SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, the area of △ADC is 2, ∠CAD=30°, ∠CBA=30°, AC⊥BC, AD is perpendicular to CD. Find the area of triangle ACB.
问题：如图所示，三角形ADC的面积为2，∠CAD=30°，∠CBA=30°，AC⊥BC，AD垂直于CD。求三角形ACB的面积。
Construction (CDL):
Shape(AD,DC,CA)
Shape(AC,CB,BA)
Conditions (CDL):
Equal(AreaOfTriangle(ADC),2)
Equal(MeasureOfAngle(CAD),30)
Equal(MeasureOfAngle(CBA),30)
PerpendicularBetweenLine(AC,BC)
PerpendicularBetweenLine(AD,CD)
Goal: Value(AreaOfTriangle(ACB))
Answer: 8
Theorem sequence:
sine_theorem(1,CAD)
mirror_similar_triangle_judgment_aa(1,CAD,ACB)
mirror_similar_triangle_property_line_ratio(1,ADC,BAC)
mirror_similar_triangle_property_area_square_ratio(1,CAD,ACB)
Problem: As shown in the diagram, Div(LengthOfLine(AD)=LengthOfLine(BD)), DE=6, DE∥BC. Find the length of line BC.
问题：如图所示，Div(LengthOfLine(AD)=LengthOfLine(BD))，DE=6，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(AD),LengthOfLine(BD)),3/2)
Equal(LengthOfLine(DE),6)
ParallelBetweenLine(DE,BC)
Goal: Value(LengthOfLine(BC))
Answer: 10
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)
line_addition(1,AD,DB)
similar_triangle_property_line_ratio(1,ADE,ABC)
similar_triangle_property_line_ratio(1,EAD,CAB)
Problem: As shown in the diagram, AB=8, BC=6, EF=2, ∠FBN=90°, DEFG is a square. Find the length of line BN.
问题：如图所示，AB=8，BC=6，EF=2，∠FBN=90°，DEFG是正方形。求直线BN的长度。
Construction (CDL):
Shape(AM,MD,DA)
Shape(AE,EM,MA)
Shape(DM,MG,GD)
Shape(ME,EF,FG,GM)
Shape(CD,DG,GN,NC)
Shape(NG,GF,FB,BN)
Collinear(ADC)
Collinear(AMGN)
Collinear(AEFB)
Collinear(DME)
Collinear(CNB)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(LengthOfLine(BC),6)
Equal(LengthOfLine(EF),2)
Equal(MeasureOfAngle(FBN),90)
Square(DEFG)
Goal: Value(LengthOfLine(BN))
Answer: 24/7
Theorem sequence:
parallel_judgment_corresponding_angle(1,FG,BN,E)
parallel_judgment_par_par(1,ED,FG,BN)
parallel_property_corresponding_angle(1,ED,BN,A)
parallel_property_corresponding_angle(2,NB,GF,A)
similar_triangle_judgment_aa(1,DAE,CAB)
similar_triangle_judgment_aa(1,AFG,ABN)
similar_triangle_property_line_ratio(1,DAE,CAB)
similar_triangle_property_line_ratio(1,AED,ABC)
similar_triangle_property_line_ratio(1,AFG,ABN)
similar_triangle_property_line_ratio(1,GAF,NAB)
line_addition(1,AE,EF)
Problem: As shown in the diagram, Div(LengthOfLine(AE)=LengthOfLine(BE)), EF=2, EF∥BC. Find the length of line BC.
