SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, Div(LengthOfLine(EO)=LengthOfLine(AD)), AF=3, AG=4, DF=6, E bisects segment AB, AF∥EO, ABCD is a ▱. Find the length of line AC.
问题：如图所示，Div(LengthOfLine(EO)=LengthOfLine(AD))，AF=3，AG=4，DF=6，E平分线段AB，AF平行于EO，四边形ABCD是平行四边形。求直线AC的长度。
Construction (CDL):
Shape(DF,FG,GO,OC,CD)
Shape(FA,AG,GF)
Shape(GE,EO,OG)
Shape(GA,AE,EG)
Shape(OE,EB,BC,CO)
Collinear(DFA)
Collinear(AEB)
Collinear(FGE)
Collinear(AGOC)
Conditions (CDL):
Equal(Div(LengthOfLine(EO),LengthOfLine(AD)),1/2)
Equal(LengthOfLine(AF),3)
Equal(LengthOfLine(AG),4)
Equal(LengthOfLine(DF),6)
IsMidpointOfLine(E,AB)
ParallelBetweenLine(AF,EO)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(AC))
Answer: 20
Theorem sequence:
line_addition(1,DF,FA)
line_addition(1,AG,GO)
parallel_property_alternate_interior_angle(1,AF,EO)
parallel_property_alternate_interior_angle(2,AF,EO)
parallel_property_alternate_interior_angle(2,BA,CD)
similar_triangle_judgment_aa(1,GFA,GEO)
similar_triangle_judgment_aa(1,DAC,EOA)
similar_triangle_property_line_ratio(1,GFA,GEO)
similar_triangle_property_line_ratio(1,FAG,EOG)
similar_triangle_property_line_ratio(1,DAC,EOA)
similar_triangle_property_line_ratio(1,CDA,AEO)
Problem: As shown in the diagram, Div(LengthOfLine(AE)=LengthOfLine(EF)), BC=10, ∠CAB=∠DOB, O is the center of ⊙O, AE⊥FE, BD is perpendicular to OD. Find the length of line OD.
问题：如图所示，Div(LengthOfLine(AE)=LengthOfLine(EF))，BC=10，∠CAB=∠DOB，O是圆O的圆心，AE⊥FE，BD垂直于OD。求直线OD的长度。
Construction (CDL):
Shape(AE,EF,FA)
Shape(EB,BO,OC,CF,FE)
Shape(OB,BD,DO)
Shape(OD,DC,CO)
Shape(BA,OAB)
Shape(CB,OBC)
Shape(AC,OCA)
Collinear(AEB)
Collinear(AFC)
Collinear(BDC)
Cocircular(O,ABC)
Conditions (CDL):
Equal(Div(LengthOfLine(AE),LengthOfLine(EF)),3/4)
Equal(LengthOfLine(BC),10)
Equal(MeasureOfAngle(CAB),MeasureOfAngle(DOB))
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AE,FE)
PerpendicularBetweenLine(BD,OD)
Goal: Value(LengthOfLine(OD))
Answer: 15/4
Theorem sequence:
circle_property_chord_perpendicular_bisect_chord(1,O,OD,BC)
line_addition(1,BD,DC)
mirror_similar_triangle_judgment_aa(1,BDO,FAE)
mirror_similar_triangle_property_line_ratio(1,BDO,FAE)
mirror_similar_triangle_property_line_ratio(1,OBD,AEF)
Problem: As shown in the diagram, Div(LengthOfLine(AE)=LengthOfLine(EC)), AD=6, DE is parallel to BC. Find the length of line AB.
问题：如图所示，Div(LengthOfLine(AE)=LengthOfLine(EC))，AD=6，DE平行于BC。求直线AB的长度。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(Div(LengthOfLine(AE),LengthOfLine(EC)),2/3)
Equal(LengthOfLine(AD),6)
ParallelBetweenLine(DE,BC)
Goal: Value(LengthOfLine(AB))
Answer: 15
Theorem sequence:
line_addition(1,AE,EC)
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)
Problem: As shown in the diagram, the area of triangle AEF is 4, Div(LengthOfLine(AE)=LengthOfLine(EB)), quadrilateral DABC is a ▱. Find the area of △CDF.
