SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, AB=5, AC=4, BC=3, BC⊥AC, CD is perpendicular to BD. Find the length of line CD.
问题：如图所示，AB=5，AC=4，BC=3，BC垂直于AC，CD⊥BD。求直线CD的长度。
Construction (CDL):
Shape(CA,AD,DC)
Shape(CD,DB,BC)
Collinear(ADB)
Conditions (CDL):
Equal(LengthOfLine(AB),5)
Equal(LengthOfLine(AC),4)
Equal(LengthOfLine(BC),3)
PerpendicularBetweenLine(BC,AC)
PerpendicularBetweenLine(CD,BD)
Goal: Value(LengthOfLine(CD))
Answer: 12/5
Theorem sequence:
line_addition(1,AD,DB)
adjacent_complementary_angle(1,ADC,CDB)
right_triangle_judgment_angle(1,ADC)
right_triangle_judgment_angle(1,CDB)
right_triangle_property_pythagorean(1,ADC)
right_triangle_property_pythagorean(1,CDB)
Problem: As shown in the diagram, ∠FBG=42°, AG is parallel to DE, AC is perpendicular to BC. Find the measure of ∠DEC.
问题：如图所示，∠FBG=42°，AG平行于DE，AC垂直于BC。求∠DEC的大小。
Construction (CDL):
Shape(AC,CB,BA)
Shape(CD,DE,EC)
Shape(FB,BG)
Collinear(ABG)
Collinear(FBCD)
Collinear(ACE)
Conditions (CDL):
Equal(MeasureOfAngle(FBG),42)
ParallelBetweenLine(AG,DE)
PerpendicularBetweenLine(AC,BC)
Goal: Value(MeasureOfAngle(DEC))
Answer: 48
Theorem sequence:
vertical_angle(1,FBG,CBA)
triangle_property_angle_sum(1,ACB)
parallel_property_alternate_interior_angle(1,AG,DE)
Problem: As shown in the diagram, AB=2, AC=2, ∠ABC=45°, ∠BCA=45°, ∠CAB=90°, O is the center of ⊙O, the diameter of circle O is AB. Find the area of △DCA.
问题：如图所示，AB=2，AC=2，∠ABC=45°，∠BCA=45°，∠CAB=90°，⊙O的圆心为O，⊙O的直径为AB。求三角形DCA的面积。
Construction (CDL):
Shape(DC,CA,ODA)
Shape(ODA,AD)
Shape(BD,DA,AO,OB)
Shape(OBD,DB)
Shape(BO,OA,OAB)
Collinear(BOA)
Collinear(BDC)
Cocircular(O,BDA)
Conditions (CDL):
Equal(LengthOfLine(AB),2)
Equal(LengthOfLine(AC),2)
Equal(MeasureOfAngle(ABC),45)
Equal(MeasureOfAngle(BCA),45)
Equal(MeasureOfAngle(CAB),90)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(AreaOfTriangle(DCA))
Answer: 1
Theorem sequence:
diameter_of_circle_property_right_angle(1,BDA,O)
flat_angle(1,BDC)
angle_addition(1,BDA,ADC)
triangle_property_angle_sum(1,BDA)
triangle_property_angle_sum(1,DCA)
isosceles_triangle_judgment_angle_equal(1,DAB)
right_triangle_judgment_angle(1,BDA)
right_triangle_property_pythagorean(1,BDA)
triangle_area_formula_sine(1,ADC)
Problem: As shown in the diagram, CD=6, ∠ADB=30°, O is the center of ⊙O, AB is parallel to CD. Find the area of the sector OBA.
问题：如图所示，CD=6，∠ADB=30°，⊙O的圆心为O，AB平行于CD。求扇形OBA的面积。
Construction (CDL):
Shape(OA,OAC,CO)
Shape(OBA,AB)
Shape(ODB,BD)
Shape(AE,EB,BA)
Shape(AO,OE,EA)
Shape(BE,ED,DB)
Shape(EO,OD,DE)
Collinear(AED)
Collinear(BEO)
Collinear(COD)
Cocircular(O,ACDB)
Conditions (CDL):
Equal(LengthOfLine(CD),6)
Equal(MeasureOfAngle(ADB),30)
IsCentreOfCircle(O,O)
ParallelBetweenLine(AB,CD)
Goal: Value(AreaOfSector(OBA))
Answer: 3*pi/2
Theorem sequence:
diameter_of_circle_judgment_pass_centre(1,COD,O)
diameter_of_circle_property_length_equal(1,CD,O)
circle_property_length_of_radius_and_diameter(1,O)
arc_property_circumference_angle_external(1,OBA,D)
sector_area_formula(1,OBA)
Problem: As shown in the diagram, Add(AreaOfSector(BIH)=AreaOfSector(AED), the radius of circle A is 2, the radius of circle B is 2, the radius of circle C is 2, A is the center of circle A, the center of ⊙B is B, C is the center of circle C. Find the value of x.
