SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, AB=2, BC=6, CD=3, AB is parallel to CD. Find the length of line BP.
问题：如图所示，AB=2，BC=6，CD=3，AB平行于CD。求直线BP的长度。
Construction (CDL):
Shape(AP,PB,BA)
Shape(PC,CD,DP)
Collinear(APD)
Collinear(BPC)
Conditions (CDL):
Equal(LengthOfLine(AB),2)
Equal(LengthOfLine(BC),6)
Equal(LengthOfLine(CD),3)
ParallelBetweenLine(AB,CD)
Goal: Value(LengthOfLine(BP))
Answer: 12/5
Theorem sequence:
parallel_property_alternate_interior_angle(1,AB,CD)
parallel_property_alternate_interior_angle(2,AB,CD)
similar_triangle_judgment_aa(1,PBA,PCD)
similar_triangle_property_line_ratio(1,PBA,PCD)
similar_triangle_property_line_ratio(1,APB,DPC)
line_addition(1,BP,PC)
Problem: As shown in the diagram, AB=8, BE=6, ABCD is a square, quadrilateral BEFG is a square. Find the area of △AEK.
问题：如图所示，AB=8，BE=6，ABCD是正方形，四边形BEFG是正方形。求三角形AEK的面积。
Construction (CDL):
Shape(DA,AK,KP,PD)
Shape(PK,KC,CP)
Shape(CK,KG,GC)
Shape(KA,AB,BG,GK)
Shape(GB,BE,EG)
Shape(GE,EF,FG)
Collinear(PKGE)
Collinear(AKC)
Collinear(DPC)
Collinear(CGB)
Collinear(ABE)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(LengthOfLine(BE),6)
Square(ABCD)
Square(BEFG)
Goal: Value(AreaOfTriangle(AEK))
Answer: 49
Theorem sequence:
triangle_property_angle_sum(1,ABC)
triangle_property_angle_sum(1,BEG)
triangle_property_angle_sum(1,AEK)
sine_theorem(1,BCA)
sine_theorem(1,BEG)
sine_theorem(1,AEK)
line_addition(1,AB,BE)
triangle_area_formula_sine(1,AEK)
Problem: As shown in the diagram, the area of triangle EDQ is 9, CE=ED, ABCD is a ▱. Find the area of triangle ABQ.
问题：如图所示，△EDQ的面积为9，CE=ED，BA和CD是平行四边形ABCD的一组对边。求三角形ABQ的面积。
Construction (CDL):
Shape(QA,AB,BQ)
Shape(QD,DA,AQ)
Shape(QB,BC,CE,EQ)
Shape(DQ,QE,ED)
Collinear(AQE)
Collinear(DQB)
Collinear(CED)
Conditions (CDL):
Equal(AreaOfTriangle(EDQ),9)
Equal(LengthOfLine(CE),LengthOfLine(ED))
Parallelogram(ABCD)
Goal: Value(AreaOfTriangle(ABQ))
Answer: 36
Theorem sequence:
parallelogram_property_opposite_line_equal(1,ABCD)
vertical_angle(1,DQE,BQA)
line_addition(1,CE,ED)
parallel_property_alternate_interior_angle(1,BA,CD)
similar_triangle_judgment_aa(1,ABQ,EDQ)
similar_triangle_property_line_ratio(1,QAB,QED)
similar_triangle_property_area_square_ratio(1,ABQ,EDQ)
Problem: As shown in the diagram, AC=12, BO=6, DO=3, CD is parallel to BA, DC is parallel to AB. Find the length of line AO.
问题：如图所示，AC=12，BO=6，DO=3，CD平行于BA，DC平行于AB。求直线AO的长度。
Construction (CDL):
Shape(CD,DO,OC)
Shape(OA,AB,BO)
Collinear(DOB)
Collinear(AOC)
Conditions (CDL):
Equal(LengthOfLine(AC),12)
Equal(LengthOfLine(BO),6)
Equal(LengthOfLine(DO),3)
ParallelBetweenLine(CD,BA)
ParallelBetweenLine(DC,AB)
Goal: Value(LengthOfLine(AO))
Answer: 8
Theorem sequence:
line_addition(1,AO,OC)
parallel_property_alternate_interior_angle(1,DC,AB)
parallel_property_alternate_interior_angle(2,DC,AB)
similar_triangle_judgment_aa(1,OCD,OAB)
similar_triangle_property_line_ratio(1,CDO,ABO)
similar_triangle_property_line_ratio(1,DOC,BOA)
Problem: As shown in the diagram, AC=6, AD=3, DB=5, ∠DEA=∠ABC. Find the length of line AE.
