SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, ∠ABC=68°, ∠DCE=65°. Find the measure of ∠DAB.
问题：如图所示，∠ABC=68°，∠DCE=65°。求∠DAB的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(CB,OBC)
Shape(DC,OCD)
Shape(AD,ODA)
Shape(AB,BC,CD,DA)
Shape(OCD,CE,ED)
Collinear(BCE)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),68)
Equal(MeasureOfAngle(DCE),65)
Goal: Value(MeasureOfAngle(DAB))
Answer: 65
Theorem sequence:
adjacent_complementary_angle(1,BCD,DCE)
arc_property_circumference_angle_external(1,ODB,C)
arc_property_circumference_angle_internal(1,ODB,A)
Problem: As shown in the diagram, ∠CDE=128°, the measure of arc OEA is equal to the measure of ⌒OAC. Find the measure of ∠ABC.
问题：如图所示，∠CDE=128°，弧OEA的角度与弧OAC的角度相等。求∠ABC的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(CB,OBC)
Shape(DC,OCD)
Shape(ED,ODE)
Shape(AE,OEA)
Shape(AB,BC,CA)
Shape(AC,CE,EA)
Shape(EC,CD,DE)
Cocircular(O,CDEAB)
Conditions (CDL):
Equal(MeasureOfAngle(CDE),128)
Equal(MeasureOfArc(OEA),MeasureOfArc(OAC))
Goal: Value(MeasureOfAngle(ABC))
Answer: 116
Theorem sequence:
arc_property_circumference_angle_external(1,OEC,D)
arc_property_circumference_angle_external(1,OEA,C)
arc_property_circumference_angle_external(1,OAC,E)
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_internal(1,OCA,E)
arc_property_circumference_angle_internal(1,OEC,A)
triangle_property_angle_sum(1,ACE)
Problem: As shown in the diagram, AB=AD, ∠BCD=68°. Find the measure of ∠ABD.
问题：如图所示，AB=AD，∠BCD=68°。求∠ABD的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(BA,OAB)
Shape(CB,OBC)
Shape(DC,OCD)
Shape(DA,AB,BD)
Shape(DB,BC,CD)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AD))
Equal(MeasureOfAngle(BCD),68)
Goal: Value(MeasureOfAngle(ABD))
Answer: 34
Theorem sequence:
arc_property_circumference_angle_external(1,ODB,C)
arc_property_circumference_angle_internal(1,ODB,A)
triangle_property_angle_sum(1,DAB)
isosceles_triangle_judgment_line_equal(1,ABD)
isosceles_triangle_property_angle_equal(1,ABD)
Problem: As shown in the diagram, AP=40, ∠APB=90°, ∠PBA=30°. Find the length of line BP.
问题：如图所示，AP=40，∠APB=90°，∠PBA=30°。求直线BP的长度。
Construction (CDL):
Shape(PB,BA,AP)
Conditions (CDL):
Equal(LengthOfLine(AP),40)
Equal(MeasureOfAngle(APB),90)
Equal(MeasureOfAngle(PBA),30)
Goal: Value(LengthOfLine(BP))
Answer: 40*sqrt(3)
Theorem sequence:
triangle_property_angle_sum(1,PBA)
sine_theorem(1,PBA)
Problem: As shown in the diagram, AB=8, ∠BAM=15°, ∠EBA=90°, ∠MBE=60°. Find the length of line MB.
问题：如图所示，AB=8，∠BAM=15°，∠EBA=90°，∠MBE=60°。求直线MB的长度。
Construction (CDL):
Shape(BA,AE,EB)
Shape(BE,EM,MB)
Collinear(AEM)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(MeasureOfAngle(BAM),15)
Equal(MeasureOfAngle(EBA),90)
Equal(MeasureOfAngle(MBE),60)
Goal: Value(LengthOfLine(MB))
Answer: 8
Theorem sequence:
triangle_property_angle_sum(1,BAE)
triangle_property_angle_sum(1,BEM)
adjacent_complementary_angle(1,AEB,BEM)
isosceles_triangle_judgment_angle_equal(1,BAM)
Problem: As shown in the diagram, ∠COA=126°, the center of ⊙O is O. Find the measure of ∠BDC.