问题：如图所示，Add(AreaOfSector(BIH)=AreaOfSector(AED)，圆A的半径为2，⊙B的半径为2，圆C的半径为2，圆A的圆心为A，B是圆B的圆心，C是⊙C的圆心。求x的值。
Construction (CDL):
Shape(DA,AE,AED)
Shape(AD,ADE,EA)
Shape(BI,BIH,HB)
Shape(BH,BHI,IB)
Shape(CG,CGF,FC)
Shape(CF,CFG,GC)
Shape(ID,AED,EF,CGF,GH,BIH)
Collinear(BIDA)
Collinear(BHGC)
Collinear(AEFC)
Cocircular(A,ED)
Cocircular(B,IH)
Cocircular(C,GF)
Conditions (CDL):
Equal(Add(AreaOfSector(BIH),AreaOfSector(AED),AreaOfSector(CGF)),x)
Equal(RadiusOfCircle(A),2)
Equal(RadiusOfCircle(B),2)
Equal(RadiusOfCircle(C),2)
IsCentreOfCircle(A,A)
IsCentreOfCircle(B,B)
IsCentreOfCircle(C,C)
Goal: Value(x)
Answer: 2*pi
Theorem sequence:
triangle_property_angle_sum(1,ACB)
arc_property_center_angle(1,AED,A)
arc_property_center_angle(1,BIH,B)
arc_property_center_angle(1,CGF,C)
sector_area_formula(1,AED)
sector_area_formula(1,BIH)
sector_area_formula(1,CGF)
Problem: As shown in the diagram, ∠DBF=110°, ∠ECD=70°, CD∥AF. Find the measure of ∠DEC.
问题：如图所示，∠DBF=110°，∠ECD=70°，CD∥AF。求∠DEC的大小。
Construction (CDL):
Shape(EC,CD,DE)
Shape(DC,CA,AB,BD)
Shape(DB,BF)
Collinear(ABF)
Collinear(EDB)
Conditions (CDL):
Equal(MeasureOfAngle(DBF),110)
Equal(MeasureOfAngle(ECD),70)
ParallelBetweenLine(CD,AF)
Goal: Value(MeasureOfAngle(DEC))
Answer: 40
Theorem sequence:
flat_angle(1,ABF)
angle_addition(1,ABD,DBF)
parallel_property_collinear_extend(3,FA,DC,B)
parallel_property_corresponding_angle(2,BA,DC,E)
triangle_property_angle_sum(1,ECD)
Problem: As shown in the diagram, the length of arc OAB is equal to the length of arc OBC, the length of ⌒OAB is equal to the length of ⌒OCD, the length of arc OCD is equal to the length of arc OBC, ∠CEB=110°, O is the center of circle O. Find the measure of ∠CDB.
问题：如图所示，⌒OAB的长度与⌒OBC的长度相等，⌒OAB的长度与⌒OCD的长度相等，⌒OCD的长度与弧OBC的长度相等，∠CEB=110°，圆O的圆心为O。求∠CDB的大小。
Construction (CDL):
Shape(ED,ODA,AE)
Shape(EA,AB,BE)
Shape(OAB,BA)
Shape(EB,BC,CE)
Shape(OBC,CB)
Shape(EC,CD,DE)
Shape(OCD,DC)
Collinear(DEB)
Collinear(AEC)
Cocircular(O,DABC)
Conditions (CDL):
Equal(LengthOfArc(OAB),LengthOfArc(OBC))
Equal(LengthOfArc(OAB),LengthOfArc(OCD))
Equal(LengthOfArc(OCD),LengthOfArc(OBC))
Equal(MeasureOfAngle(CEB),110)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CDB))
Answer: 35
Theorem sequence:
congruent_arc_judgment_length_equal(1,OCD,OAB)
congruent_arc_judgment_length_equal(1,OCD,OBC)
congruent_arc_property_measure_equal(1,OCD,OAB)
congruent_arc_property_measure_equal(1,OCD,OBC)
arc_property_circumference_angle_external(1,OBC,D)
arc_property_circumference_angle_external(1,OCD,B)
arc_property_circumference_angle_external(1,OAB,C)
triangle_property_angle_sum(1,EBC)
Problem: As shown in the diagram, OA=1, ∠BOA=90°, the center of circle O is O. Find the area of △BOA.