问题：如图所示，△AEF的面积为4，Div(LengthOfLine(AE)=LengthOfLine(EB))，DC和AB是▱DABC的一组对边。求三角形CDF的面积。
Construction (CDL):
Shape(DA,AF,FD)
Shape(DF,FC,CD)
Shape(AE,EF,FA)
Shape(FE,EB,BC,CF)
Collinear(DFE)
Collinear(AFC)
Collinear(AEB)
Conditions (CDL):
Equal(AreaOfTriangle(AEF),4)
Equal(Div(LengthOfLine(AE),LengthOfLine(EB)),1/2)
Parallelogram(DABC)
Goal: Value(AreaOfTriangle(CDF))
Answer: 36
Theorem sequence:
line_addition(1,AE,EB)
parallelogram_property_opposite_line_equal(1,ABCD)
parallel_property_alternate_interior_angle(2,DC,AB)
vertical_angle(1,DFC,EFA)
similar_triangle_judgment_aa(1,DFC,EFA)
similar_triangle_property_line_ratio(1,FCD,FAE)
similar_triangle_property_area_square_ratio(1,DFC,EFA)
Problem: As shown in the diagram, AB=10, AC=8, AE=5, ∠ADE=90°, ∠BCA=90°. Find the length of line AD.
问题：如图所示，AB=10，AC=8，AE=5，∠ADE=90°，∠BCA=90°。求直线AD的长度。
Construction (CDL):
Shape(AD,DE,EA)
Shape(ED,DB,BC,CE)
Collinear(AEC)
Collinear(ADB)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(LengthOfLine(AC),8)
Equal(LengthOfLine(AE),5)
Equal(MeasureOfAngle(ADE),90)
Equal(MeasureOfAngle(BCA),90)
Goal: Value(LengthOfLine(AD))
Answer: 4
Theorem sequence:
mirror_similar_triangle_judgment_aa(1,EAD,BCA)
mirror_similar_triangle_property_line_ratio(1,EAD,BCA)
mirror_similar_triangle_property_line_ratio(1,DEA,CAB)
Problem: As shown in the diagram, Div(LengthOfLine(FD)=LengthOfLine(FC)), CE=2, AD and BC are opposite sides of the parallelogram ABCD. Find the length of line AD.
问题：如图所示，Div(LengthOfLine(FD)=LengthOfLine(FC))，CE=2，ABCD是▱。求直线AD的长度。
Construction (CDL):
Shape(BC,CF,FA,AB)
Shape(AF,FD,DA)
Shape(FC,CE,EF)
Collinear(AFE)
Collinear(DFC)
Collinear(BCE)
Conditions (CDL):
Equal(Div(LengthOfLine(FD),LengthOfLine(FC)),2)
Equal(LengthOfLine(CE),2)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(AD))
Answer: 4
Theorem sequence:
parallel_property_collinear_extend(1,CB,DA,E)
parallel_property_alternate_interior_angle(1,EC,DA)
parallel_property_alternate_interior_angle(2,EC,DA)
similar_triangle_judgment_aa(1,FDA,FCE)
similar_triangle_property_line_ratio(1,FDA,FCE)
similar_triangle_property_line_ratio(1,AFD,EFC)
Problem: As shown in the diagram, the area of △ADE is 4, Div(LengthOfLine(AD)=LengthOfLine(AB)), DE∥BC. Find the area of triangle ABC.
问题：如图所示，△ADE的面积为4，Div(LengthOfLine(AD)=LengthOfLine(AB))，DE∥BC。求三角形ABC的面积。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(AreaOfTriangle(ADE),4)
Equal(Div(LengthOfLine(AD),LengthOfLine(AB)),1/3)
ParallelBetweenLine(DE,BC)
Goal: Value(AreaOfTriangle(ABC))
Answer: 36
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,CAB,EAD)
similar_triangle_property_area_square_ratio(1,ABC,ADE)
Problem: As shown in the diagram, the area of △ADE is 2, Div(LengthOfLine(DE)=LengthOfLine(BC)), DE∥BC. Find the area of triangle ABC minus the area of △ADE.