问题：如图所示，∠COA=126°，O是⊙O的圆心。求∠BDC的大小。
Construction (CDL):
Shape(AE,ED,ODA)
Shape(EB,BD,DE)
Shape(DB,OBD)
Shape(CO,OA,OAC)
Shape(BE,EC,OCB)
Shape(OC,CE,EO)
Collinear(AOEB)
Collinear(DEC)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(COA),126)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BDC))
Answer: 27
Theorem sequence:
adjacent_complementary_angle(1,EOC,COA)
arc_property_center_angle(1,OCB,O)
arc_property_circumference_angle_external(1,OCB,D)
Problem: As shown in the diagram, the length of arc OAB is equal to the length of arc OCA, ∠CAB=40°. Find the measure of ∠ABC.
问题：如图所示，⌒OAB的长度与弧OCA的长度相等，∠CAB=40°。求∠ABC的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(CB,OBC)
Shape(AC,OCA)
Shape(BC,CA,AB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(LengthOfArc(OAB),LengthOfArc(OCA))
Equal(MeasureOfAngle(CAB),40)
Goal: Value(MeasureOfAngle(ABC))
Answer: 70
Theorem sequence:
congruent_arc_judgment_length_equal(1,OAB,OCA)
congruent_arc_property_chord_equal(1,OAB,OCA)
isosceles_triangle_judgment_line_equal(1,ABC)
isosceles_triangle_property_angle_equal(1,ABC)
triangle_property_angle_sum(1,BCA)
Problem: As shown in the diagram, AE=ED, ∠DAB=20°, the center of ⊙O is O. Find the measure of ∠OCA.
问题：如图所示，AE=ED，∠DAB=20°，O是⊙O的圆心。求∠OCA的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(BA,OAB)
Shape(EC,CA,AE)
Shape(ED,ODC,CE)
Shape(EA,AO,OE)
Shape(DE,EO,OB,OBD)
Collinear(CEO)
Collinear(AED)
Collinear(AOB)
Cocircular(O,ABDC)
Conditions (CDL):
Equal(LengthOfLine(AE),LengthOfLine(ED))
Equal(MeasureOfAngle(DAB),20)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(OCA))
Answer: 55
Theorem sequence:
circle_property_chord_perpendicular_bisect_chord(2,O,OE,DA)
triangle_property_angle_sum(1,EAO)
triangle_property_angle_sum(1,CAO)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OC,O)
isosceles_triangle_judgment_line_equal(1,OCA)
isosceles_triangle_property_angle_equal(1,OCA)
Problem: As shown in the diagram, ∠DOB=110°, the center of circle O is O. Find the measure of ∠BCD.
问题：如图所示，∠DOB=110°，O是⊙O的圆心。求∠BCD的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(BA,OAB)
Shape(CB,OBC)
Shape(DC,OCD)
Shape(DA,AB,BO,OD)
Shape(DO,OB,BC,CD)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(MeasureOfAngle(DOB),110)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCD))
Answer: 125
Theorem sequence:
arc_property_center_angle(1,OBD,O)
arc_property_circumference_angle_internal(1,OBD,C)
Problem: As shown in the diagram, OH=1, ∠AHO=90°, ∠COA=60°, the center of circle O is O. Find the length of line AB.
问题：如图所示，OH=1，∠AHO=90°，∠COA=60°，O是圆O的圆心。求直线AB的长度。
Construction (CDL):
Shape(OA,AH,HO)
Shape(CH,HA,OAC)
Shape(BH,HC,OCB)
Shape(OH,HB,OBA)
Collinear(OHC)
Collinear(AHB)
Cocircular(O,ACB)
Conditions (CDL):
Equal(LengthOfLine(OH),1)
Equal(MeasureOfAngle(AHO),90)
Equal(MeasureOfAngle(COA),60)
IsCentreOfCircle(O,O)
Goal: Value(LengthOfLine(AB))
Answer: 2*sqrt(3)
Theorem sequence:
circle_property_chord_perpendicular_bisect_chord(1,O,OH,AB)
triangle_property_angle_sum(1,OAH)
sine_theorem(1,HOA)
line_addition(1,AH,HB)
Problem: As shown in the diagram, ∠ACD=25°, O is the center of ⊙O. Find the measure of ∠DOB.