问题：如图所示，OA=1，∠BOA=90°，O是⊙O的圆心。求三角形BOA的面积。
Construction (CDL):
Shape(BO,OA,AB)
Shape(OAB,BA)
Shape(OB,OBA,AO)
Cocircular(O,AB)
Conditions (CDL):
Equal(LengthOfLine(OA),1)
Equal(MeasureOfAngle(BOA),90)
IsCentreOfCircle(O,O)
Goal: Value(AreaOfTriangle(BOA))
Answer: 1/2
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
triangle_area_formula_sine(1,OAB)
Problem: As shown in the diagram, AB=2, AC=2, ∠AED=30°, O is the center of circle O, AB is the diameter of circle O. Find the length of line CD.
问题：如图所示，AB=2，AC=2，∠AED=30°，O是圆O的圆心，⊙O的直径为AB。求直线CD的长度。
Construction (CDL):
Shape(DC,CA,ODA)
Shape(ODA,AD)
Shape(DA,AF,FD)
Shape(FA,AE,EF)
Shape(OAE,EA)
Shape(DF,FO,OB,BD)
Shape(OF,FE,OEB,BO)
Shape(OBD,DB)
Collinear(AFOB)
Collinear(DFE)
Collinear(CDB)
Cocircular(O,DAEB)
Conditions (CDL):
Equal(LengthOfLine(AB),2)
Equal(LengthOfLine(AC),2)
Equal(MeasureOfAngle(AED),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(LengthOfLine(CD))
Answer: sqrt(3)
Theorem sequence:
arc_property_circumference_angle_external(1,ODA,E)
arc_property_circumference_angle_external(1,ODA,B)
isosceles_triangle_judgment_line_equal(1,ABC)
isosceles_triangle_property_angle_equal(1,ABC)
diameter_of_circle_property_right_angle(1,BDA,O)
triangle_property_angle_sum(1,DCA)
sine_theorem(1,DCA)
sine_theorem(1,ADC)
sine_theorem(1,ABD)
Problem: As shown in the diagram, the length of arc OAB is 20*pi, OB=24, the center of circle O is O. Find the area of the sector OAB.
问题：如图所示，⌒OAB的长度为20*pi，OB=24，O是⊙O的圆心。求扇形OAB的面积。
Construction (CDL):
Shape(BO,OA,OAB)
Cocircular(O,AB)
Conditions (CDL):
Equal(LengthOfArc(OAB),20*pi)
Equal(LengthOfLine(OB),24)
IsCentreOfCircle(O,O)
Goal: Value(AreaOfSector(OAB))
Answer: 240*pi
Theorem sequence:
radius_of_circle_property_length_equal(1,OB,O)
arc_length_formula(1,OAB)
sector_area_formula(1,OAB)
Problem: As shown in the diagram, the radius of circle A is 1, the radius of circle B is 1, the radius of circle C is 1, the radius of circle D is 1, the center of circle A is A, B is the center of circle B, C is the center of circle C, the center of ⊙D is D. Find the sum of the area of the sector AFE, the area of the sector BNM, the area of the sector CIL and the area of the sector DGH.