问题：如图所示，AC=6，AD=3，DB=5，∠DEA=∠ABC。求直线AE的长度。
Construction (CDL):
Shape(AD,DE,EA)
Shape(ED,DB,BC,CE)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(LengthOfLine(AC),6)
Equal(LengthOfLine(AD),3)
Equal(LengthOfLine(DB),5)
Equal(MeasureOfAngle(DEA),MeasureOfAngle(ABC))
Goal: Value(LengthOfLine(AE))
Answer: 4
Theorem sequence:
line_addition(1,AD,DB)
mirror_similar_triangle_judgment_aa(1,DEA,CAB)
mirror_similar_triangle_property_line_ratio(1,DEA,CAB)
mirror_similar_triangle_property_line_ratio(1,EAD,BCA)
Problem: As shown in the diagram, Div(LengthOfLine(DE)=LengthOfLine(EA)), CD=10, EF is parallel to AB, DABC is a parallelogram. Find the length of line EF.
问题：如图所示，Div(LengthOfLine(DE)=LengthOfLine(EA))，CD=10，EF∥AB，DABC是▱。求直线EF的长度。
Construction (CDL):
Shape(DE,EF,FD)
Shape(FE,EA,AB,BF)
Shape(CD,DF,FB,BC)
Collinear(DEA)
Collinear(DFB)
Conditions (CDL):
Equal(Div(LengthOfLine(DE),LengthOfLine(EA)),2/3)
Equal(LengthOfLine(CD),10)
ParallelBetweenLine(EF,AB)
Parallelogram(DABC)
Goal: Value(LengthOfLine(EF))
Answer: 4
Theorem sequence:
parallelogram_property_opposite_line_equal(1,ABCD)
line_addition(1,DE,EA)
parallel_property_corresponding_angle(1,EF,AB,D)
parallel_property_corresponding_angle(2,BA,FE,D)
similar_triangle_judgment_aa(1,DAB,DEF)
similar_triangle_property_line_ratio(1,DAB,DEF)
similar_triangle_property_line_ratio(1,BDA,FDE)
Problem: As shown in the diagram, AB=7, AC=11, AN=NC, BE=EC, ∠CNE=Add(MeasureOfAngle(NFE),MeasureOfAngle(FEN))°, AD is the angle bisector of ∠CAB, triangleNFE is an isosceles triangle, AD∥FE, DA∥EF. Find the length of line FC.
问题：如图所示，AB=7，AC=11，AN=NC，BE=EC，∠CNE=Add(MeasureOfAngle(NFE),MeasureOfAngle(FEN))°，AD平分∠CAB，△NFE为等腰△，AD平行于FE，DA平行于EF。求直线FC的长度。
Construction (CDL):
Shape(AB,BD,DA)
Shape(FA,AD,DE,EF)
Shape(FE,EN,NF)
Shape(NE,EC,CN)
Collinear(AFNC)
Collinear(BDEC)
Conditions (CDL):
Equal(LengthOfLine(AB),7)
Equal(LengthOfLine(AC),11)
Equal(LengthOfLine(AN),LengthOfLine(NC))
Equal(LengthOfLine(BE),LengthOfLine(EC))
Equal(MeasureOfAngle(CNE),Add(MeasureOfAngle(NFE),MeasureOfAngle(FEN)))
IsBisectorOfAngle(AD,CAB)
IsoscelesTriangle(NFE)
ParallelBetweenLine(AD,FE)
ParallelBetweenLine(DA,EF)
Goal: Value(LengthOfLine(FC))
Answer: 9
Theorem sequence:
midsegment_of_triangle_judgment_midpoint(1,NE,CAB)
midsegment_of_triangle_property_length(1,NE,CAB)
line_addition(1,AF,FN)
line_addition(1,AF,FC)
line_addition(1,AN,NC)
Problem: As shown in the diagram, AB=3, BD=2, ∠ABD=90°, ∠CDB=90°, ∠DAB=∠DBC. Find the length of line CD.
问题：如图所示，AB=3，BD=2，∠ABD=90°，∠CDB=90°，∠DAB=∠DBC。求直线CD的长度。
Construction (CDL):
Shape(CD,DB,BC)
Shape(DA,AB,BD)
Conditions (CDL):
Equal(LengthOfLine(AB),3)
Equal(LengthOfLine(BD),2)
Equal(MeasureOfAngle(ABD),90)
Equal(MeasureOfAngle(CDB),90)
Equal(MeasureOfAngle(DAB),MeasureOfAngle(DBC))
Goal: Value(LengthOfLine(CD))
Answer: 4/3
Theorem sequence:
mirror_similar_triangle_judgment_aa(1,CDB,DAB)
mirror_similar_triangle_property_line_ratio(1,CDB,DAB)
mirror_similar_triangle_property_line_ratio(1,BCD,ABD)
Problem: As shown in the diagram, AD=3, BD=5, DE∥BC. Find the ratio of the length of line DE to the length of line BC.