问题：如图所示，∠ACD=25°，O是圆O的圆心。求∠DOB的大小。
Construction (CDL):
Shape(BE,EC,OCB)
Shape(DO,OB,OBD)
Shape(OD,DE,EO)
Shape(AE,ED,ODA)
Shape(CA,OAC)
Shape(CE,EA,AC)
Collinear(CED)
Collinear(AEOB)
Cocircular(O,BDAC)
Conditions (CDL):
Equal(MeasureOfAngle(ACD),25)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(DOB))
Answer: 130
Theorem sequence:
arc_property_circumference_angle_external(1,ODA,C)
arc_property_center_angle(1,ODA,O)
adjacent_complementary_angle(1,AOD,DOB)
Problem: As shown in the diagram, ∠BCD=30°, the diameter of circle O is AB. Find the measure of ∠DBA.
问题：如图所示，∠BCD=30°，AB是圆O的直径。求∠DBA的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AE,EC)
Shape(CE,EB,BC)
Shape(CB,OBC)
Shape(DE,EA,OAD)
Shape(ED,DB,BE)
Shape(BD,ODB)
Collinear(CED)
Collinear(AOEB)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(BCD),30)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(DBA))
Answer: 60
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, ∠BAC=55°, the center of circle O is O. Find the measure of ∠BOC.
问题：如图所示，∠BAC=55°，O是⊙O的圆心。求∠BOC的大小。
Construction (CDL):
Shape(CO,OE,EA,OAC)
Shape(OC,CE,EO)
Shape(EC,CB,BE)
Shape(BC,OCB)
Shape(AB,OBA)
Shape(EB,BA,AE)
Collinear(OEB)
Collinear(AEC)
Cocircular(O,CBA)
Conditions (CDL):
Equal(MeasureOfAngle(BAC),55)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BOC))
Answer: 110
Theorem sequence:
arc_property_circumference_angle_external(1,OCB,A)
arc_property_center_angle(1,OCB,O)
Problem: As shown in the diagram, ∠APB=36°, the center of circle O is O, PA is the tangent to ⊙O, PB is the tangent to circle O. Find the measure of ∠BCA.
问题：如图所示，∠APB=36°，O是⊙O的圆心，圆O的切线为PA，圆O的切线为PB。求∠BCA的大小。
Construction (CDL):
Shape(AP,PB,OAB)
Shape(BO,OA,OAB)
Shape(CB,OBC)
Shape(AC,OCA)
Shape(AO,OB,BC,CA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(APB),36)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
IsTangentOfCircle(PB,O)
Goal: Value(MeasureOfAngle(BCA))
Answer: 72
Theorem sequence:
tangent_of_circle_property_perpendicular(1,PA,O,O)
tangent_of_circle_property_perpendicular(2,PB,O,O)
quadrilateral_property_angle_sum(1,APBO)
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, PA=10, EA is the tangent to ⊙O, EC is the tangent to circle O, the tangent to circle O is FB, the tangent to circle O is FC, PA is the tangent to circle O, PB is the tangent to circle O. Find the perimeter of △PFE.
问题：如图所示，PA=10，EA是⊙O的切线，EC是⊙O的切线，FB是⊙O的切线，⊙O的切线为FC，PA是⊙O的切线，圆O的切线为PB。求三角形PFE的周长。
Construction (CDL):
Shape(EP,PF,FE)
Shape(AE,EC,OAC)
Shape(CF,FB,OCB)
Shape(O)
Collinear(AEP)
Collinear(BFP)
Collinear(ECF)
Cocircular(O,ACB)
Conditions (CDL):
Equal(LengthOfLine(PA),10)
IsTangentOfCircle(EA,O)
IsTangentOfCircle(EC,O)
IsTangentOfCircle(FB,O)
IsTangentOfCircle(FC,O)
IsTangentOfCircle(PA,O)
IsTangentOfCircle(PB,O)
Goal: Value(PerimeterOfTriangle(PFE))
Answer: 20
Theorem sequence:
tangent_of_circle_property_length_equal(1,EA,EC,O)
tangent_of_circle_property_length_equal(1,FC,FB,O)
tangent_of_circle_property_length_equal(1,PA,PB,O)
line_addition(1,AE,EP)
line_addition(1,BF,FP)
line_addition(1,EC,CF)
triangle_perimeter_formula(1,EPF)
Problem: As shown in the diagram, OB=3, ∠BPA=30°, O is the center of circle O, the tangent to circle O is PA. Find the length of line AP.
