(LengthOfLine(OC),3)
Equal(RadiusOfCircle(O),5)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AC,OC)
Goal: Value(LengthOfLine(AB))
Answer: 8
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
right_triangle_judgment_angle(1,ACO)
right_triangle_property_pythagorean(1,ACO)
circle_property_chord_perpendicular_bisect_chord(1,O,OC,AB)
line_addition(1,AC,CB)
Problem: As shown in the diagram, AB=10, CD=8, the center of circle O is O, CE is perpendicular to OE. Find the length of line AE.
问题：如图所示，AB=10，CD=8，O是⊙O的圆心，CE垂直于OE。求直线AE的长度。
Construction (CDL):
Shape(OAD,DE,EA)
Shape(ODB,BO,OE,ED)
Shape(OBC,CO,OB)
Shape(CE,EO,OC)
Shape(OCA,AE,EC)
Collinear(AEOB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(LengthOfLine(CD),8)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(CE,OE)
Goal: Value(LengthOfLine(AE))
Answer: 2
Theorem sequence:
diameter_of_circle_judgment_pass_centre(1,AOB,O)
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,OA,O)
radius_of_circle_property_length_equal(1,OC,O)
circle_property_chord_perpendicular_bisect_chord(1,O,OE,CD)
line_addition(1,CE,ED)
right_triangle_judgment_angle(1,CEO)
right_triangle_property_pythagorean(1,CEO)
line_addition(1,AE,EO)
Problem: As shown in the diagram, AB=10, AP=4, OP=5, O is the center of ⊙O, BC⊥OC. Find the radius of ⊙O.
问题：如图所示，AB=10，AP=4，OP=5，⊙O的圆心为O，BC⊥OC。求⊙O的半径。
Construction (CDL):
Shape(OAB,BC,CO,OA)
Shape(OBA,AP,PC,CB)
Shape(OC,CP,PO)
Shape(OP,PA,AO)
Collinear(BCPA)
Cocircular(O,AB)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(LengthOfLine(AP),4)
Equal(LengthOfLine(OP),5)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(BC,OC)
Goal: Value(RadiusOfCircle(O))
Answer: 7
Theorem sequence:
circle_property_chord_perpendicular_bisect_chord(1,O,OC,BA)
line_addition(1,BC,CA)
line_addition(1,CP,PA)
right_triangle_judgment_angle(1,OCP)
right_triangle_judgment_angle(1,OCA)
right_triangle_property_pythagorean(1,OCP)
right_triangle_property_pythagorean(1,OCA)
radius_of_circle_property_length_equal(1,OA,O)
Problem: As shown in the diagram, AB=4, BC=5, DE=3, O is the center of ⊙O, NM is perpendicular to BM, ON⊥EN, quadrilateral ADQP is a rectangle. Find the length of line EF.
问题：如图所示，AB=4，BC=5，DE=3，O是⊙O的圆心，NM⊥BM，ON垂直于EN，ADQP是长方形。求直线EF的长度。
Construction (CDL):
Shape(OCB,BM,MC)
Shape(OBE,EN,NM,MB)
Shape(OEF,FN,NO,ON,NE)
Shape(OFC,CM,MN,NF)
Shape(OBE,BA,AD,DE)
Shape(OFC,FQ,QP,PC)
Collinear(ABMCP)
Collinear(DENFQ)
Collinear(MNO)
Cocircular(O,BEFC)
Conditions (CDL):
Equal(LengthOfLine(AB),4)
Equal(LengthOfLine(BC),5)
Equal(LengthOfLine(DE),3)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(NM,BM)
PerpendicularBetweenLine(ON,EN)
Rectangle(ADQP)
Goal: Value(LengthOfLine(EF))
Answer: 7
Theorem sequence:
adjacent_complementary_angle(1,CMN,NMA)
adjacent_complementary_angle(1,FNO,OND)
circle_property_chord_perpendicular_bisect_chord(1,O,OM,CB)
circle_property_chord_perpendicular_bisect_chord(1,O,ON,FE)
line_addition(1,BM,MC)
parallel_judgment_ipsilateral_internal_angle(1,AM,DN)
parallel_judgment_ipsilateral_internal_angle(1,MN,AD)
parallelogram_judgment_parallel_and_parallel(1,NMAD)
parallelogram_property_opposite_line_equal(1,MADN)
line_addition(1,AB,BM)
line_addition(1,DE,EN)
line_addition(1,EN,NF)
Problem: As shown in the diagram, EB=2, ∠EOD=60°, O is the center of ⊙O, OE⊥CE. Find the length of line CD.