问题：如图所示，AD=3，BD=5，DE平行于BC。求直线DE的长度与直线BC的长度之比。
Construction (CDL):
Shape(AD,DE,EA)
Shape(ED,DB,BC,CE)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(LengthOfLine(AD),3)
Equal(LengthOfLine(BD),5)
ParallelBetweenLine(DE,BC)
Goal: Value(Div(LengthOfLine(DE),LengthOfLine(BC)))
Answer: 3/8
Theorem sequence:
line_addition(1,AD,DB)
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,ADE,ABC)
similar_triangle_property_line_ratio(1,EAD,CAB)
Problem: As shown in the diagram, the area of △AEF is 6, Div(LengthOfLine(AE)=LengthOfLine(EB)), AD and BC are opposite sides of the parallelogram DABC. Find the area of triangle CDF.
问题：如图所示，△AEF的面积为6，Div(LengthOfLine(AE)=LengthOfLine(EB))，四边形DABC是▱。求△CDF的面积。
Construction (CDL):
Shape(DA,AF,FD)
Shape(FA,AE,EF)
Shape(CD,DF,FC)
Shape(CF,FE,EB,BC)
Collinear(AFC)
Collinear(DFE)
Collinear(AEB)
Conditions (CDL):
Equal(AreaOfTriangle(AEF),6)
Equal(Div(LengthOfLine(AE),LengthOfLine(EB)),1/2)
Parallelogram(DABC)
Goal: Value(AreaOfTriangle(CDF))
Answer: 54
Theorem sequence:
parallelogram_property_opposite_line_equal(1,ABCD)
vertical_angle(1,DFC,EFA)
line_addition(1,AE,EB)
parallel_property_alternate_interior_angle(2,DC,AB)
similar_triangle_judgment_aa(1,EFA,DFC)
similar_triangle_property_line_ratio(1,FAE,FCD)
similar_triangle_property_area_square_ratio(1,EFA,DFC)
Problem: As shown in the diagram, AB=6, BC=4, CE bisects ∠BCD, DC and AB are opposite sides of the parallelogram DABC. Find the ratio of the length of line OB to the length of line BD.
问题：如图所示，AB=6，BC=4，CE是∠BCD的角平分线，DABC是▱。求直线OB的长度与直线BD的长度之比。
Construction (CDL):
Shape(DA,AE,EO,OD)
Shape(CD,DO,OC)
Shape(OE,EB,BO)
Shape(CO,OB,BC)
Collinear(DOB)
Collinear(EOC)
Collinear(AEB)
Conditions (CDL):
Equal(LengthOfLine(AB),6)
Equal(LengthOfLine(BC),4)
IsBisectorOfAngle(CE,BCD)
Parallelogram(DABC)
Goal: Value(Div(LengthOfLine(OB),LengthOfLine(BD)))
Answer: 2/5
Theorem sequence:
line_addition(1,DO,OB)
parallelogram_property_opposite_line_equal(1,ABCD)
parallel_property_collinear_extend(3,BA,CD,E)
parallel_property_alternate_interior_angle(1,DC,AB)
parallel_property_alternate_interior_angle(2,BE,CD)
isosceles_triangle_judgment_angle_equal(1,BCE)
similar_triangle_judgment_aa(1,OCD,OEB)
similar_triangle_property_line_ratio(1,OCD,OEB)
similar_triangle_property_line_ratio(1,CDO,EBO)
Problem: As shown in the diagram, AD=6, DE=3, ABCD is a parallelogram. Find the ratio of the length of line AM to the length of line MC.
问题：如图所示，AD=6，DE=3，AD和BC是▱ABCD的一组对边。求直线AM的长度与直线MC的长度之比。
Construction (CDL):
Shape(EA,AM,ME)
Shape(MA,AB,BM)
Shape(DE,EM,MC,CD)
Shape(CM,MB,BC)
Collinear(AED)
Collinear(AMC)
Collinear(EMB)
Conditions (CDL):
Equal(LengthOfLine(AD),6)
Equal(LengthOfLine(DE),3)
Parallelogram(ABCD)
Goal: Value(Div(LengthOfLine(AM),LengthOfLine(MC)))
Answer: 1/2
Theorem sequence:
parallelogram_property_opposite_line_equal(1,BCDA)
vertical_angle(1,AME,CMB)
line_addition(1,AE,ED)
parallel_property_collinear_extend(3,AD,BC,E)
parallel_property_alternate_interior_angle(2,AE,BC)
similar_triangle_judgment_aa(1,CMB,AME)
similar_triangle_property_line_ratio(1,MBC,MEA)
similar_triangle_property_line_ratio(1,BCM,EAM)
Problem: As shown in the diagram, AC=sqrt(2), ABCD is a square. Find the area of quadrilateral ABCD.