问题：如图所示，OB=3，∠BPA=30°，O是圆O的圆心，PA是圆O的切线。求直线AP的长度。
Construction (CDL):
Shape(BO,OA,OAB)
Shape(AO,OB,OBA)
Shape(BP,PA,OBA)
Collinear(OBP)
Cocircular(O,BA)
Conditions (CDL):
Equal(LengthOfLine(OB),3)
Equal(MeasureOfAngle(BPA),30)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
Goal: Value(LengthOfLine(AP))
Answer: 3*sqrt(3)
Theorem sequence:
tangent_of_circle_property_perpendicular(2,PA,O,O)
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OA,O)
triangle_property_angle_sum(1,AOP)
sine_theorem(1,AOP)
Problem: As shown in the diagram, ∠BCA=55°, O is the center of ⊙O, the tangent to circle O is PA, PB is the tangent to circle O. Find the measure of ∠APB.
问题：如图所示，∠BCA=55°，O是圆O的圆心，PA是圆O的切线，圆O的切线为PB。求∠APB的大小。
Construction (CDL):
Shape(AP,PB,OAB)
Shape(BO,OA,OAB)
Shape(AO,OB,BC,CA)
Shape(CB,OBC)
Shape(AC,OCA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),55)
IsCentreOfCircle(O,O)
IsTangentOfCircle(PA,O)
IsTangentOfCircle(PB,O)
Goal: Value(MeasureOfAngle(APB))
Answer: 70
Theorem sequence:
tangent_of_circle_property_perpendicular(1,PA,O,O)
tangent_of_circle_property_perpendicular(2,PB,O,O)
quadrilateral_property_angle_sum(1,APBO)
arc_property_center_angle(1,OAB,O)
arc_property_circumference_angle_external(1,OAB,C)
Problem: As shown in the diagram, AB=15, AC=35, ∠ABC=90°. Find the measure of ∠BCA.
问题：如图所示，AB=15，AC=35，∠ABC=90°。求∠BCA的大小。
Construction (CDL):
Shape(AB,BC,CA)
Conditions (CDL):
Equal(LengthOfLine(AB),15)
Equal(LengthOfLine(AC),35)
Equal(MeasureOfAngle(ABC),90)
Goal: Value(MeasureOfAngle(BCA))
Answer: 180*asin(3/7)/pi
Theorem sequence:
sine_theorem(1,ABC)
Problem: As shown in the diagram, AC=13, ∠ABC=90°, Sin(MeasureOfAngle(CAB))=5/13. Find the length of line BC.
问题：如图所示，AC=13，∠ABC=90°，Sin(MeasureOfAngle(CAB))=5/13。求直线BC的长度。
Construction (CDL):
Shape(AB,BC,CA)
Conditions (CDL):
Equal(LengthOfLine(AC),13)
Equal(MeasureOfAngle(ABC),90)
Equal(Sin(MeasureOfAngle(CAB)),5/13)
Goal: Value(LengthOfLine(BC))
Answer: 5
Theorem sequence:
sine_theorem(1,CAB)
Problem: As shown in the diagram, Div(LengthOfLine(BC)=LengthOfLine(CA)), BC=6, ∠BCA is the right angle of the right triangle BCA. Find the length of line AB.
问题：如图所示，Div(LengthOfLine(BC)=LengthOfLine(CA))，BC=6，∠BCA是直角三角形BCA的直角。求直线AB的长度。
Construction (CDL):
Shape(BC,CA,AB)
Conditions (CDL):
Equal(Div(LengthOfLine(BC),LengthOfLine(CA)),3/4)
Equal(LengthOfLine(BC),6)
RightTriangle(BCA)
Goal: Value(LengthOfLine(AB))
Answer: 10
Theorem sequence:
right_triangle_property_pythagorean(1,BCA)
Problem: As shown in the diagram, Div(LengthOfLine(BC)=LengthOfLine(CA)), BC=5, BC and CA are the legs of the right triangle BCA. Find the length of line AC.