问题：如图所示，⊙A的半径为1，圆B的半径为1，圆C的半径为1，⊙D的半径为1，A是⊙A的圆心，B是圆B的圆心，⊙C的圆心为C，D是圆D的圆心。求扇形AFE的面积、扇形BNM的面积、扇形CIL的面积与扇形DGH的面积之和。
Construction (CDL):
Shape(FA,FE,AEF)
Shape(EA,AF,AFE)
Shape(HD,DG,DGH)
Shape(GD,DH,DHG)
Shape(MB,BN,BNM)
Shape(NB,BM,BMN)
Shape(LC,CI,CIL)
Shape(IC,CL,CLI)
Shape(AFE,FM,BNM,NL,CIL,IH,DGH,GE)
Collinear(AFMB)
Collinear(BNLC)
Collinear(CIHD)
Collinear(DGEA)
Cocircular(A,FE)
Cocircular(B,NM)
Cocircular(D,GH)
Cocircular(C,IL)
Conditions (CDL):
Equal(RadiusOfCircle(A),1)
Equal(RadiusOfCircle(B),1)
Equal(RadiusOfCircle(C),1)
Equal(RadiusOfCircle(D),1)
IsCentreOfCircle(A,A)
IsCentreOfCircle(B,B)
IsCentreOfCircle(C,C)
IsCentreOfCircle(D,D)
Goal: Value(Add(AreaOfSector(AFE),AreaOfSector(BNM),AreaOfSector(CIL),AreaOfSector(DGH)))
Answer: pi
Theorem sequence:
quadrilateral_property_angle_sum(1,ABCD)
arc_property_center_angle(1,AFE,A)
arc_property_center_angle(1,BNM,B)
arc_property_center_angle(1,DGH,D)
arc_property_center_angle(1,CIL,C)
sector_area_formula(1,AFE)
sector_area_formula(1,BNM)
sector_area_formula(1,DGH)
sector_area_formula(1,CIL)
Problem: As shown in the diagram, ∠CAB=90°, the radius of circle A is 4, A is the center of ⊙A. Find the length of arc ABC.
问题：如图所示，∠CAB=90°，⊙A的半径为4，圆A的圆心为A。求⌒ABC的长度。
Construction (CDL):
Shape(CA,AB,ABC)
Cocircular(A,BC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),90)
Equal(RadiusOfCircle(A),4)
IsCentreOfCircle(A,A)
Goal: Value(LengthOfArc(ABC))
Answer: 2*pi
Theorem sequence:
arc_property_center_angle(1,ABC,A)
arc_length_formula(1,ABC)
Problem: As shown in the diagram, ∠BOA=30°, O is the center of ⊙O. Find the measure of ∠BPA.
问题：如图所示，∠BOA=30°，O是⊙O的圆心。求∠BPA的大小。
Construction (CDL):
Shape(PA,AC,CP)
Shape(AP,OPA)
Shape(BC,CA,OAB)
Shape(OC,CB,BO)
Shape(PC,CO,OB,OBP)
Collinear(PCB)
Collinear(OCA)
Cocircular(O,PAB)
Conditions (CDL):
Equal(MeasureOfAngle(BOA),30)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BPA))
Answer: 15
Theorem sequence:
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,P)
Problem: As shown in the diagram, BE=2, ∠CAB=30°, O is the center of circle O, the diameter of ⊙O is AB, AE⊥CE. Find the length of line AC.
问题：如图所示，BE=2，∠CAB=30°，O是圆O的圆心，AB是圆O的直径，AE垂直于CE。求直线AC的长度。
Construction (CDL):
Shape(AC,OCA)
Shape(AO,OE,EC,CA)
Shape(CE,EB,BC)
Shape(CB,OBC)
Shape(DO,OA,OAD)
Shape(OD,DE,EO)
Shape(BE,ED,ODB)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(LengthOfLine(BE),2)
Equal(MeasureOfAngle(CAB),30)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
PerpendicularBetweenLine(AE,CE)
Goal: Value(LengthOfLine(AC))
Answer: 4*sqrt(3)
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
adjacent_complementary_angle(1,AEC,CEB)
triangle_property_angle_sum(1,ABC)
triangle_property_angle_sum(1,CEB)
sine_theorem(1,BCE)
sine_theorem(1,CAB)
Problem: As shown in the diagram, ∠CAB=26°, O is the center of ⊙O, the diameter of ⊙O is AB. Find the measure of ∠ADC.
问题：如图所示，∠CAB=26°，O是圆O的圆心，AB是圆O的直径。求∠ADC的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AO,OE,EC)
Shape(CE,EB,BC)
Shape(CB,OBC)
Shape(DA,OAD)
Shape(AD,DE,EO,OA)
Shape(BE,ED,OAB)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,CADB)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),26)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ADC))
Answer: 64
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,CAB)
arc_property_circumference_angle_external(1,OCA,D)
arc_property_circumference_angle_external(1,OCA,B)
Problem: As shown in the diagram, AB=3, AC=3, AD=3, BC=2, A is the center of circle A, DC is parallel to AB. Find the length of line BD.