问题：如图所示，三角形ADE的面积为2，Div(LengthOfLine(DE)=LengthOfLine(BC))，DE∥BC。求△ABC的面积减去△ADE的面积。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(AreaOfTriangle(ADE),2)
Equal(Div(LengthOfLine(DE),LengthOfLine(BC)),1/2)
ParallelBetweenLine(DE,BC)
Goal: Value(Sub(AreaOfTriangle(ABC),AreaOfTriangle(ADE)))
Answer: 6
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,ABC,ADE)
similar_triangle_property_area_square_ratio(1,ABC,ADE)
Problem: As shown in the diagram, AB=10, AE=5, DE=6, ∠DEA=∠ABC. Find the length of line BC.
问题：如图所示，AB=10，AE=5，DE=6，∠DEA=∠ABC。求直线BC的长度。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(LengthOfLine(AE),5)
Equal(LengthOfLine(DE),6)
Equal(MeasureOfAngle(DEA),MeasureOfAngle(ABC))
Goal: Value(LengthOfLine(BC))
Answer: 12
Theorem sequence:
mirror_similar_triangle_judgment_aa(1,DEA,CAB)
mirror_similar_triangle_property_line_ratio(1,DEA,CAB)
mirror_similar_triangle_property_line_ratio(1,ADE,ABC)
Problem: As shown in the diagram, Div(LengthOfLine(AD)=LengthOfLine(DE)), BD=4, ∠DCA=∠DBE. Find the length of line DC.
问题：如图所示，Div(LengthOfLine(AD)=LengthOfLine(DE))，BD=4，∠DCA=∠DBE。求直线DC的长度。
Construction (CDL):
Shape(AB,BD,DA)
Shape(AD,DC,CA)
Shape(DB,BE,ED)
Collinear(ADE)
Collinear(BDC)
Conditions (CDL):
Equal(Div(LengthOfLine(AD),LengthOfLine(DE)),3/5)
Equal(LengthOfLine(BD),4)
Equal(MeasureOfAngle(DCA),MeasureOfAngle(DBE))
Goal: Value(LengthOfLine(DC))
Answer: 12/5
Theorem sequence:
vertical_angle(1,ADC,EDB)
similar_triangle_judgment_aa(1,EDB,ADC)
similar_triangle_property_line_ratio(1,EDB,ADC)
similar_triangle_property_line_ratio(1,BED,CAD)
Problem: As shown in the diagram, the area of △FDE is 3, Div(LengthOfLine(CD)=LengthOfLine(DE)), ABCD is a ▱. Find the area of △FAB.
问题：如图所示，三角形FDE的面积为3，Div(LengthOfLine(CD)=LengthOfLine(DE))，四边形ABCD是▱。求三角形FAB的面积。
Construction (CDL):
Shape(AB,BF,FA)
Shape(FD,DE,EF)
Shape(FB,BC,CD,DF)
Collinear(EFB)
Collinear(EDC)
Collinear(AFD)
Conditions (CDL):
Equal(AreaOfTriangle(FDE),3)
Equal(Div(LengthOfLine(CD),LengthOfLine(DE)),2)
Parallelogram(ABCD)
Goal: Value(AreaOfTriangle(FAB))
Answer: 12
Theorem sequence:
parallel_property_collinear_extend(1,DC,AB,E)
parallel_property_alternate_interior_angle(1,ED,AB)
parallel_property_alternate_interior_angle(2,ED,AB)
similar_triangle_judgment_aa(1,FAB,FDE)
parallelogram_property_opposite_line_equal(1,ABCD)
similar_triangle_property_line_ratio(1,FAB,FDE)
similar_triangle_property_area_square_ratio(1,FAB,FDE)
Problem: As shown in the diagram, Add(MeasureOfAngle(ABC)=MeasureOfAngle(CDE)), AD=3, AE=2, CD=1. Find the length of line BE.