问题：如图所示，AC=sqrt(2)，四边形ABCD是正方形。求ABCD的面积。
Construction (CDL):
Shape(DA,AO,OD)
Shape(DO,OC,CD)
Shape(OA,AB,BO)
Shape(CO,OB,BC)
Collinear(DOB)
Collinear(AOC)
Conditions (CDL):
Equal(LengthOfLine(AC),sqrt(2))
Square(ABCD)
Goal: Value(AreaOfQuadrilateral(ABCD))
Answer: 1
Theorem sequence:
rectangle_property_diagonal_equal(1,ABCD)
kite_area_formula_diagonal(1,ABCD)
Problem: As shown in the diagram, Div(LengthOfLine(AD)=LengthOfLine(BD)), DE=6, DE is parallel to BC. Find the length of line BC.
问题：如图所示，Div(LengthOfLine(AD)=LengthOfLine(BD))，DE=6，DE∥BC。求直线BC的长度。
Construction (CDL):
Shape(AD,DE,EA)
Shape(ED,DB,BC,CE)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(Div(LengthOfLine(AD),LengthOfLine(BD)),3)
Equal(LengthOfLine(DE),6)
ParallelBetweenLine(DE,BC)
Goal: Value(LengthOfLine(BC))
Answer: 8
Theorem sequence:
line_addition(1,AD,DB)
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,ABC,ADE)
similar_triangle_property_line_ratio(1,CAB,EAD)
Problem: As shown in the diagram, AB=AC, AD=DB, AE=EC, DE=2, tan(BCA)=2, DGFE is a square. Find the length of line AC.
问题：如图所示，AB=AC，AD=DB，AE=EC，DE=2，tan(BCA)=2，DGFE是正方形。求直线AC的长度。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BG,GD)
Shape(ED,DG,GF,FE)
Shape(EF,FC,CE)
Collinear(ADB)
Collinear(AEC)
Collinear(BGFC)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(LengthOfLine(AD),LengthOfLine(DB))
Equal(LengthOfLine(AE),LengthOfLine(EC))
Equal(LengthOfLine(DE),2)
Equal(Tan(BCA),2)
Square(DGFE)
Goal: Value(LengthOfLine(AC))
Answer: 2*sqrt(5)
Theorem sequence:
midsegment_of_triangle_judgment_midpoint(1,DE,ABC)
midsegment_of_triangle_property_length(1,DE,ABC)
line_addition(1,AE,EC)
parallel_property_corresponding_angle(2,GD,FE,C)
cosine_theorem(1,CAB)
sine_theorem(1,EFC)
Problem: As shown in the diagram, Div(LengthOfLine(EC)=LengthOfLine(AB)), EF=4, AD and BC are opposite sides of the ▱ DABC. Find the length of line BF.
问题：如图所示，Div(LengthOfLine(EC)=LengthOfLine(AB))，EF=4，DABC是平行四边形。求直线BF的长度。
Construction (CDL):
Shape(DA,AF,FE,ED)
Shape(FA,AB,BF)
Shape(FB,BC,CF)
Shape(EF,FC,CE)
Collinear(AFC)
Collinear(EFB)
Collinear(DEC)
Conditions (CDL):
Equal(Div(LengthOfLine(EC),LengthOfLine(AB)),2/3)
Equal(LengthOfLine(EF),4)
Parallelogram(DABC)
Goal: Value(LengthOfLine(BF))
Answer: 6
Theorem sequence:
parallel_property_collinear_extend(3,DC,AB,E)
parallel_property_alternate_interior_angle(1,EC,AB)
parallel_property_alternate_interior_angle(2,EC,AB)
similar_triangle_judgment_aa(1,FCE,FAB)
similar_triangle_property_line_ratio(1,FCE,FAB)
similar_triangle_property_line_ratio(1,CEF,ABF)
Problem: As shown in the diagram, AD=2, DB=4, the perimeter of triangle ADE is 10, DE is parallel to BC. Find the perimeter of △ABC.