问题：如图所示，Div(LengthOfLine(AE)=LengthOfLine(BE))，EF=2，EF∥BC。求直线BC的长度。
Construction (CDL):
Shape(AE,EF,FA)
Shape(EB,BC,CF,FE)
Collinear(AEB)
Collinear(AFC)
Conditions (CDL):
Equal(Div(LengthOfLine(AE),LengthOfLine(BE)),1/2)
Equal(LengthOfLine(EF),2)
ParallelBetweenLine(EF,BC)
Goal: Value(LengthOfLine(BC))
Answer: 6
Theorem sequence:
parallel_property_corresponding_angle(1,EF,BC,A)
parallel_property_corresponding_angle(2,CB,FE,A)
similar_triangle_judgment_aa(1,AEF,ABC)
line_addition(1,AE,EB)
similar_triangle_property_line_ratio(1,AEF,ABC)
similar_triangle_property_line_ratio(1,FAE,CAB)
Problem: As shown in the diagram, Div(LengthOfLine(DE)=LengthOfLine(EA)), EF=6, EF is parallel to AB, ABCD is a ▱. Find the length of line CD.
问题：如图所示，Div(LengthOfLine(DE)=LengthOfLine(EA))，EF=6，EF∥AB，AD和BC是▱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),6)
ParallelBetweenLine(EF,AB)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(CD))
Answer: 14
Theorem sequence:
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)
line_addition(1,DE,EA)
parallelogram_property_opposite_line_equal(1,ABCD)
Problem: As shown in the diagram, Add(MeasureOfAngle(BDA)=MeasureOfAngle(EDC)), BD=3, CE=2, △ABC is an equilateral △. Find the length of line AB.
问题：如图所示，Add(MeasureOfAngle(BDA)=MeasureOfAngle(EDC))，BD=3，CE=2，三角形ABC为等边三角形。求直线AB的长度。
Construction (CDL):
Shape(AB,BD,DA)
Shape(AD,DE,EA)
Shape(ED,DC,CE)
Collinear(BDC)
Collinear(AEC)
Conditions (CDL):
Equal(Add(MeasureOfAngle(BDA),MeasureOfAngle(EDC)),120)
Equal(LengthOfLine(BD),3)
Equal(LengthOfLine(CE),2)
EquilateralTriangle(ABC)
Goal: Value(LengthOfLine(AB))
Answer: 9
Theorem sequence:
equilateral_triangle_property_angle(1,BCA)
equilateral_triangle_property_angle(1,CAB)
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)
Problem: As shown in the diagram, BC=4, DE=1, quadrilateral ABCD is a square. Find the length of line DF.
问题：如图所示，BC=4，DE=1，ABCD是正方形。求直线DF的长度。
Construction (CDL):
Shape(AB,BE,ED,DA)
Shape(FD,DE,EF)
Shape(EB,BC,CE)
Collinear(ADF)
Collinear(DEC)
Collinear(BEF)
Conditions (CDL):
Equal(LengthOfLine(BC),4)
Equal(LengthOfLine(DE),1)
Square(ABCD)
Goal: Value(LengthOfLine(DF))
Answer: 4/3
Theorem sequence:
line_addition(1,DE,EC)
parallel_property_collinear_extend(2,AD,BC,F)
parallel_property_alternate_interior_angle(1,DF,BC)
parallel_property_alternate_interior_angle(2,DF,BC)
similar_triangle_judgment_aa(1,EFD,EBC)
similar_triangle_property_line_ratio(1,EFD,EBC)
similar_triangle_property_line_ratio(1,FDE,BCE)
Problem: As shown in the diagram, BC=4, EF=3, EF is parallel to BC. Find the ratio of the length of line AF to the length of line AC.
问题：如图所示，BC=4，EF=3，EF平行于BC。求直线AF的长度与直线AC的长度之比。
Construction (CDL):
Shape(AE,EF,FA)
Shape(EB,BC,CF,FE)
Collinear(AEB)
Collinear(AFC)
Conditions (CDL):
Equal(LengthOfLine(BC),4)
Equal(LengthOfLine(EF),3)
ParallelBetweenLine(EF,BC)
Goal: Value(Div(LengthOfLine(AF),LengthOfLine(AC)))
Answer: 3/4
Theorem sequence:
parallel_property_corresponding_angle(1,EF,BC,A)
parallel_property_corresponding_angle(2,CB,FE,A)
similar_triangle_judgment_aa(1,ABC,AEF)
similar_triangle_property_line_ratio(1,ABC,AEF)
similar_triangle_property_line_ratio(1,BCA,EFA)
Problem: As shown in the diagram, AD=1, DB=2, DE is parallel to 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(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, Div(LengthOfLine(AD)=LengthOfLine(DB)), DE=6, DE is parallel to BC. Find the length of line BC.