问题：如图所示，Add(MeasureOfAngle(ABC)=MeasureOfAngle(CDE))，AD=3，AE=2，CD=1。求直线BE的长度。
Construction (CDL):
Shape(AE,ED,DA)
Shape(EB,BC,CD,DE)
Collinear(AEB)
Collinear(ADC)
Conditions (CDL):
Equal(Add(MeasureOfAngle(ABC),MeasureOfAngle(CDE)),Add(MeasureOfAngle(BCA),MeasureOfAngle(DEB)))
Equal(LengthOfLine(AD),3)
Equal(LengthOfLine(AE),2)
Equal(LengthOfLine(CD),1)
Goal: Value(LengthOfLine(BE))
Answer: 4
Theorem sequence:
quadrilateral_property_angle_sum(1,EBCD)
adjacent_complementary_angle(1,AED,DEB)
adjacent_complementary_angle(1,CDE,EDA)
mirror_similar_triangle_judgment_aa(1,AED,ABC)
mirror_similar_triangle_property_line_ratio(1,EDA,CAB)
mirror_similar_triangle_property_line_ratio(1,DAE,BCA)
line_addition(1,AE,EB)
line_addition(1,AD,DC)
Problem: As shown in the diagram, the area of △ADE is 4, Div(LengthOfLine(AD)=LengthOfLine(AB)), DE∥BC. Find the area of △ABC.
问题：如图所示，△ADE的面积为4，Div(LengthOfLine(AD)=LengthOfLine(AB))，DE∥BC。求△ABC的面积。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(AreaOfTriangle(ADE),4)
Equal(Div(LengthOfLine(AD),LengthOfLine(AB)),1/2)
ParallelBetweenLine(DE,BC)
Goal: Value(AreaOfTriangle(ABC))
Answer: 16
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,CAB,EAD)
similar_triangle_property_area_square_ratio(1,ABC,ADE)
Problem: As shown in the diagram, AE=5, BC=8, DE=4, ∠ABC=∠DEA. Find the length of line AB.
问题：如图所示，AE=5，BC=8，DE=4，∠ABC=∠DEA。求直线AB的长度。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Collinear(AEC)
Collinear(ADB)
Conditions (CDL):
Equal(LengthOfLine(AE),5)
Equal(LengthOfLine(BC),8)
Equal(LengthOfLine(DE),4)
Equal(MeasureOfAngle(ABC),MeasureOfAngle(DEA))
Goal: Value(LengthOfLine(AB))
Answer: 10
Theorem sequence:
mirror_similar_triangle_judgment_aa(1,DEA,CAB)
mirror_similar_triangle_property_line_ratio(1,DEA,CAB)
mirror_similar_triangle_property_line_ratio(1,ADE,ABC)
Problem: As shown in the diagram, Div(LengthOfLine(DE)=LengthOfLine(EA)), EF=3, EF∥AB, AD and BC are opposite sides of the parallelogram ABCD. Find the length of line CD.
问题：如图所示，Div(LengthOfLine(DE)=LengthOfLine(EA))，EF=3，EF∥AB，AD和BC是平行四边形ABCD的一组对边。求直线CD的长度。
Construction (CDL):
Shape(DE,EF,FD)
Shape(EA,AB,BF,FE)
Shape(DF,FB,BC,CD)
Collinear(DEA)
Collinear(DFB)
Conditions (CDL):
Equal(Div(LengthOfLine(DE),LengthOfLine(EA)),3/4)
Equal(LengthOfLine(EF),3)
ParallelBetweenLine(EF,AB)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(CD))
Answer: 7
Theorem sequence:
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,DEF,DAB)
similar_triangle_property_line_ratio(1,DEF,DAB)
similar_triangle_property_line_ratio(1,FDE,BDA)
parallelogram_property_opposite_line_equal(1,ABCD)
Problem: As shown in the diagram, Div(LengthOfLine(AE)=LengthOfLine(EC)), DE=4, DE∥BC. Find the length of line BC.