问题：如图所示，AD=2，DB=4，△ADE的周长为10，DE∥BC。求△ABC的周长。
Construction (CDL):
Shape(EA,AD,DE)
Shape(CE,ED,DB,BC)
Collinear(AEC)
Collinear(ADB)
Conditions (CDL):
Equal(LengthOfLine(AD),2)
Equal(LengthOfLine(DB),4)
Equal(PerimeterOfTriangle(ADE),10)
ParallelBetweenLine(DE,BC)
Goal: Value(PerimeterOfTriangle(ABC))
Answer: 30
Theorem sequence:
line_addition(1,AD,DB)
parallel_property_corresponding_angle(1,DE,BC,A)
similar_triangle_judgment_aa(1,EAD,CAB)
similar_triangle_property_line_ratio(1,EAD,CAB)
similar_triangle_property_perimeter_ratio(1,EAD,CAB)
Problem: As shown in the diagram, AB=3, BC=5, ∠ADB=90°, ∠BAC=90°, ∠CDA=90°. Find the length of line CD.
问题：如图所示，AB=3，BC=5，∠ADB=90°，∠BAC=90°，∠CDA=90°。求直线CD的长度。
Construction (CDL):
Shape(AC,CD,DA)
Shape(AD,DB,BA)
Collinear(CDB)
Conditions (CDL):
Equal(LengthOfLine(AB),3)
Equal(LengthOfLine(BC),5)
Equal(MeasureOfAngle(ADB),90)
Equal(MeasureOfAngle(BAC),90)
Equal(MeasureOfAngle(CDA),90)
Goal: Value(LengthOfLine(CD))
Answer: 16/5
Theorem sequence:
line_addition(1,CD,DB)
right_triangle_judgment_angle(1,CDA)
right_triangle_judgment_angle(1,ADB)
right_triangle_judgment_angle(1,BAC)
right_triangle_property_pythagorean(1,CDA)
right_triangle_property_pythagorean(1,ADB)
right_triangle_property_pythagorean(1,BAC)
Problem: As shown in the diagram, AC=12, BC=5, DE=5, ∠ACB=90°, ∠AFD=90°, ∠AGE=90°. Find the length of line DF.
问题：如图所示，AC=12，BC=5，DE=5，∠ACB=90°，∠AFD=90°，∠AGE=90°。求直线DF的长度。
Construction (CDL):
Shape(FD,DE,EF)
Shape(EA,AG,GE)
Shape(BF,FE,EG,GC,CB)
Collinear(DEG)
Collinear(AEFB)
Collinear(AGC)
Conditions (CDL):
Equal(LengthOfLine(AC),12)
Equal(LengthOfLine(BC),5)
Equal(LengthOfLine(DE),5)
Equal(MeasureOfAngle(ACB),90)
Equal(MeasureOfAngle(AFD),90)
Equal(MeasureOfAngle(AGE),90)
Goal: Value(LengthOfLine(DF))
Answer: 60/13
Theorem sequence:
vertical_angle(1,DEF,GEA)
right_triangle_judgment_angle(1,ACB)
right_triangle_property_pythagorean(1,ACB)
mirror_similar_triangle_judgment_aa(1,DEF,AGE)
mirror_similar_triangle_property_angle_equal(1,DEF,AGE)
mirror_similar_triangle_judgment_aa(1,BAC,EFD)
mirror_similar_triangle_property_line_ratio(1,BAC,EFD)
mirror_similar_triangle_property_line_ratio(1,CBA,FDE)
Problem: As shown in the diagram, the ratio of similar triangles OBC and ODA is 2, AD∥BC. Find the ratio of the area of triangle AOD to the area of triangle OBC.
问题：如图所示，△OBC和△ODA的相似比为2，AD∥BC。求三角形AOD的面积与三角形OBC的面积之比。
Construction (CDL):
Shape(DA,AO,OD)
Shape(OA,AB,BO)
Shape(OB,BC,CO)
Shape(DO,OC,CD)
Collinear(AOC)
Collinear(DOB)
Conditions (CDL):
Equal(RatioOfSimilarTriangle(OBC,ODA),2)
ParallelBetweenLine(AD,BC)
Goal: Value(Div(AreaOfTriangle(AOD),AreaOfTriangle(OBC)))
Answer: 1/4
Theorem sequence:
parallel_property_alternate_interior_angle(1,AD,BC)
parallel_property_alternate_interior_angle(2,AD,BC)
similar_triangle_judgment_aa(1,OBC,ODA)
similar_triangle_property_area_square_ratio(1,OBC,ODA)
Problem: As shown in the diagram, AB=12, AE=4, BC=10, EC=4, ∠DEA=∠ABC. Find the ratio of mirror similar △ADE and △ABC.