问题：如图所示，Div(LengthOfLine(BC)=LengthOfLine(CA))，BC=5，△BCA为直角△。求直线AC的长度。
Construction (CDL):
Shape(BC,CA,AB)
Conditions (CDL):
Equal(Div(LengthOfLine(BC),LengthOfLine(CA)),1/sqrt(3))
Equal(LengthOfLine(BC),5)
RightTriangle(BCA)
Goal: Value(LengthOfLine(AC))
Answer: 5*sqrt(3)
Theorem sequence:
Problem: As shown in the diagram, AC=4, BD=3, CE=6, quadrilateral ACDB is similar to quadrilateral CEFD. Find the length of line BF.
问题：如图所示，AC=4，BD=3，CE=6，ACDB与CEFD相似。求直线BF的长度。
Construction (CDL):
Shape(AC,CD,DB,BA)
Shape(CE,EF,FD,DC)
Collinear(ACE)
Collinear(BDF)
Conditions (CDL):
Equal(LengthOfLine(AC),4)
Equal(LengthOfLine(BD),3)
Equal(LengthOfLine(CE),6)
SimilarBetweenQuadrilateral(ACDB,CEFD)
Goal: Value(LengthOfLine(BF))
Answer: 15/2
Theorem sequence:
similar_quadrilateral_property_line_ratio(1,ACDB,CEFD)
similar_quadrilateral_property_line_ratio(1,DBAC,FDCE)
line_addition(1,BD,DF)
Problem: As shown in the diagram, AD=8, AE=6, DB=4, DE∥BC. Find the length of line EC.
问题：如图所示，AD=8，AE=6，DB=4，DE平行于BC。求直线EC的长度。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(LengthOfLine(AD),8)
Equal(LengthOfLine(AE),6)
Equal(LengthOfLine(DB),4)
ParallelBetweenLine(DE,BC)
Goal: Value(LengthOfLine(EC))
Answer: 3
Theorem sequence:
parallel_property_corresponding_angle(1,DE,BC,A)
parallel_property_corresponding_angle(2,CB,ED,A)
similar_triangle_judgment_aa(1,ADE,ABC)
similar_triangle_property_line_ratio(1,DEA,BCA)
similar_triangle_property_line_ratio(1,EAD,CAB)
line_addition(1,AD,DB)
line_addition(1,AE,EC)
Problem: As shown in the diagram, AB=3, AD=1, DE∥BC. Find the ratio of the length of line AE to the length of line EC.
问题：如图所示，AB=3，AD=1，DE∥BC。求直线AE的长度与直线EC的长度之比。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(LengthOfLine(AB),3)
Equal(LengthOfLine(AD),1)
ParallelBetweenLine(DE,BC)
Goal: Value(Div(LengthOfLine(AE),LengthOfLine(EC)))
Answer: 1/2
Theorem sequence:
parallel_property_corresponding_angle(1,DE,BC,A)
parallel_property_corresponding_angle(2,CB,ED,A)
similar_triangle_judgment_aa(1,ABC,ADE)
similar_triangle_property_line_ratio(1,BCA,DEA)
similar_triangle_property_line_ratio(1,CAB,EAD)
line_addition(1,AE,EC)
Problem: As shown in the diagram, AB=8, ∠ADC=90°, ∠CBA=30°, ∠DCA=45°. Find the length of line AC.
问题：如图所示，AB=8，∠ADC=90°，∠CBA=30°，∠DCA=45°。求直线AC的长度。
Construction (CDL):
Shape(AD,DC,CA)
Shape(AC,CB,BA)
Collinear(DCB)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(MeasureOfAngle(ADC),90)
Equal(MeasureOfAngle(CBA),30)
Equal(MeasureOfAngle(DCA),45)
Goal: Value(LengthOfLine(AC))
Answer: 4*sqrt(2)
Theorem sequence:
sine_theorem(1,ADC)
sine_theorem(1,ADB)
Problem: As shown in the diagram, AB=CD, BD=32, ∠BDA=90°, ∠DOB=110°, O bisects segment AB, O bisects segment CD. Find the length of line AB.