问题：如图所示，Div(LengthOfLine(AD)=LengthOfLine(DB))，DE=6，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(AD),LengthOfLine(DB)),3/2)
Equal(LengthOfLine(DE),6)
ParallelBetweenLine(DE,BC)
Goal: Value(LengthOfLine(BC))
Answer: 10
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, AC=2, AD=1, ∠CAD=90°, ∠DCA=∠DBC. Find the length of line BC.
问题：如图所示，AC=2，AD=1，∠CAD=90°，∠DCA=∠DBC。求直线BC的长度。
Construction (CDL):
Shape(AD,DC,CA)
Shape(DB,BC,CD)
Collinear(ADB)
Conditions (CDL):
Equal(LengthOfLine(AC),2)
Equal(LengthOfLine(AD),1)
Equal(MeasureOfAngle(CAD),90)
Equal(MeasureOfAngle(DCA),MeasureOfAngle(DBC))
Goal: Value(LengthOfLine(BC))
Answer: 2*sqrt(5)
Theorem sequence:
right_triangle_judgment_angle(1,CAD)
right_triangle_property_pythagorean(1,CAD)
mirror_similar_triangle_judgment_aa(1,DCA,CAB)
mirror_similar_triangle_property_line_ratio(1,CAD,BCA)
mirror_similar_triangle_property_line_ratio(1,ADC,ABC)
Problem: As shown in the diagram, AD=5, DB=10, DE=4, DE is parallel to BC. Find the length of line BC.
问题：如图所示，AD=5，DB=10，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(LengthOfLine(AD),5)
Equal(LengthOfLine(DB),10)
Equal(LengthOfLine(DE),4)
ParallelBetweenLine(DE,BC)
Goal: Value(LengthOfLine(BC))
Answer: 12
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 △ABC is 48, Div(LengthOfLine(AD)=LengthOfLine(AB)), DE is parallel to BC. Find the area of triangle ADE.
问题：如图所示，△ABC的面积为48，Div(LengthOfLine(AD)=LengthOfLine(AB))，DE∥BC。求三角形ADE的面积。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(AreaOfTriangle(ABC),48)
Equal(Div(LengthOfLine(AD),LengthOfLine(AB)),3/4)
ParallelBetweenLine(DE,BC)
Goal: Value(AreaOfTriangle(ADE))
Answer: 27
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,EAD,CAB)
similar_triangle_property_area_square_ratio(1,ADE,ABC)
Problem: As shown in the diagram, AB=6, BD=4, ∠DAB=∠DCA. Find the length of line CD.
问题：如图所示，AB=6，BD=4，∠DAB=∠DCA。求直线CD的长度。
Construction (CDL):
Shape(AB,BD,DA)
Shape(AD,DC,CA)
Collinear(BDC)
Conditions (CDL):
Equal(LengthOfLine(AB),6)
Equal(LengthOfLine(BD),4)
Equal(MeasureOfAngle(DAB),MeasureOfAngle(DCA))
Goal: Value(LengthOfLine(CD))
Answer: 5
Theorem sequence:
mirror_similar_triangle_judgment_aa(1,DAB,ABC)
mirror_similar_triangle_property_line_ratio(1,DAB,ABC)
mirror_similar_triangle_property_line_ratio(1,ABD,CAB)
line_addition(1,BD,DC)
Problem: As shown in the diagram, AD=3, BC=6, DB=2, DE∥BC. Find the length of line DE.