问题：如图所示，AB=12，AE=4，BC=10，EC=4，∠DEA=∠ABC。求三角形ADE和三角形ABC的镜像相似比。
Construction (CDL):
Shape(AD,DE,EA)
Shape(ED,DB,BC,CE)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(LengthOfLine(AB),12)
Equal(LengthOfLine(AE),4)
Equal(LengthOfLine(BC),10)
Equal(LengthOfLine(EC),4)
Equal(MeasureOfAngle(DEA),MeasureOfAngle(ABC))
Goal: Value(RatioOfMirrorSimilarTriangle(ADE,ABC))
Answer: 1/3
Theorem sequence:
mirror_similar_triangle_judgment_aa(1,DEA,CAB)
mirror_similar_triangle_property_line_ratio(1,DEA,CAB)
Problem: As shown in the diagram, Div(LengthOfLine(AE)=LengthOfLine(ED)), CD=3, DABC is a parallelogram. Find the length of line AF.
问题：如图所示，Div(LengthOfLine(AE)=LengthOfLine(ED))，CD=3，AD和BC是▱DABC的一组对边。求直线AF的长度。
Construction (CDL):
Shape(FA,AE,EF)
Shape(EA,AB,BC,CE)
Shape(DE,EC,CD)
Collinear(FAB)
Collinear(FEC)
Collinear(AED)
Conditions (CDL):
Equal(Div(LengthOfLine(AE),LengthOfLine(ED)),2)
Equal(LengthOfLine(CD),3)
Parallelogram(DABC)
Goal: Value(LengthOfLine(AF))
Answer: 6
Theorem sequence:
parallel_property_collinear_extend(2,BA,CD,F)
vertical_angle(1,AEF,DEC)
parallel_property_alternate_interior_angle(1,AF,CD)
similar_triangle_judgment_aa(1,FAE,CDE)
similar_triangle_property_line_ratio(1,FAE,CDE)
similar_triangle_property_line_ratio(1,EFA,ECD)
Problem: As shown in the diagram, AB=10, BC=26, ∠BAC=90°. Find the ratio of the length of line AB to the length of line AC.
问题：如图所示，AB=10，BC=26，∠BAC=90°。求直线AB的长度与直线AC的长度之比。
Construction (CDL):
Shape(BA,AC,CB)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(LengthOfLine(BC),26)
Equal(MeasureOfAngle(BAC),90)
Goal: Value(Div(LengthOfLine(AB),LengthOfLine(AC)))
Answer: 5/12
Theorem sequence:
right_triangle_judgment_angle(1,BAC)
right_triangle_property_pythagorean(1,BAC)
Problem: As shown in the diagram, BC=8, ∠ABC=150°, ∠BEC=90°. Find the length of line EC.
问题：如图所示，BC=8，∠ABC=150°，∠BEC=90°。求直线EC的长度。
Construction (CDL):
Shape(CB,BE,EC)
Shape(AB,BC)
Collinear(ABE)
Conditions (CDL):
Equal(LengthOfLine(BC),8)
Equal(MeasureOfAngle(ABC),150)
Equal(MeasureOfAngle(BEC),90)
Goal: Value(LengthOfLine(EC))
Answer: 4
Theorem sequence:
flat_angle(1,ABE)
angle_addition(1,ABC,CBE)
sine_theorem(1,CBE)
Problem: As shown in the diagram, AB=10, BC=5, ∠ACB=90°. Find the ratio of the length of line BC to the length of line AC.
问题：如图所示，AB=10，BC=5，∠ACB=90°。求直线BC的长度与直线AC的长度之比。
Construction (CDL):
Shape(BA,AC,CB)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(LengthOfLine(BC),5)
Equal(MeasureOfAngle(ACB),90)
Goal: Value(Div(LengthOfLine(BC),LengthOfLine(AC)))
Answer: sqrt(3)/3
Theorem sequence:
right_triangle_judgment_angle(1,ACB)
right_triangle_property_pythagorean(1,ACB)
Problem: As shown in the diagram, Div(LengthOfLine(AB)=LengthOfLine(BC)), AE=EC, BD=6, DC=6, ∠ABC=90°. Find the length of line EC.
问题：如图所示，Div(LengthOfLine(AB)=LengthOfLine(BC))，AE=EC，BD=6，DC=6，∠ABC=90°。求直线EC的长度。
Construction (CDL):
Shape(AB,BD,DE,EA)
Shape(ED,DC,CE)
Collinear(AEC)
Collinear(BDC)
Conditions (CDL):
Equal(Div(LengthOfLine(AB),LengthOfLine(BC)),1/2)
Equal(LengthOfLine(AE),LengthOfLine(EC))
Equal(LengthOfLine(BD),6)
Equal(LengthOfLine(DC),6)
Equal(MeasureOfAngle(ABC),90)
Goal: Value(LengthOfLine(EC))
Answer: 3*sqrt(5)
Theorem sequence:
line_addition(1,AE,EC)
line_addition(1,BD,DC)
right_triangle_judgment_angle(1,ABC)
right_triangle_property_pythagorean(1,ABC)
Problem: As shown in the diagram, Div(LengthOfLine(DF)=LengthOfLine(CF)), AE=8, DF=8, EF=6, ∠ABC=45°, ∠BEA=90°, ∠DFC=90°. Find the length of line BC.