问题：如图所示，Div(LengthOfLine(AE)=LengthOfLine(EC))，DE=4，DE平行于BC。求直线BC的长度。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(Div(LengthOfLine(AE),LengthOfLine(EC)),2/3)
Equal(LengthOfLine(DE),4)
ParallelBetweenLine(DE,BC)
Goal: Value(LengthOfLine(BC))
Answer: 10
Theorem sequence:
line_addition(1,AE,EC)
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,DEA,BCA)
Problem: As shown in the diagram, AD=5, BD=2, AD bisects ∠CAB. Find the length of line AE.
问题：如图所示，AD=5，BD=2，AD是∠CAB的角平分线。求直线AE的长度。
Construction (CDL):
Shape(BA,OAB)
Shape(DB,OBD)
Shape(CE,ED,ODC)
Shape(AC,OCA)
Shape(AB,BE,EA)
Shape(AE,EC,CA)
Shape(EB,BD,DE)
Collinear(BEC)
Collinear(AED)
Cocircular(O,BDCA)
Conditions (CDL):
Equal(LengthOfLine(AD),5)
Equal(LengthOfLine(BD),2)
IsBisectorOfAngle(AD,CAB)
Goal: Value(LengthOfLine(AE))
Answer: 21/5
Theorem sequence:
arc_property_circumference_angle_external(1,ODC,B)
arc_property_circumference_angle_external(1,ODC,A)
mirror_similar_triangle_judgment_aa(1,EBD,BDA)
mirror_similar_triangle_property_line_ratio(1,EBD,BDA)
mirror_similar_triangle_property_line_ratio(1,BDE,ABD)
line_addition(1,AE,ED)
Problem: As shown in the diagram, AB=2, CD=3, AB is parallel to CD. Find the ratio of the area of △AOB to the area of △OCD.
问题：如图所示，AB=2，CD=3，AB平行于CD。求三角形AOB的面积与三角形OCD的面积之比。
Construction (CDL):
Shape(AO,OB,BA)
Shape(OC,CD,DO)
Collinear(AOD)
Collinear(BOC)
Conditions (CDL):
Equal(LengthOfLine(AB),2)
Equal(LengthOfLine(CD),3)
ParallelBetweenLine(AB,CD)
Goal: Value(Div(AreaOfTriangle(AOB),AreaOfTriangle(OCD)))
Answer: 4/9
Theorem sequence:
parallel_property_alternate_interior_angle(1,AB,CD)
parallel_property_alternate_interior_angle(2,AB,CD)
similar_triangle_judgment_aa(1,OBA,OCD)
similar_triangle_property_line_ratio(1,OBA,OCD)
similar_triangle_property_area_square_ratio(1,OBA,OCD)
Problem: As shown in the diagram, AG=2, BF=4, ∠GEF=90°, E bisects segment AB, quadrilateral ABCD is a square. Find the length of line GF.
问题：如图所示，AG=2，BF=4，∠GEF=90°，E平分线段AB，ABCD是正方形。求直线GF的长度。
Construction (CDL):
Shape(DG,GF,FC,CD)
Shape(FG,GE,EF)
Shape(GA,AE,EG)
Shape(FE,EB,BF)
Collinear(DGA)
Collinear(AEB)
Collinear(CFB)
Conditions (CDL):
Equal(LengthOfLine(AG),2)
Equal(LengthOfLine(BF),4)
Equal(MeasureOfAngle(GEF),90)
IsMidpointOfLine(E,AB)
Square(ABCD)
Goal: Value(LengthOfLine(GF))
Answer: 6
Theorem sequence:
triangle_property_angle_sum(1,GAE)
angle_addition(1,AEG,GEF)
adjacent_complementary_angle(1,AEF,FEB)
similar_triangle_judgment_aa(1,EGA,FEB)
similar_triangle_property_line_ratio(1,EGA,FEB)
similar_triangle_property_line_ratio(1,GAE,EBF)
similar_triangle_property_line_ratio(1,AEG,BFE)
right_triangle_judgment_angle(1,GEF)
right_triangle_judgment_angle(1,GAE)
right_triangle_property_pythagorean(1,GEF)
right_triangle_property_pythagorean(1,GAE)
Problem: As shown in the diagram, the area of △ADE is 1, Div(LengthOfLine(AD)=LengthOfLine(DB)), DE∥BC. Find the area of triangle ABC minus the area of triangle ADE.