问题：如图所示，EB=2，∠EOD=60°，圆O的圆心为O，OE⊥CE。求直线CD的长度。
Construction (CDL):
Shape(OAD,DO,OA)
Shape(OD,DE,EO)
Shape(ODB,BE,ED)
Shape(OBC,CB)
Shape(CE,EB,BC)
Shape(AO,OE,EC,CA)
Shape(OCA,AC)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(LengthOfLine(EB),2)
Equal(MeasureOfAngle(EOD),60)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(OE,CE)
Goal: Value(LengthOfLine(CD))
Answer: 4*sqrt(3)
Theorem sequence:
arc_property_center_angle(1,ODB,O)
arc_property_circumference_angle_external(1,ODB,C)
adjacent_complementary_angle(1,AEC,CEB)
adjacent_complementary_angle(1,DEO,OEC)
triangle_property_angle_sum(1,CEB)
sine_theorem(1,EBC)
circle_property_chord_perpendicular_bisect_chord(1,O,OE,DC)
line_addition(1,CE,ED)
Problem: As shown in the diagram, AB=10, CD=6, the center of ⊙O is 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,CO,OA)
Shape(OCB,BE,EC)
Shape(OC,CE,EO)
Shape(OBD,DE,EB)
Shape(ODA,AO,OE,ED)
Collinear(CED)
Collinear(AOEB)
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:
diameter_of_circle_judgment_pass_centre(1,AOB,O)
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,OC,O)
radius_of_circle_property_length_equal(1,OB,O)
circle_property_chord_perpendicular_bisect_chord(1,O,OE,CD)
line_addition(1,CE,ED)
line_addition(1,OE,EB)
right_triangle_judgment_angle(1,CEO)
right_triangle_property_pythagorean(1,CEO)
Problem: As shown in the diagram, OE=3, the radius of circle O is 5, the center of circle O is O, CE⊥OE. Find the length of line CD.
问题：如图所示，OE=3，⊙O的半径为5，⊙O的圆心为O，CE垂直于OE。求直线CD的长度。
Construction (CDL):
Shape(OAC,CO,OA)
Shape(OCB,BE,EC)
Shape(OC,CE,EO)
Shape(OBD,DE,EB)
Shape(ODA,AO,OE,ED)
Collinear(CED)
Collinear(AOEB)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(LengthOfLine(OE),3)
Equal(RadiusOfCircle(O),5)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(CE,OE)
Goal: Value(LengthOfLine(CD))
Answer: 8
Theorem sequence:
radius_of_circle_property_length_equal(1,OC,O)
circle_property_chord_perpendicular_bisect_chord(1,O,OE,CD)
right_triangle_judgment_angle(1,CEO)
right_triangle_property_pythagorean(1,CEO)
line_addition(1,CE,ED)
Problem: As shown in the diagram, ∠MEC=40°, the center of ⊙E is E, M bisects segment CD. Find the measure of ∠MBD.
问题：如图所示，∠MEC=40°，E是⊙E的圆心，M是线段CD的中点。求∠MBD的大小。
Construction (CDL):
Shape(EAC,CE,EA)
Shape(CM,ME,EC)
Shape(ECB,BM,MC)
Shape(EBD,DB)
Shape(MB,BD,DM)
Shape(EM,MD,DE)
Shape(EDA,AE,ED)
Collinear(CMD)
Collinear(AEMB)
Cocircular(E,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(MEC),40)
IsCentreOfCircle(E,E)
IsMidpointOfLine(M,CD)
Goal: Value(MeasureOfAngle(MBD))
Answer: 70
Theorem sequence:
radius_of_circle_property_length_equal(1,EC,E)
radius_of_circle_property_length_equal(1,EB,E)
radius_of_circle_property_length_equal(1,ED,E)
mirror_congruent_triangle_judgment_sss(1,CME,DEM)
mirror_congruent_triangle_property_angle_equal(1,ECM,EMD)
isosceles_triangle_judgment_line_equal(1,EBD)
isosceles_triangle_property_angle_equal(1,EBD)
triangle_property_angle_sum(1,EBD)
Problem: As shown in the diagram, AB=8, the center of circle O is O, AE⊥ME, PF⊥NF. Find the length of line EF.