问题：如图所示，AB=CD，BD=32，∠BDA=90°，∠DOB=110°，O平分线段AB，O平分线段CD。求直线AB的长度。
Construction (CDL):
Shape(AO,OD,DA)
Shape(OC,CB,BO)
Shape(DO,OB,BD)
Collinear(AOB)
Collinear(DOC)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(CD))
Equal(LengthOfLine(BD),32)
Equal(MeasureOfAngle(BDA),90)
Equal(MeasureOfAngle(DOB),110)
IsMidpointOfLine(O,AB)
IsMidpointOfLine(O,CD)
Goal: Value(LengthOfLine(AB))
Answer: 32/sin(11*pi/36)
Theorem sequence:
adjacent_complementary_angle(1,AOD,DOB)
line_addition(1,AO,OB)
line_addition(1,DO,OC)
triangle_property_angle_sum(1,AOD)
isosceles_triangle_judgment_line_equal(1,ODA)
isosceles_triangle_property_angle_equal(1,ODA)
sine_theorem(1,BDA)
Problem: As shown in the diagram, BE=40, CM=60, ∠ABE=70°, ∠CMB=90°, ∠EAB=45°. Find the length of line CE.
问题：如图所示，BE=40，CM=60，∠ABE=70°，∠CMB=90°，∠EAB=45°。求直线CE的长度。
Construction (CDL):
Shape(AM,MN,NA)
Shape(CN,NE,EC)
Shape(NM,MB,BE,EN)
Collinear(CNM)
Collinear(ANE)
Collinear(CEB)
Collinear(AMB)
Conditions (CDL):
Equal(LengthOfLine(BE),40)
Equal(LengthOfLine(CM),60)
Equal(MeasureOfAngle(ABE),70)
Equal(MeasureOfAngle(CMB),90)
Equal(MeasureOfAngle(EAB),45)
Goal: Value(LengthOfLine(CE))
Answer: -40+60/cos(pi/9)
Theorem sequence:
sine_theorem(1,CMB)
line_addition(1,CE,EB)
Problem: As shown in the diagram, AB=10, BC=12, BD=6, BD⊥AD. Find the value of sin(AB).
问题：如图所示，AB=10，BC=12，BD=6，BD垂直于AD。求sin(AB)的值。
Construction (CDL):
Shape(AB,BD,DA)
Shape(AD,DC,CA)
Cocircular(BDC)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(LengthOfLine(BC),12)
Equal(LengthOfLine(BD),6)
PerpendicularBetweenLine(BD,AD)
Goal: Value(Sin(MeasureOfAngle(ABD)))
Answer: 4/5
Theorem sequence:
right_triangle_judgment_angle(1,BDA)
right_triangle_property_pythagorean(1,BDA)
sine_theorem(1,ABD)
Problem: As shown in the diagram, CN=1, DC=3, ∠CEN=45°, DC is parallel to EN, ∠NCD is the right angle of the right triangle NCD. Find the value of tan(CP).
问题：如图所示，CN=1，DC=3，∠CEN=45°，DC∥EN，△NCD为直角△。求tan(CP)的值。
Construction (CDL):
Shape(DP,PC,CD)
Shape(CP,PN,NC)
Shape(PE,EN,NP)
Collinear(DPN)
Collinear(CPE)
Conditions (CDL):
Equal(LengthOfLine(CN),1)
Equal(LengthOfLine(DC),3)
Equal(MeasureOfAngle(CEN),45)
ParallelBetweenLine(DC,EN)
RightTriangle(NCD)
Goal: Value(Tan(MeasureOfAngle(CPN)))
Answer: tan(pi*(asin(sqrt(10)/10)/pi+1/4))
Theorem sequence:
right_triangle_property_pythagorean(1,NCD)
sine_theorem(1,NCD)
parallel_property_alternate_interior_angle(1,DC,EN)
triangle_property_angle_sum(1,PEN)
adjacent_complementary_angle(1,CPN,NPE)
Problem: As shown in the diagram, ∠ADC=55°, the center of ⊙O is O, AB is the diameter of ⊙O. Find the measure of ∠CAB.