问题：如图所示，AB=3，AC=3，AD=3，BC=2，A是圆A的圆心，DC∥AB。求直线BD的长度。
Construction (CDL):
Shape(FD,ADF)
Shape(DF,FA,AD)
Shape(DA,AE,ED)
Shape(EA,AB,BE)
Shape(DE,EC,CD)
Shape(CE,EB,BC)
Shape(CB,ABC)
Shape(DC,ACD)
Shape(BA,AF,AFB)
Collinear(FAB)
Collinear(AEC)
Collinear(DEB)
Cocircular(A,DFBC)
Conditions (CDL):
Equal(LengthOfLine(AB),3)
Equal(LengthOfLine(AC),3)
Equal(LengthOfLine(AD),3)
Equal(LengthOfLine(BC),2)
IsCentreOfCircle(A,A)
ParallelBetweenLine(DC,AB)
Goal: Value(LengthOfLine(BD))
Answer: 4*sqrt(2)
Theorem sequence:
radius_of_circle_property_length_equal(1,AB,A)
radius_of_circle_property_length_equal(1,AF,A)
diameter_of_circle_judgment_pass_centre(1,FAB,A)
diameter_of_circle_property_right_angle(1,BDF,A)
parallel_property_alternate_interior_angle(1,DC,AB)
arc_property_circumference_angle_external(1,ADF,B)
arc_property_circumference_angle_external(1,ABC,D)
congruent_arc_judgment_measure_equal(1,ADF,ABC)
congruent_arc_property_chord_equal(1,ADF,ABC)
line_addition(1,FA,AB)
right_triangle_judgment_angle(1,BDF)
right_triangle_property_pythagorean(1,BDF)
Problem: As shown in the diagram, ∠COB=160°, O is the center of ⊙O, AD is perpendicular to OD, OE is perpendicular to AE. Find the measure of ∠DOE.
问题：如图所示，∠COB=160°，O是圆O的圆心，AD⊥OD，OE⊥AE。求∠DOE的大小。
Construction (CDL):
Shape(BO,OD,DB)
Shape(AD,DO,OA)
Shape(AO,OE,EA)
Shape(EO,OC,CE)
Shape(BD,DA,OAB)
Shape(AE,EC,OCA)
Shape(CO,OB,OBC)
Collinear(BDA)
Collinear(AEC)
Cocircular(O,BCA)
Conditions (CDL):
Equal(MeasureOfAngle(COB),160)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AD,OD)
PerpendicularBetweenLine(OE,AE)
Goal: Value(MeasureOfAngle(DOE))
Answer: 100
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
quadrilateral_property_angle_sum(1,ADOE)
Problem: As shown in the diagram, ∠COB=80°, O is the center of ⊙O, AB is the diameter of ⊙O, BE⊥DE. Find the measure of ∠BAD.
问题：如图所示，∠COB=80°，⊙O的圆心为O，AB是⊙O的直径，BE垂直于DE。求∠BAD的大小。
Construction (CDL):
Shape(AO,OC,OCA)
Shape(CO,OE,EC)
Shape(CE,EB,OBC)
Shape(BE,ED,ODB)
Shape(EO,OD,DE)
Shape(OA,AD,DO)
Shape(DA,OAD)
Collinear(AOEB)
Collinear(CED)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(COB),80)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
PerpendicularBetweenLine(BE,DE)
Goal: Value(MeasureOfAngle(BAD))
Answer: 40
Theorem sequence:
adjacent_complementary_angle(1,BED,DEO)
adjacent_complementary_angle(1,DEO,OEC)
radius_of_circle_property_length_equal(1,OD,O)
radius_of_circle_property_length_equal(1,OC,O)
mirror_congruent_triangle_judgment_hl(1,DEO,COE)
mirror_congruent_triangle_property_angle_equal(1,ODE,OEC)
arc_property_center_angle(1,ODB,O)
arc_property_circumference_angle_external(1,ODB,A)
Problem: As shown in the diagram, ∠APD=76°, ∠PAC=40°, O is the center of circle O. Find the measure of ∠PBD.