问题：如图所示，△ADE的面积为1，Div(LengthOfLine(AD)=LengthOfLine(DB))，DE∥BC。求△ABC的面积减去三角形ADE的面积。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(AreaOfTriangle(ADE),1)
Equal(Div(LengthOfLine(AD),LengthOfLine(DB)),1/2)
ParallelBetweenLine(DE,BC)
Goal: Value(Sub(AreaOfTriangle(ABC),AreaOfTriangle(ADE)))
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,CAB,EAD)
similar_triangle_property_area_square_ratio(1,ABC,ADE)
Problem: As shown in the diagram, AE=5, BC=8, DE=4, ∠ABC=∠DEA. Find the length of line AB.
问题：如图所示，AE=5，BC=8，DE=4，∠ABC=∠DEA。求直线AB的长度。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(LengthOfLine(AE),5)
Equal(LengthOfLine(BC),8)
Equal(LengthOfLine(DE),4)
Equal(MeasureOfAngle(ABC),MeasureOfAngle(DEA))
Goal: Value(LengthOfLine(AB))
Answer: 10
Theorem sequence:
mirror_similar_triangle_judgment_aa(1,DEA,CAB)
mirror_similar_triangle_property_line_ratio(1,DEA,CAB)
mirror_similar_triangle_property_line_ratio(1,ADE,ABC)
Problem: As shown in the diagram, AD=1, DB=2, DE∥BC. Find the ratio of the length of line DE to the length of line BC.
问题：如图所示，AD=1，DB=2，DE∥BC。求直线DE的长度与直线BC的长度之比。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(LengthOfLine(AD),1)
Equal(LengthOfLine(DB),2)
ParallelBetweenLine(DE,BC)
Goal: Value(Div(LengthOfLine(DE),LengthOfLine(BC)))
Answer: 1/3
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, AI=3, BI=IG, BE is the angle bisector of ∠ABC, quadrilateral ABCD is a ▱, AF⊥DF, BH is perpendicular to AH. Find the length of line AE.
问题：如图所示，AI=3，BI=IG，BE是∠ABC的角平分线，AD和BC是▱ABCD的一组对边，AF垂直于DF，BH⊥AH。求直线AE的长度。
Construction (CDL):
Shape(AB,BI,IA)
Shape(AI,IG,GA)
Shape(AG,GE,EA)
Shape(EG,GF,FD,DE)
Shape(GI,IH,HC,CF,FG)
Shape(IB,BH,HI)
Collinear(AIH)
Collinear(BIGE)
Collinear(AGF)
Collinear(AED)
Collinear(DFC)
Collinear(BHC)
Conditions (CDL):
Equal(LengthOfLine(AI),3)
Equal(LengthOfLine(BI),LengthOfLine(IG))
IsBisectorOfAngle(BE,ABC)
Parallelogram(ABCD)
PerpendicularBetweenLine(AF,DF)
PerpendicularBetweenLine(BH,AH)
Goal: Value(LengthOfLine(AE))
Answer: 3*sqrt(3)
Theorem sequence:
parallel_property_collinear_extend(3,AD,BC,E)
parallel_property_collinear_extend(3,CB,DA,H)
parallel_property_alternate_interior_angle(1,HB,DA)
parallel_property_alternate_interior_angle(2,AE,BC)
isosceles_triangle_judgment_angle_equal(1,ABE)
parallelogram_property_opposite_angle_equal(1,BCDA)
mirror_similar_triangle_judgment_aa(1,ABH,AFD)
mirror_similar_triangle_property_angle_equal(1,ABH,AFD)
mirror_congruent_triangle_judgment_aas(2,IAB,GEA)
mirror_congruent_triangle_property_line_equal(1,ABI,AGE)
mirror_congruent_triangle_property_line_equal(1,BIA,EAG)
right_triangle_judgment_angle(1,EAI)
median_of_triangle_judgment(1,AG,AIE)
right_triangle_property_length_of_median(1,EAI,G)
right_triangle_property_pythagorean(1,EAI)
Problem: As shown in the diagram, AD=1, DB=2, DE=2, DE is parallel to BC. Find the length of line BC.