问题：如图所示，∠ADC=55°，⊙O的圆心为O，AB是⊙O的直径。求∠CAB的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(DA,OAD)
Shape(BE,ED,ODB)
Shape(AD,DE,EA)
Shape(CE,EB,BC)
Shape(AE,EC,CA)
Shape(CB,OBC)
Collinear(CED)
Collinear(AOEB)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(ADC),55)
IsCentreOfCircle(O,O)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(CAB))
Answer: 35
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
arc_property_circumference_angle_external(1,OCA,D)
arc_property_circumference_angle_external(1,OCA,B)
triangle_property_angle_sum(1,BCA)
Problem: As shown in the diagram, AB=AC, ∠CAB=56°. Find the measure of ∠CDA.
问题：如图所示，AB=AC，∠CAB=56°。求∠CDA的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(CB,OBC)
Shape(DC,OCD)
Shape(AD,ODA)
Shape(AB,BC,CA)
Shape(AC,CD,DA)
Cocircular(O,ABCD)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(MeasureOfAngle(CAB),56)
Goal: Value(MeasureOfAngle(CDA))
Answer: 118
Theorem sequence:
triangle_property_angle_sum(1,ABC)
isosceles_triangle_judgment_line_equal(1,ABC)
isosceles_triangle_property_angle_equal(1,ABC)
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_internal(1,OCA,D)
Problem: As shown in the diagram, BC=8, ∠CAB=30°, O is the center of circle O. Find the radius of circle O.
问题：如图所示，BC=8，∠CAB=30°，O是⊙O的圆心。求圆O的半径。
Construction (CDL):
Shape(BA,OAB)
Shape(CB,OBC)
Shape(OC,OCA,AE,EO)
Shape(AB,BE,EA)
Shape(EB,BC,CE)
Shape(OE,EC,CO)
Collinear(AEC)
Collinear(BEO)
Cocircular(O,ABC)
Conditions (CDL):
Equal(LengthOfLine(BC),8)
Equal(MeasureOfAngle(CAB),30)
IsCentreOfCircle(O,O)
Goal: Value(RadiusOfCircle(O))
Answer: 8
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,OBC,O)
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OC,O)
triangle_property_angle_sum(1,BCO)
isosceles_triangle_judgment_line_equal(1,OBC)
isosceles_triangle_property_angle_equal(1,OBC)
isosceles_triangle_judgment_angle_equal(1,COB)
Problem: As shown in the diagram, AC=6, ∠ABC=30°, O is the center of circle O. Find the diameter of ⊙O.
问题：如图所示，AC=6，∠ABC=30°，⊙O的圆心为O。求⊙O的直径。
Construction (CDL):
Shape(BA,OAB)
Shape(CB,OBC)
Shape(AC,OCA)
Shape(AB,BC,CD,DO)
Shape(AO,OC,CA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(LengthOfLine(AC),6)
Equal(MeasureOfAngle(ABC),30)
IsCentreOfCircle(O,O)
Goal: Value(DiameterOfCircle(O))
Answer: 12
Theorem sequence:
arc_property_circumference_angle_external(1,OCA,B)
arc_property_center_angle(1,OCA,O)
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OC,O)
triangle_property_angle_sum(1,AOC)
isosceles_triangle_judgment_line_equal(1,OCA)
isosceles_triangle_property_angle_equal(1,OCA)
isosceles_triangle_judgment_angle_equal(1,CAO)
circle_property_length_of_radius_and_diameter(1,O)
Problem: As shown in the diagram, ∠ABC=45°, AD is the diameter of circle O. Find the measure of ∠CAD.
问题：如图所示，∠ABC=45°，圆O的直径为AD。求∠CAD的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(DE,EB,OBD)
Shape(CD,ODC)
Shape(AC,OCA)
Shape(BE,EA,AB)
Shape(CE,ED,DC)
Shape(EC,CA,AE)
Collinear(BEC)
Collinear(AOED)
Cocircular(O,BDCA)
Conditions (CDL):
Equal(MeasureOfAngle(ABC),45)
IsDiameterOfCircle(AD,O)
Goal: Value(MeasureOfAngle(CAD))
Answer: 45
Theorem sequence:
diameter_of_circle_property_right_angle(1,DCA,O)
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_external(1,OCA,D)
triangle_property_angle_sum(1,DCA)
Problem: As shown in the diagram, ∠BCA=50°, O is the center of ⊙O, triangle ABD is mirror congruent to triangle ADE. Find the measure of ∠EAB.