问题：如图所示，∠APD=76°，∠PAC=40°，⊙O的圆心为O。求∠PBD的大小。
Construction (CDL):
Shape(DB,OBD)
Shape(DP,PB,BD)
Shape(PA,AC,CP)
Shape(AP,PD,ODA)
Shape(BP,PC,OCB)
Shape(CA,OAC)
Collinear(DPC)
Collinear(APB)
Cocircular(O,DACB)
Conditions (CDL):
Equal(MeasureOfAngle(APD),76)
Equal(MeasureOfAngle(PAC),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(PBD))
Answer: 36
Theorem sequence:
adjacent_complementary_angle(1,CPA,APD)
triangle_property_angle_sum(1,PAC)
arc_property_circumference_angle_external(1,ODA,C)
arc_property_circumference_angle_external(1,ODA,B)
Problem: As shown in the diagram, ∠CAB=50°, O is the center of ⊙O. Find the measure of ∠COB.
问题：如图所示，∠CAB=50°，⊙O的圆心为O。求∠COB的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AB,BO,OC,CA)
Shape(OB,BC,CO)
Shape(CB,OBC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),50)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COB))
Answer: 100
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,OBC,O)
Problem: As shown in the diagram, ∠CAB=90°, the radius of circle A is 3, A is the center of ⊙A. Find the length of arc ABC.
问题：如图所示，∠CAB=90°，圆A的半径为3，A是⊙A的圆心。求⌒ABC的长度。
Construction (CDL):
Shape(CA,AB,ABC)
Cocircular(A,BC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),90)
Equal(RadiusOfCircle(A),3)
IsCentreOfCircle(A,A)
Goal: Value(LengthOfArc(ABC))
Answer: 3*pi/2
Theorem sequence:
arc_property_center_angle(1,ABC,A)
arc_length_formula(1,ABC)
Problem: As shown in the diagram, ⌒ODC is congruent to ⌒OCE, ∠ACB=90°, ∠CBA=34°, ∠FEO=90°, the center of circle O is O. Find the measure of ∠CFE.
问题：如图所示，弧ODC全等于弧OCE，∠ACB=90°，∠CBA=34°，∠FEO=90°，⊙O的圆心为O。求∠CFE的大小。
Construction (CDL):
Shape(DB,OBD)
Shape(CO,OD,ODC)
Shape(BD,DO,OB)
Shape(EO,OC,OCE)
Shape(BO,OE,OEB)
Shape(DA,AC,ODC)
Shape(CF,FE,OCE)
Collinear(BOC)
Collinear(BDA)
Collinear(ACF)
Cocircular(O,DCEB)
Conditions (CDL):
CongruentBetweenArc(ODC,OCE)
Equal(MeasureOfAngle(ACB),90)
Equal(MeasureOfAngle(CBA),34)
Equal(MeasureOfAngle(FEO),90)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(CFE))
Answer: 112
Theorem sequence:
arc_property_circumference_angle_external(1,ODC,B)
congruent_arc_property_measure_equal(1,ODC,OCE)
arc_property_center_angle(1,OCE,O)
adjacent_complementary_angle(1,ACO,OCF)
quadrilateral_property_angle_sum(1,OCFE)
Problem: As shown in the diagram, ∠ACB=60°, O is the center of circle O. Find the measure of ∠AOB.
问题：如图所示，∠ACB=60°，O是⊙O的圆心。求∠AOB的大小。
Construction (CDL):
Shape(OB,BD,DO)
Shape(DA,AC,CD)
Shape(BO,OD,DC,OCB)
Shape(CA,OAC)
Shape(AD,DB,OBA)
Collinear(BDC)
Collinear(ODA)
Cocircular(O,BAC)
Conditions (CDL):
Equal(MeasureOfAngle(ACB),60)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(AOB))
Answer: 120
Theorem sequence:
arc_property_circumference_angle_external(1,OBA,C)
arc_property_center_angle(1,OBA,O)
Problem: As shown in the diagram, ∠FBM=20°, the center of circle A is A, DE perpendicular bisects AM. Find the measure of ∠DMF.