问题：如图所示，AD=1，DB=2，DE=2，DE平行于BC。求直线BC的长度。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(LengthOfLine(AD),1)
Equal(LengthOfLine(DB),2)
Equal(LengthOfLine(DE),2)
ParallelBetweenLine(DE,BC)
Goal: Value(LengthOfLine(BC))
Answer: 6
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, AB=9, BD=3, ∠ADE=60°, triangle ABC is an equilateral triangle. Find the length of line AE.
问题：如图所示，AB=9，BD=3，∠ADE=60°，△ABC为等边△。求直线AE的长度。
Construction (CDL):
Shape(AB,BD,DA)
Shape(AD,DE,EA)
Shape(ED,DC,CE)
Collinear(BDC)
Collinear(AEC)
Conditions (CDL):
Equal(LengthOfLine(AB),9)
Equal(LengthOfLine(BD),3)
Equal(MeasureOfAngle(ADE),60)
EquilateralTriangle(ABC)
Goal: Value(LengthOfLine(AE))
Answer: 7
Theorem sequence:
equilateral_triangle_property_angle(1,BCA)
equilateral_triangle_property_angle(1,CAB)
angle_addition(1,BDA,ADE)
adjacent_complementary_angle(1,BDE,EDC)
triangle_property_angle_sum(1,ABD)
similar_triangle_judgment_aa(1,DAB,EDC)
similar_triangle_property_line_ratio(1,DAB,EDC)
similar_triangle_property_line_ratio(1,ABD,DCE)
line_addition(1,BD,DC)
line_addition(1,AE,EC)
Problem: As shown in the diagram, AP=PB, PB=5, PD=3, PQ=PB, ∠BPA=Add(MeasureOfAngle(PQA),MeasureOfAngle(QAP))°, ∠BPA=Mul(MeasureOfAngle(BCD),2)°. Find the value of AD*DC.
问题：如图所示，AP=PB，PB=5，PD=3，PQ=PB，∠BPA=Add(MeasureOfAngle(PQA),MeasureOfAngle(QAP))°，∠BPA=Mul(MeasureOfAngle(BCD),2)°。求AD*DC的值。
Construction (CDL):
Shape(PA,AD,DP)
Shape(DA,AB,BD)
Shape(CD,DB,BC)
Shape(QA,AP,PQ)
Collinear(ADC)
Collinear(QPDB)
Conditions (CDL):
Equal(LengthOfLine(AP),LengthOfLine(PB))
Equal(LengthOfLine(PB),5)
Equal(LengthOfLine(PD),3)
Equal(LengthOfLine(PQ),LengthOfLine(PB))
Equal(MeasureOfAngle(BPA),Add(MeasureOfAngle(PQA),MeasureOfAngle(QAP)))
Equal(MeasureOfAngle(BPA),Mul(MeasureOfAngle(BCD),2))
Goal: Value(Mul(LengthOfLine(AD),LengthOfLine(DC)))
Answer: 16
Theorem sequence:
line_addition(1,QP,PD)
line_addition(1,PD,DB)
isosceles_triangle_judgment_line_equal(1,PQA)
isosceles_triangle_property_angle_equal(1,PQA)
vertical_angle(1,ADQ,CDB)
mirror_similar_triangle_judgment_aa(1,ADQ,BCD)
mirror_similar_triangle_property_line_ratio(1,ADQ,BCD)
mirror_similar_triangle_property_line_ratio(1,QAD,CDB)
Problem: As shown in the diagram, Div(LengthOfLine(AD)=LengthOfLine(AB)), the perimeter of △ADE is 2, DE∥BC. Find the perimeter of triangle ABC.