问题：如图所示，AD=3，BC=6，DB=2，DE平行于BC。求直线DE的长度。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(LengthOfLine(AD),3)
Equal(LengthOfLine(BC),6)
Equal(LengthOfLine(DB),2)
ParallelBetweenLine(DE,BC)
Goal: Value(LengthOfLine(DE))
Answer: 18/5
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)
line_addition(1,AD,DB)
Problem: As shown in the diagram, AD=5, DB=10, DE=3, DE is parallel to BC. Find the length of line BC.
问题：如图所示，AD=5，DB=10，DE=3，DE∥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),5)
Equal(LengthOfLine(DB),10)
Equal(LengthOfLine(DE),3)
ParallelBetweenLine(DE,BC)
Goal: Value(LengthOfLine(BC))
Answer: 9
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)
line_addition(1,AD,DB)
Problem: As shown in the diagram, the area of △ABC is equal to the area of triangle DEC, AB=6, EF=4, DE is parallel to AB. Find the length of line DF.
问题：如图所示，△ABC的面积与△DEC的面积相等，AB=6，EF=4，DE平行于AB。求直线DF的长度。
Construction (CDL):
Shape(AB,BE,EF,FA)
Shape(DF,FC,CD)
Shape(FE,EC,CF)
Collinear(AFC)
Collinear(EFD)
Collinear(BEC)
Conditions (CDL):
Equal(AreaOfTriangle(ABC),AreaOfTriangle(DEC))
Equal(LengthOfLine(AB),6)
Equal(LengthOfLine(EF),4)
ParallelBetweenLine(DE,AB)
Goal: Value(LengthOfLine(DF))
Answer: 5
Theorem sequence:
parallel_property_corresponding_angle(2,BA,ED,C)
triangle_area_formula_sine(1,BCA)
triangle_area_formula_sine(1,ECD)
similar_triangle_judgment_aa(1,FEC,ABC)
similar_triangle_property_line_ratio(1,FEC,ABC)
similar_triangle_property_line_ratio(1,CFE,CAB)
line_addition(1,EF,FD)
Problem: As shown in the diagram, AB=4, CD=12, AB is perpendicular to FB, BF is perpendicular to EF, FD⊥CD. Find the length of line EF.
问题：如图所示，AB=4，CD=12，AB⊥FB，BF垂直于EF，FD垂直于CD。求直线EF的长度。
Construction (CDL):
Shape(AB,BE,EA)
Shape(EB,BF,FE)
Shape(EF,FD,DE)
Shape(CE,ED,DC)
Collinear(AED)
Collinear(BEC)
Collinear(BFD)
Conditions (CDL):
Equal(LengthOfLine(AB),4)
Equal(LengthOfLine(CD),12)
PerpendicularBetweenLine(AB,FB)
PerpendicularBetweenLine(BF,EF)
PerpendicularBetweenLine(FD,CD)
Goal: Value(LengthOfLine(EF))
Answer: 3
Theorem sequence:
parallel_judgment_corresponding_angle(1,FE,DC,B)
parallel_judgment_ipsilateral_internal_angle(1,BA,FE)
parallel_property_corresponding_angle(2,CD,EF,B)
parallel_property_corresponding_angle(2,BA,FE,D)
similar_triangle_judgment_aa(1,EBF,CBD)
similar_triangle_judgment_aa(1,ABD,EFD)
similar_triangle_judgment_aa(1,BDC,BFE)
similar_triangle_property_line_ratio(1,EBF,CBD)
similar_triangle_property_line_ratio(1,BFE,BDC)
similar_triangle_property_line_ratio(1,ABD,EFD)
similar_triangle_property_line_ratio(1,DAB,DEF)
similar_triangle_property_line_ratio(1,CBD,EBF)
line_addition(1,BF,FD)
Problem: As shown in the diagram, AD=3, BC=5, BD=4, DE∥BC. Find the length of line DE.