问题：如图所示，Div(LengthOfLine(DF)=LengthOfLine(CF))，AE=8，DF=8，EF=6，∠ABC=45°，∠BEA=90°，∠DFC=90°。求直线BC的长度。
Construction (CDL):
Shape(AB,BE,EA)
Shape(DA,AE,EF,FD)
Shape(DF,FC,CD)
Collinear(BEFC)
Conditions (CDL):
Equal(Div(LengthOfLine(DF),LengthOfLine(CF)),1/3)
Equal(LengthOfLine(AE),8)
Equal(LengthOfLine(DF),8)
Equal(LengthOfLine(EF),6)
Equal(MeasureOfAngle(ABC),45)
Equal(MeasureOfAngle(BEA),90)
Equal(MeasureOfAngle(DFC),90)
Goal: Value(LengthOfLine(BC))
Answer: 38
Theorem sequence:
triangle_property_angle_sum(1,ABE)
sine_theorem(1,EAB)
line_addition(1,BE,EF)
line_addition(1,BF,FC)
Problem: As shown in the diagram, Div(LengthOfLine(BO)=LengthOfLine(AB)), CD=1, ∠ACO=60°, ∠DOB=90°, AC is parallel to BD. Find the length of line AB.
问题：如图所示，Div(LengthOfLine(BO)=LengthOfLine(AB))，CD=1，∠ACO=60°，∠DOB=90°，AC平行于BD。求直线AB的长度。
Construction (CDL):
Shape(OB,BD,DO)
Shape(BA,AC,CD,DB)
Collinear(CDO)
Collinear(ABO)
Conditions (CDL):
Equal(Div(LengthOfLine(BO),LengthOfLine(AB)),Div(LengthOfLine(DO),LengthOfLine(CD)))
Equal(LengthOfLine(CD),1)
Equal(MeasureOfAngle(ACO),60)
Equal(MeasureOfAngle(DOB),90)
ParallelBetweenLine(AC,BD)
Goal: Value(LengthOfLine(AB))
Answer: sqrt(3)
Theorem sequence:
triangle_property_angle_sum(1,OAC)
line_addition(1,CD,DO)
line_addition(1,AB,BO)
sine_theorem(1,OAC)
sine_theorem(1,ACO)
sine_theorem(1,COA)
Problem: As shown in the diagram, BC=38, ∠ABC=90°, ∠CAB=30°. Find the length of line AC.
问题：如图所示，BC=38，∠ABC=90°，∠CAB=30°。求直线AC的长度。
Construction (CDL):
Shape(CA,AB,BC)
Conditions (CDL):
Equal(LengthOfLine(BC),38)
Equal(MeasureOfAngle(ABC),90)
Equal(MeasureOfAngle(CAB),30)
Goal: Value(LengthOfLine(AC))
Answer: 76
Theorem sequence:
sine_theorem(1,CAB)
Problem: As shown in the diagram, AB=Add(LengthOfLine(AF),LengthOfLine(CD)), AC=20, CD=2, ∠AFC=90°, ∠EAC=30°, ∠EAF=90°. Find the length of line AB.
问题：如图所示，AB=Add(LengthOfLine(AF),LengthOfLine(CD))，AC=20，CD=2，∠AFC=90°，∠EAC=30°，∠EAF=90°。求直线AB的长度。
Construction (CDL):
Shape(AF,FC,CA)
Shape(FB,BD,DC,CF)
Shape(EA,AC)
Collinear(AFB)
Conditions (CDL):
Equal(LengthOfLine(AB),Add(LengthOfLine(AF),LengthOfLine(CD)))
Equal(LengthOfLine(AC),20)
Equal(LengthOfLine(CD),2)
Equal(MeasureOfAngle(AFC),90)
Equal(MeasureOfAngle(EAC),30)
Equal(MeasureOfAngle(EAF),90)
Goal: Value(LengthOfLine(AB))
Answer: 12
Theorem sequence:
angle_addition(1,EAC,CAF)
triangle_property_angle_sum(1,AFC)
sine_theorem(1,AFC)
Problem: As shown in the diagram, AC=2, ∠ACB=90°, ∠BAC=30°. Find the length of line AB.