问题：如图所示，Div(LengthOfLine(AD)=LengthOfLine(AB))，△ADE的周长为2，DE平行于BC。求三角形ABC的周长。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(Div(LengthOfLine(AD),LengthOfLine(AB)),1/3)
Equal(PerimeterOfTriangle(ADE),2)
ParallelBetweenLine(DE,BC)
Goal: Value(PerimeterOfTriangle(ABC))
Answer: 6
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,CAB,EAD)
similar_triangle_property_perimeter_ratio(1,ABC,ADE)
Problem: As shown in the diagram, AG=1, BF=2, ∠GEF=90°, E bisects segment AB, ABCD is a square. Find the length of line GF.
问题：如图所示，AG=1，BF=2，∠GEF=90°，E是线段AB的中点，ABCD是正方形。求直线GF的长度。
Construction (CDL):
Shape(DG,GF,FC,CD)
Shape(FG,GE,EF)
Shape(GA,AE,EG)
Shape(FE,EB,BF)
Collinear(DGA)
Collinear(AEB)
Collinear(CFB)
Conditions (CDL):
Equal(LengthOfLine(AG),1)
Equal(LengthOfLine(BF),2)
Equal(MeasureOfAngle(GEF),90)
IsMidpointOfLine(E,AB)
Square(ABCD)
Goal: Value(LengthOfLine(GF))
Answer: 3
Theorem sequence:
triangle_property_angle_sum(1,GAE)
angle_addition(1,AEG,GEF)
adjacent_complementary_angle(1,AEF,FEB)
similar_triangle_judgment_aa(1,EGA,FEB)
similar_triangle_property_line_ratio(1,EGA,FEB)
similar_triangle_property_line_ratio(1,GAE,EBF)
similar_triangle_property_line_ratio(1,AEG,BFE)
right_triangle_judgment_angle(1,GEF)
right_triangle_judgment_angle(1,GAE)
right_triangle_property_pythagorean(1,GEF)
right_triangle_property_pythagorean(1,GAE)
Problem: As shown in the diagram, AB=3, AD=1, DE=2, DE is parallel to BC. Find the length of line BC.
问题：如图所示，AB=3，AD=1，DE=2，DE平行于BC。求直线BC的长度。
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)
Equal(LengthOfLine(DE),2)
ParallelBetweenLine(DE,BC)
Goal: Value(LengthOfLine(BC))
Answer: 6
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,ADE,ABC)
similar_triangle_property_line_ratio(1,EAD,CAB)
Problem: As shown in the diagram, AC=6, BC=3, ∠DBC=∠CAB. Find the length of line CD.
问题：如图所示，AC=6，BC=3，∠DBC=∠CAB。求直线CD的长度。
Construction (CDL):
Shape(CD,DB,BC)
Shape(DA,AB,BD)
Collinear(CDA)
Conditions (CDL):
Equal(LengthOfLine(AC),6)
Equal(LengthOfLine(BC),3)
Equal(MeasureOfAngle(DBC),MeasureOfAngle(CAB))
Goal: Value(LengthOfLine(CD))
Answer: 3/2
Theorem sequence:
mirror_similar_triangle_judgment_aa(1,DBC,BCA)
mirror_similar_triangle_property_line_ratio(1,DBC,BCA)
mirror_similar_triangle_property_line_ratio(1,BCD,ABC)
Problem: As shown in the diagram, AD=3, BD=5, DC=2, ∠DCA=∠BED. Find the length of line DE.
问题：如图所示，AD=3，BD=5，DC=2，∠DCA=∠BED。求直线DE的长度。
Construction (CDL):
Shape(AB,BD,DA)
Shape(AD,DC,CA)
Shape(DB,BE,ED)
Collinear(ADE)
Collinear(BDC)
Conditions (CDL):
Equal(LengthOfLine(AD),3)
Equal(LengthOfLine(BD),5)
Equal(LengthOfLine(DC),2)
Equal(MeasureOfAngle(DCA),MeasureOfAngle(BED))
Goal: Value(LengthOfLine(DE))
Answer: 10/3
Theorem sequence:
vertical_angle(1,ADC,EDB)
mirror_similar_triangle_judgment_aa(1,ADC,BED)
mirror_similar_triangle_property_line_ratio(1,ADC,BED)
mirror_similar_triangle_property_line_ratio(1,CAD,EDB)