问题：如图所示，AD=3，BC=5，BD=4，DE平行于BC。求直线DE的长度。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(LengthOfLine(AD),3)
Equal(LengthOfLine(BC),5)
Equal(LengthOfLine(BD),4)
ParallelBetweenLine(DE,BC)
Goal: Value(LengthOfLine(DE))
Answer: 15/7
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)
line_addition(1,AD,DB)
Problem: As shown in the diagram, the area of triangle ADC is 1, AC=2, AD=1, ∠DCA=∠DBC. Find the area of △BCD.
问题：如图所示，△ADC的面积为1，AC=2，AD=1，∠DCA=∠DBC。求△BCD的面积。
Construction (CDL):
Shape(AD,DC,CA)
Shape(DB,BC,CD)
Collinear(BDA)
Conditions (CDL):
Equal(AreaOfTriangle(ADC),1)
Equal(LengthOfLine(AC),2)
Equal(LengthOfLine(AD),1)
Equal(MeasureOfAngle(DCA),MeasureOfAngle(DBC))
Goal: Value(AreaOfTriangle(BCD))
Answer: 3
Theorem sequence:
mirror_similar_triangle_judgment_aa(1,DCA,CAB)
mirror_similar_triangle_property_line_ratio(1,DCA,CAB)
mirror_similar_triangle_property_line_ratio(1,CAD,BCA)
triangle_area_formula_sine(1,ADC)
sine_theorem(1,CAB)
line_addition(1,BD,DA)
triangle_area_formula_sine(1,BCD)
Problem: As shown in the diagram, AD=2, AE=3, CE=1, ∠DEA=∠DBC. Find the length of line BD.
问题：如图所示，AD=2，AE=3，CE=1，∠DEA=∠DBC。求直线BD的长度。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(LengthOfLine(AD),2)
Equal(LengthOfLine(AE),3)
Equal(LengthOfLine(CE),1)
Equal(MeasureOfAngle(DEA),MeasureOfAngle(DBC))
Goal: Value(LengthOfLine(BD))
Answer: 4
Theorem sequence:
line_addition(1,AD,DB)
line_addition(1,AE,EC)
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(AD)=LengthOfLine(DB)), DE=3, DE is parallel to BC. Find the length of line BC.
问题：如图所示，Div(LengthOfLine(AD)=LengthOfLine(DB))，DE=3，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(AD),LengthOfLine(DB)),1/2)
Equal(LengthOfLine(DE),3)
ParallelBetweenLine(DE,BC)
Goal: Value(LengthOfLine(BC))
Answer: 9
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, Div(LengthOfLine(AO)=LengthOfLine(OD)), AB∥CD. Find the ratio of the area of triangle AOB to the area of triangle OCD.
问题：如图所示，Div(LengthOfLine(AO)=LengthOfLine(OD))，AB∥CD。求△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(COB)
Conditions (CDL):
Equal(Div(LengthOfLine(AO),LengthOfLine(OD)),2/3)
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,OCD,OBA)
similar_triangle_property_line_ratio(1,CDO,BAO)
similar_triangle_property_area_square_ratio(1,OCD,OBA)
Problem: As shown in the diagram, AE=BC, CE=5, CF=4, DC∥AB, DC∥EF. Find the ratio of the length of line DC to the length of line AB.