问题：如图所示，AC=2，∠ACB=90°，∠BAC=30°。求直线AB的长度。
Construction (CDL):
Shape(BA,AC,CB)
Conditions (CDL):
Equal(LengthOfLine(AC),2)
Equal(MeasureOfAngle(ACB),90)
Equal(MeasureOfAngle(BAC),30)
Goal: Value(LengthOfLine(AB))
Answer: 4*sqrt(3)/3
Theorem sequence:
triangle_property_angle_sum(1,BAC)
sine_theorem(1,ACB)
Problem: As shown in the diagram, BC=35, ∠ACB=90°, ∠BAC=30°. Find the length of line AB.
问题：如图所示，BC=35，∠ACB=90°，∠BAC=30°。求直线AB的长度。
Construction (CDL):
Shape(BA,AC,CB)
Conditions (CDL):
Equal(LengthOfLine(BC),35)
Equal(MeasureOfAngle(ACB),90)
Equal(MeasureOfAngle(BAC),30)
Goal: Value(LengthOfLine(AB))
Answer: 70
Theorem sequence:
sine_theorem(1,BAC)
Problem: As shown in the diagram, CD=8, ∠ABC=60°, ∠ADC=90°, ∠CAB=60°, ∠CDB=90°. Find the length of line AB.
问题：如图所示，CD=8，∠ABC=60°，∠ADC=90°，∠CAB=60°，∠CDB=90°。求直线AB的长度。
Construction (CDL):
Shape(CA,AD,DC)
Shape(CD,DB,BC)
Collinear(ADB)
Conditions (CDL):
Equal(LengthOfLine(CD),8)
Equal(MeasureOfAngle(ABC),60)
Equal(MeasureOfAngle(ADC),90)
Equal(MeasureOfAngle(CAB),60)
Equal(MeasureOfAngle(CDB),90)
Goal: Value(LengthOfLine(AB))
Answer: 16*sqrt(3)/3
Theorem sequence:
triangle_property_angle_sum(1,CAB)
sine_theorem(1,CAD)
sine_theorem(1,ABC)
Problem: As shown in the diagram, AC=3, ∠ACB=90°, ∠BAC=30°. Find the length of line AB.
问题：如图所示，AC=3，∠ACB=90°，∠BAC=30°。求直线AB的长度。
Construction (CDL):
Shape(BA,AC,CB)
Conditions (CDL):
Equal(LengthOfLine(AC),3)
Equal(MeasureOfAngle(ACB),90)
Equal(MeasureOfAngle(BAC),30)
Goal: Value(LengthOfLine(AB))
Answer: 2*sqrt(3)
Theorem sequence:
triangle_property_angle_sum(1,BAC)
sine_theorem(1,ACB)
Problem: As shown in the diagram, AB=10, CA=5, ∠BCA=90°. Find the ratio of the length of line CA to the length of line BC.
问题：如图所示，AB=10，CA=5，∠BCA=90°。求直线CA的长度与直线BC的长度之比。
Construction (CDL):
Shape(AB,BC,CA)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(LengthOfLine(CA),5)
Equal(MeasureOfAngle(BCA),90)
Goal: Value(Div(LengthOfLine(CA),LengthOfLine(BC)))
Answer: sqrt(3)/3
Theorem sequence:
right_triangle_judgment_angle(1,BCA)
right_triangle_property_pythagorean(1,BCA)
Problem: As shown in the diagram, AB=80, ∠ACB=90°, ∠BAC=30°. Find the length of line BC.
问题：如图所示，AB=80，∠ACB=90°，∠BAC=30°。求直线BC的长度。
Construction (CDL):
Shape(BA,AC,CB)
Conditions (CDL):
Equal(LengthOfLine(AB),80)
Equal(MeasureOfAngle(ACB),90)
Equal(MeasureOfAngle(BAC),30)
Goal: Value(LengthOfLine(BC))
Answer: 40
Theorem sequence:
sine_theorem(1,BAC)
Problem: As shown in the diagram, Div(LengthOfLine(BC)=LengthOfLine(AC)), BC=1200, ∠ACB=90°. Find the length of line AB.
问题：如图所示，Div(LengthOfLine(BC)=LengthOfLine(AC))，BC=1200，∠ACB=90°。求直线AB的长度。
Construction (CDL):
Shape(BA,AC,CB)
Conditions (CDL):
Equal(Div(LengthOfLine(BC),LengthOfLine(AC)),3/4)
Equal(LengthOfLine(BC),1200)
Equal(MeasureOfAngle(ACB),90)
Goal: Value(LengthOfLine(AB))
Answer: 2000
Theorem sequence:
right_triangle_judgment_angle(1,ACB)
right_triangle_property_pythagorean(1,ACB)