问题：如图所示，AB=8，圆O的圆心为O，AE垂直于ME，PF⊥NF。求直线EF的长度。
Construction (CDL):
Shape(OAP,PE,EA)
Shape(OPB,BF,FP)
Shape(MA,AE,EM)
Shape(NM,ME,EF,FN)
Shape(BN,NF,FB)
Shape(FE,EP,PF)
Shape(OM,MN,NO)
Shape(OBA,AM,MO,ON,NB)
Collinear(AMNB)
Collinear(AEP)
Collinear(PFB)
Collinear(OME)
Collinear(ONF)
Cocircular(O,APB)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AE,ME)
PerpendicularBetweenLine(PF,NF)
Goal: Value(LengthOfLine(EF))
Answer: 4
Theorem sequence:
circle_property_chord_perpendicular_bisect_chord(1,O,OE,AP)
circle_property_chord_perpendicular_bisect_chord(1,O,OF,PB)
midsegment_of_triangle_judgment_midpoint(1,FE,PBA)
midsegment_of_triangle_property_length(1,FE,PBA)
Problem: As shown in the diagram, AB=16, OB=10, O is the center of circle O, AH is perpendicular to OH. Find the length of line CH.
问题：如图所示，AB=16，OB=10，⊙O的圆心为O，AH垂直于OH。求直线CH的长度。
Construction (CDL):
Shape(OAC,CH,HA)
Shape(OCB,BH,HC)
Shape(OH,HB,BO)
Shape(OBA,AH,HO,OB)
Collinear(AHB)
Collinear(OHC)
Cocircular(O,ACB)
Conditions (CDL):
Equal(LengthOfLine(AB),16)
Equal(LengthOfLine(OB),10)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AH,OH)
Goal: Value(LengthOfLine(CH))
Answer: 4
Theorem sequence:
circle_property_chord_perpendicular_bisect_chord(1,O,OH,AB)
line_addition(1,AH,HB)
right_triangle_judgment_angle(1,OHB)
right_triangle_property_pythagorean(1,OHB)
radius_of_circle_property_length_equal(1,OC,O)
radius_of_circle_property_length_equal(1,OB,O)
line_addition(1,OH,HC)
Problem: As shown in the diagram, OH=1, ∠HOA=60°, O is the center of ⊙O, AH⊥OH. Find the radius of circle O.
问题：如图所示，OH=1，∠HOA=60°，O是圆O的圆心，AH⊥OH。求圆O的半径。
Construction (CDL):
Shape(OAC,CH,HA)
Shape(OCB,BH,HC)
Shape(OBA,AO,OH,HB)
Shape(OA,AH,HO)
Collinear(AHB)
Collinear(OHC)
Cocircular(O,ACB)
Conditions (CDL):
Equal(LengthOfLine(OH),1)
Equal(MeasureOfAngle(HOA),60)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AH,OH)
Goal: Value(RadiusOfCircle(O))
Answer: 2
Theorem sequence:
triangle_property_angle_sum(1,OAH)
sine_theorem(1,OAH)
radius_of_circle_property_length_equal(1,OA,O)
Problem: As shown in the diagram, AB=8, BC=6, the center of ⊙O is O, ABCD is a rectangle. Find the diameter of ⊙O.
问题：如图所示，AB=8，BC=6，圆O的圆心为O，四边形ABCD是长方形。求圆O的直径。
Construction (CDL):
Shape(OAB,BA)
Shape(OBC,CB)
Shape(OCD,DC)
Shape(ODA,AD)
Shape(AB,BC,CO,OA)
Shape(CD,DA,AO,OC)
Collinear(AOC)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(LengthOfLine(BC),6)
IsCentreOfCircle(O,O)
Rectangle(ABCD)
Goal: Value(DiameterOfCircle(O))
Answer: 10
Theorem sequence:
right_triangle_judgment_angle(1,ABC)
right_triangle_property_pythagorean(1,ABC)
diameter_of_circle_judgment_pass_centre(1,AOC,O)
diameter_of_circle_property_length_equal(1,AC,O)