问题：如图所示，AE=BC，CE=5，CF=4，DC∥AB，DC∥EF。求直线DC的长度与直线AB的长度之比。
Construction (CDL):
Shape(DE,EC,CD)
Shape(CE,EF,FC)
Shape(EA,AB,BE)
Shape(EB,BF,FE)
Collinear(AEC)
Collinear(CFB)
Collinear(DEB)
Conditions (CDL):
Equal(LengthOfLine(AE),LengthOfLine(BC))
Equal(LengthOfLine(CE),5)
Equal(LengthOfLine(CF),4)
ParallelBetweenLine(DC,AB)
ParallelBetweenLine(DC,EF)
Goal: Value(Div(LengthOfLine(DC),LengthOfLine(AB)))
Answer: 1/4
Theorem sequence:
parallel_property_alternate_interior_angle(1,DC,AB)
parallel_property_alternate_interior_angle(2,DC,AB)
parallel_property_alternate_interior_angle(2,DC,EF)
similar_triangle_judgment_aa(1,ECD,EAB)
similar_triangle_judgment_aa(1,FCE,BCA)
similar_triangle_property_line_ratio(1,ECD,EAB)
similar_triangle_property_line_ratio(1,DEC,BEA)
similar_triangle_property_line_ratio(1,FCE,BCA)
similar_triangle_property_line_ratio(1,EFC,ABC)
line_addition(1,AE,EC)
Problem: As shown in the diagram, AB=6, DE=4, EF=6, ABCD is a ▱. Find the length of line BF.
问题：如图所示，AB=6，DE=4，EF=6，AD和BC是▱ABCD的一组对边。求直线BF的长度。
Construction (CDL):
Shape(AB,BF,FA)
Shape(EF,FD,DE)
Shape(FB,BC,CD,DF)
Collinear(AFD)
Collinear(BFE)
Collinear(EDC)
Conditions (CDL):
Equal(LengthOfLine(AB),6)
Equal(LengthOfLine(DE),4)
Equal(LengthOfLine(EF),6)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(BF))
Answer: 9
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)
similar_triangle_property_line_ratio(1,FAB,FDE)
similar_triangle_property_line_ratio(1,ABF,DEF)
Problem: As shown in the diagram, the area of triangle ADE is 3, DE is the midsegment of △ ABC. Find the area of triangle ABC minus the area of △ADE.
问题：如图所示，△ADE的面积为3，DE是△ABC的中位线。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),3)
IsMidsegmentOfTriangle(DE,ABC)
Goal: Value(Sub(AreaOfTriangle(ABC),AreaOfTriangle(ADE)))
Answer: 9
Theorem sequence:
midsegment_of_triangle_property_parallel(1,DE,ABC)
parallel_property_corresponding_angle(1,DE,BC,A)
parallel_property_corresponding_angle(2,CB,ED,A)
similar_triangle_judgment_aa(1,ADE,ABC)
line_addition(1,AE,EC)
similar_triangle_property_line_ratio(1,DEA,BCA)
similar_triangle_property_area_square_ratio(1,ADE,ABC)
Problem: As shown in the diagram, the area of △ADE is 3, Div(LengthOfLine(AD)=LengthOfLine(AB)), DE is parallel to BC. Find the area of △ABC.
问题：如图所示，三角形ADE的面积为3，Div(LengthOfLine(AD)=LengthOfLine(AB))，DE平行于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),3)
Equal(Div(LengthOfLine(AD),LengthOfLine(AB)),1/3)
ParallelBetweenLine(DE,BC)
Goal: Value(AreaOfTriangle(ABC))
Answer: 27
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, AD=2, DB=4, DE is parallel to BC. Find the ratio of the length of line DE to the length of line BC.
问题：如图所示，AD=2，DB=4，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),2)
Equal(LengthOfLine(DB),4)
ParallelBetweenLine(DE,BC)
Goal: Value(Div(LengthOfLine(DE),LengthOfLine(BC)))
Answer: 1/3
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,ADE,ABC)
similar_triangle_property_line_ratio(1,EAD,CAB)
line_addition(1,AD,DB)
Problem: As shown in the diagram, Div(LengthOfLine(DE)=LengthOfLine(EC)), FB=12, ABCD is a parallelogram. Find the length of line DF.