问题：如图所示，∠FBM=20°，A是⊙A的圆心，DE是AM的垂直平分线。求∠DMF的大小。
Construction (CDL):
Shape(AE,ED,DA)
Shape(EG,GI,ID,DE)
Shape(GM,MI,IG)
Shape(IM,MF,FI)
Shape(IF,FD,DI)
Shape(BG,GE,EB)
Shape(BM,MG,GB)
Shape(MB,OBM)
Shape(FM,OMF)
Shape(CD,DF,OFC)
Shape(BE,EA,AD,DC,OCB)
Collinear(BEDC)
Collinear(BGF)
Collinear(AEGM)
Collinear(ADF)
Collinear(MID)
Cocircular(A,BMFC)
Conditions (CDL):
Equal(MeasureOfAngle(FBM),20)
IsCentreOfCircle(A,A)
IsPerpendicularBisectorOfLine(DE,AM)
Goal: Value(MeasureOfAngle(DMF))
Answer: 30
Theorem sequence:
arc_property_circumference_angle_external(1,AMF,B)
arc_property_center_angle(1,AMF,A)
radius_of_circle_property_length_equal(1,AM,A)
radius_of_circle_property_length_equal(1,AF,A)
isosceles_triangle_judgment_line_equal(1,AMF)
isosceles_triangle_property_angle_equal(1,AMF)
triangle_property_angle_sum(1,AMF)
perpendicular_bisector_property_distance_equal(1,DE,AM)
isosceles_triangle_judgment_line_equal(1,DAM)
isosceles_triangle_property_angle_equal(1,DAM)
angle_addition(1,AMD,DMF)
Problem: As shown in the diagram, ∠EAF=40°, ∠FCD=80°, AB∥CD. Find the measure of ∠FEA.
问题：如图所示，∠EAF=40°，∠FCD=80°，AB∥CD。求∠FEA的大小。
Construction (CDL):
Shape(EA,AF,FE)
Shape(EF,FB)
Shape(BF,FC)
Shape(CF,FA)
Shape(FC,CD)
Collinear(AFB)
Collinear(EFC)
Conditions (CDL):
Equal(MeasureOfAngle(EAF),40)
Equal(MeasureOfAngle(FCD),80)
ParallelBetweenLine(AB,CD)
Goal: Value(MeasureOfAngle(FEA))
Answer: 40
Theorem sequence:
parallel_property_collinear_extend(3,AB,CD,F)
parallel_property_corresponding_angle(1,FB,CD,E)
adjacent_complementary_angle(1,AFE,EFB)
triangle_property_angle_sum(1,AFE)
Problem: As shown in the diagram, ∠BDC=20°, the center of ⊙O is O, AC is the diameter of ⊙O. Find the measure of ∠ACB.
问题：如图所示，∠BDC=20°，O是圆O的圆心，⊙O的直径为AC。求∠ACB的大小。
Construction (CDL):
Shape(DC,CE,ED)
Shape(EC,CB,BE)
Shape(EB,BA,AO,OE)
Shape(CD,ODC)
Shape(BC,OCB)
Shape(AB,OBA)
Shape(DE,EO,OA,OAD)
Collinear(DEB)
Collinear(CEOA)
Cocircular(O,ADCB)
Conditions (CDL):
Equal(MeasureOfAngle(BDC),20)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AC,O)
Goal: Value(MeasureOfAngle(ACB))
Answer: 70
Theorem sequence:
arc_property_circumference_angle_external(1,OCB,D)
arc_property_circumference_angle_external(1,OCB,A)
diameter_of_circle_property_right_angle(1,CBA,O)
triangle_property_angle_sum(1,ACB)
Problem: As shown in the diagram, ∠BCA=40°, the center of ⊙O is O. Find the measure of ∠BOA.
问题：如图所示，∠BCA=40°，圆O的圆心为O。求∠BOA的大小。
Construction (CDL):
Shape(CA,AO,OB,BC)
Shape(BO,OA,OAB)
Shape(CB,OBC)
Shape(AC,OCA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),40)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOA))
Answer: 80
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,C)
arc_property_center_angle(1,OAB,O)
Problem: As shown in the diagram, ∠EAB=20°, O is the center of circle O, OC∥AB. Find the measure of ∠CEB.
问题：如图所示，∠EAB=20°，O是⊙O的圆心，OC平行于AB。求∠CEB的大小。
Construction (CDL):
Shape(OE,EC,CO)
Shape(EA,AB,BE)
Shape(CE,EB,OBC)
Shape(BA,OAB)
Shape(AE,EO,OC,OCA)
Collinear(AEC)
Collinear(OEB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(EAB),20)
IsCentreOfCircle(O,O)
ParallelBetweenLine(OC,AB)
Goal: Value(MeasureOfAngle(CEB))
Answer: 60
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,OBC,O)
parallel_property_alternate_interior_angle(2,OC,AB)
triangle_property_angle_sum(1,OEC)
adjacent_complementary_angle(1,OEC,CEB)
Problem: As shown in the diagram, AB=10, ∠BCA=45°, the center of ⊙O is O, the diameter of ⊙O is AD. Find the diameter of circle O.