问题：如图所示，∠BCA=50°，圆O的圆心为O，三角形ABD与三角形ADE是镜像全等三角形。求∠EAB的大小。
Construction (CDL):
Shape(OAB,BA)
Shape(AB,BP,PA)
Shape(PB,BD,DP)
Shape(PD,DM,MQ,QP)
Shape(AQ,QE,EA)
Shape(AP,PQ,QA)
Shape(QM,ME,EQ)
Shape(MD,DC,CM)
Shape(AE,OEA)
Shape(EM,MC,OCE)
Shape(CD,DB,OBC)
Shape(O)
Collinear(BDC)
Collinear(AQMC)
Collinear(DME)
Collinear(BPQE)
Collinear(APD)
Cocircular(O,ABCE)
Conditions (CDL):
Equal(MeasureOfAngle(BCA),50)
IsCentreOfCircle(O,O)
MirrorCongruentBetweenTriangle(ABD,ADE)
Goal: Value(MeasureOfAngle(EAB))
Answer: 80
Theorem sequence:
arc_property_circumference_angle_external(1,OAB,C)
arc_property_circumference_angle_external(1,OAB,E)
mirror_congruent_triangle_property_line_equal(1,DAB,DEA)
sine_theorem(1,ABE)
triangle_property_angle_sum(1,ABE)
Problem: As shown in the diagram, ∠COB=120°, O is the center of circle O. Find the measure of ∠BAC.
问题：如图所示，∠COB=120°，圆O的圆心为O。求∠BAC的大小。
Construction (CDL):
Shape(BO,OC,OCB)
Shape(OB,BC,CO)
Shape(CB,BA,AC)
Shape(AB,OBA)
Shape(CA,OAC)
Cocircular(O,BAC)
Conditions (CDL):
Equal(MeasureOfAngle(COB),120)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BAC))
Answer: 120
Theorem sequence:
arc_property_center_angle(1,OBC,O)
round_arc(1,OBC,OCB)
arc_property_circumference_angle_external(1,OCB,A)
Problem: As shown in the diagram, Add(MeasureOfAngle(CAB)=MeasureOfAngle(COB)), the radius of circle O is 5, the center of circle O is O. Find the length of line BC.
问题：如图所示，Add(MeasureOfAngle(CAB)=MeasureOfAngle(COB))，圆O的半径为5，圆O的圆心为O。求直线BC的长度。
Construction (CDL):
Shape(OAB,BA)
Shape(AB,BO,OC,CA)
Shape(OB,BC,CO)
Shape(AC,OCA)
Shape(CB,OBC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(Add(MeasureOfAngle(CAB),MeasureOfAngle(COB)),180)
Equal(RadiusOfCircle(O),5)
IsCentreOfCircle(O,O)
Goal: Value(LengthOfLine(BC))
Answer: 5*sqrt(3)
Theorem sequence:
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OC,O)
cosine_theorem(1,OBC)
Problem: As shown in the diagram, AC=6, CD=3, the radius of circle O is 5, the center of circle O is O, CD is perpendicular to BD. Find the length of line BC.
问题：如图所示，AC=6，CD=3，⊙O的半径为5，⊙O的圆心为O，CD⊥BD。求直线BC的长度。
Construction (CDL):
Shape(OCA,AC)
Shape(CA,AM,MC)
Shape(CM,MD,DC)
Shape(CD,DB,BC)
Shape(CB,OBC)
Shape(MO,OB,BD,DM)
Shape(OM,MA,OAB,BO)
Collinear(OMC)
Collinear(AMDB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(LengthOfLine(AC),6)
Equal(LengthOfLine(CD),3)
Equal(RadiusOfCircle(O),5)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(CD,BD)
Goal: Value(LengthOfLine(BC))
Answer: 5
Theorem sequence:
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OC,O)
adjacent_complementary_angle(1,ADC,CDB)
sine_theorem(1,CAD)
arc_property_center_angle(1,OBC,O)
arc_property_circumference_angle_external(1,OBC,A)
cosine_theorem(1,OBC)