问题：如图所示，Div(LengthOfLine(DE)=LengthOfLine(EC))，FB=12，ABCD是平行四边形。求直线DF的长度。
Construction (CDL):
Shape(DA,AF,FD)
Shape(ED,DF,FE)
Shape(CE,EF,FB,BC)
Shape(FA,AB,BF)
Collinear(DEC)
Collinear(DFB)
Collinear(EFA)
Conditions (CDL):
Equal(Div(LengthOfLine(DE),LengthOfLine(EC)),1/2)
Equal(LengthOfLine(FB),12)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(DF))
Answer: 4
Theorem sequence:
parallel_property_collinear_extend(3,DC,AB,E)
parallel_property_alternate_interior_angle(1,BA,CD)
parallel_property_alternate_interior_angle(2,DE,AB)
similar_triangle_judgment_aa(1,FED,FAB)
similar_triangle_property_line_ratio(1,FED,FAB)
similar_triangle_property_line_ratio(1,EDF,ABF)
line_addition(1,DE,EC)
parallelogram_property_opposite_line_equal(1,ABCD)
Problem: As shown in the diagram, Div(LengthOfLine(AD)=LengthOfLine(AC)), DE=10, ∠ADE=∠BCE. Find the length of line BC.
问题：如图所示，Div(LengthOfLine(AD)=LengthOfLine(AC))，DE=10，∠ADE=∠BCE。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(AD),LengthOfLine(AC)),2/3)
Equal(LengthOfLine(DE),10)
Equal(MeasureOfAngle(ADE),MeasureOfAngle(BCE))
Goal: Value(LengthOfLine(BC))
Answer: 15
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,ADE,ABC)
Problem: As shown in the diagram, AB=AC, AE=3/2, BD=2, CD=3, CE=4, ∠FDE=∠FBD, triangleABC is an isosceles triangle. Find the length of line AF.
问题：如图所示，AB=AC，AE=3/2，BD=2，CD=3，CE=4，∠FDE=∠FBD，AB和AC是等腰△ABC的腰。求直线AF的长度。
Construction (CDL):
Shape(AF,FD,DE,EA)
Shape(FB,BD,DF)
Shape(ED,DC,CE)
Collinear(AFB)
Collinear(AEC)
Collinear(BDC)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(AC))
Equal(LengthOfLine(AE),3/2)
Equal(LengthOfLine(BD),2)
Equal(LengthOfLine(CD),3)
Equal(LengthOfLine(CE),4)
Equal(MeasureOfAngle(FDE),MeasureOfAngle(FBD))
IsoscelesTriangle(ABC)
Goal: Value(LengthOfLine(AF))
Answer: 4
Theorem sequence:
isosceles_triangle_property_angle_equal(1,ABC)
triangle_property_angle_sum(1,FBD)
angle_addition(1,BDF,FDE)
adjacent_complementary_angle(1,BDE,EDC)
similar_triangle_judgment_aa(1,DFB,EDC)
similar_triangle_property_line_ratio(1,DFB,EDC)
similar_triangle_property_line_ratio(1,FBD,DCE)
line_addition(1,AF,FB)
line_addition(1,AE,EC)
Problem: As shown in the diagram, the area of triangle AOB is 100, Div(LengthOfLine(BO)=LengthOfLine(CO)), Div(LengthOfLine(OA)=LengthOfLine(DO)). Find the area of △OCD.
问题：如图所示，三角形AOB的面积为100，Div(LengthOfLine(BO)=LengthOfLine(CO))，Div(LengthOfLine(OA)=LengthOfLine(DO))。求△OCD的面积。
Construction (CDL):
Shape(AO,OB,BA)
Shape(OC,CD,DO)
Collinear(AOD)
Collinear(COB)
Conditions (CDL):
Equal(AreaOfTriangle(AOB),100)
Equal(Div(LengthOfLine(BO),LengthOfLine(CO)),1/2)
Equal(Div(LengthOfLine(OA),LengthOfLine(DO)),1/2)
Goal: Value(AreaOfTriangle(OCD))
Answer: 400
Theorem sequence:
vertical_angle(1,AOB,DOC)
similar_triangle_judgment_sas(1,OBA,OCD)
similar_triangle_property_line_ratio(1,BAO,CDO)
similar_triangle_property_area_square_ratio(1,OBA,OCD)