问题：如图所示，AB=10，∠BCA=45°，⊙O的圆心为O，AD是圆O的直径。求圆O的直径。
Construction (CDL):
Shape(CA,AO,OE,EC)
Shape(EO,OA,AB,BE)
Shape(CE,ED,ODC)
Shape(DE,EB,BD)
Shape(DB,OBD)
Shape(AC,OCA)
Shape(BA,OAB)
Collinear(AOED)
Collinear(CEB)
Cocircular(O,ABDC)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(MeasureOfAngle(BCA),45)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AD,O)
Goal: Value(DiameterOfCircle(O))
Answer: 10*sqrt(2)
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,C)
arc_property_circumference_angle_external(1,OAB,D)
diameter_of_circle_property_right_angle(1,ABD,O)
triangle_property_angle_sum(1,ABD)
isosceles_triangle_judgment_angle_equal(1,BDA)
right_triangle_judgment_angle(1,ABD)
right_triangle_property_pythagorean(1,ABD)
diameter_of_circle_property_length_equal(1,AD,O)
Problem: As shown in the diagram, ∠EOA=50°, the center of circle O is O, AE⊥OE. Find the measure of ∠BDF.
问题：如图所示，∠EOA=50°，圆O的圆心为O，AE垂直于OE。求∠BDF的大小。
Construction (CDL):
Shape(OAC,CE,EA)
Shape(EC,CF,FE)
Shape(FC,OCB,BF)
Shape(OA,AE,EO)
Shape(OBD,DB)
Shape(FB,BD,DF)
Shape(ODA,AO,OE,EF,FD)
Collinear(AEFB)
Collinear(OEC)
Collinear(DFC)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(EOA),50)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AE,OE)
Goal: Value(MeasureOfAngle(BDF))
Answer: 25
Theorem sequence:
arc_property_center_angle(1,OAC,O)
circle_property_chord_perpendicular_bisect_arc(1,OAB,OEC)
congruent_arc_judgment_length_equal(1,OAC,OCB)
congruent_arc_property_measure_equal(1,OAC,OCB)
arc_property_circumference_angle_external(1,OCB,D)
Problem: As shown in the diagram, AC=6, BP=2, ∠MCO=90°, M is the center of ⊙M, O is the center of ⊙O. Find the length of line AB.
问题：如图所示，AC=6，BP=2，∠MCO=90°，圆M的圆心为M，圆O的圆心为O。求直线AB的长度。
Construction (CDL):
Shape(MDC,CM,MD)
Shape(MCP,PC)
Shape(CP,OPC)
Shape(OPC,PD,DM,MC)
Shape(MPD,DP)
Shape(MCP,CO,OP)
Shape(PO,OA,AP)
Shape(PA,AE,OEP)
Shape(OAE,EA)
Shape(MPD,PB,BD)
Shape(BP,OEP,EB)
Shape(OCA,AO,OC)
Collinear(BDMC)
Collinear(COA)
Collinear(BEA)
Collinear(DPA)
Collinear(BPO)
Cocircular(M,DCP)
Cocircular(O,AEPC)
Conditions (CDL):
Equal(LengthOfLine(AC),6)
Equal(LengthOfLine(BP),2)
Equal(MeasureOfAngle(MCO),90)
IsCentreOfCircle(M,M)
IsCentreOfCircle(O,O)
Goal: Value(LengthOfLine(AB))
Answer: 2*sqrt(13)
Theorem sequence:
line_addition(1,CO,OA)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OP,O)
radius_of_circle_property_length_equal(1,OC,O)
line_addition(1,BP,PO)
right_triangle_judgment_angle(1,BCO)
right_triangle_property_pythagorean(1,BCO)
right_triangle_judgment_angle(1,BCA)
right_triangle_property_pythagorean(1,BCA)
Problem: As shown in the diagram, ∠BAC=70°, the center of circle O is O. Find the measure of ∠BOC.
问题：如图所示，∠BAC=70°，圆O的圆心为O。求∠BOC的大小。
Construction (CDL):
Shape(OAC,CA)
Shape(OCB,BC)
Shape(OBA,AB)
Shape(OC,CB,BO)
Shape(AC,CO,OB,BA)
Cocircular(O,ACB)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),70)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOC))
Answer: 140
Theorem sequence:
arc_property_circumference_angle_external(1,OCB,A)
arc_property_center_angle(1,OCB,O)