SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, AB=3, AD=2, DE=4, EF=15/2, BD is parallel to CE, FA∥BD, FA is parallel to CE. Find the length of line BE.
问题：如图所示，AB=3，AD=2，DE=4，EF=15/2，BD平行于CE，FA平行于BD，FA∥CE。求直线BE的长度。
Construction (CDL):
Shape(AF,FB,BA)
Shape(AB,BD,DA)
Shape(BC,CE,EB)
Shape(BE,ED,DB)
Collinear(ABC)
Collinear(FBE)
Collinear(ADE)
Conditions (CDL):
Equal(LengthOfLine(AB),3)
Equal(LengthOfLine(AD),2)
Equal(LengthOfLine(DE),4)
Equal(LengthOfLine(EF),15/2)
ParallelBetweenLine(BD,CE)
ParallelBetweenLine(FA,BD)
ParallelBetweenLine(FA,CE)
Goal: Value(LengthOfLine(BE))
Answer: 5
Theorem sequence:
line_addition(1,AD,DE)
parallel_property_corresponding_angle(1,DB,AF,E)
similar_triangle_judgment_aa(1,BED,FEA)
similar_triangle_property_line_ratio(1,BED,FEA)
similar_triangle_property_line_ratio(1,DBE,AFE)
Problem: As shown in the diagram, AB=6, AC=5, AE=1, DE∥BC. Find the length of line AD.
问题：如图所示，AB=6，AC=5，AE=1，DE平行于BC。求直线AD的长度。
Construction (CDL):
Shape(EA,AD,DE)
Shape(CE,ED,DB,BC)
Collinear(AEC)
Collinear(ADB)
Conditions (CDL):
Equal(LengthOfLine(AB),6)
Equal(LengthOfLine(AC),5)
Equal(LengthOfLine(AE),1)
ParallelBetweenLine(DE,BC)
Goal: Value(LengthOfLine(AD))
Answer: 6/5
Theorem sequence:
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_line_ratio(1,DEA,BCA)
Problem: As shown in the diagram, AE=2, BE=5, OD=3, AD is parallel to BC, EF∥AD, EF is parallel to BC. Find the length of line BD.
问题：如图所示，AE=2，BE=5，OD=3，AD平行于BC，EF平行于AD，EF∥BC。求直线BD的长度。
Construction (CDL):
Shape(AE,EO,OA)
Shape(OE,EB,BO)
Shape(OB,BC,CO)
Shape(FO,OC,CF)
Shape(DO,OF,FD)
Shape(AO,OD,DA)
Collinear(EOF)
Collinear(AEB)
Collinear(DFC)
Collinear(AOC)
Collinear(DOB)
Conditions (CDL):
Equal(LengthOfLine(AE),2)
Equal(LengthOfLine(BE),5)
Equal(LengthOfLine(OD),3)
ParallelBetweenLine(AD,BC)
ParallelBetweenLine(EF,AD)
ParallelBetweenLine(EF,BC)
Goal: Value(LengthOfLine(BD))
Answer: 21/2
Theorem sequence:
line_addition(1,AE,EB)
line_addition(1,DO,OB)
parallel_property_collinear_extend(3,EF,BC,O)
parallel_property_alternate_interior_angle(2,AD,BC)
parallel_property_alternate_interior_angle(2,EO,BC)
similar_triangle_judgment_aa(1,EBO,ABD)
similar_triangle_property_line_ratio(1,EBO,ABD)
similar_triangle_property_line_ratio(1,OEB,DAB)
Problem: As shown in the diagram, AB=3, AP=4, CD=6, AB is parallel to CD, BA is parallel to DC. Find the length of line DP.
问题：如图所示，AB=3，AP=4，CD=6，AB平行于CD，BA∥DC。求直线DP的长度。
Construction (CDL):
Shape(BA,AP,PB)
Shape(PC,CD,DP)
Collinear(APD)
Collinear(BPC)
Conditions (CDL):
Equal(LengthOfLine(AB),3)
Equal(LengthOfLine(AP),4)
Equal(LengthOfLine(CD),6)
ParallelBetweenLine(AB,CD)
ParallelBetweenLine(BA,DC)
Goal: Value(LengthOfLine(DP))
Answer: 8
Theorem sequence:
vertical_angle(1,APB,DPC)
parallel_property_alternate_interior_angle(1,AB,CD)
similar_triangle_judgment_aa(1,BAP,CDP)
similar_triangle_property_line_ratio(1,BAP,CDP)
similar_triangle_property_line_ratio(1,PBA,PCD)
Problem: As shown in the diagram, AB=5, AC=10, DB=AE, DE∥BC. Find the length of line AE.
问题：如图所示，AB=5，AC=10，DB=AE，DE平行于BC。求直线AE的长度。
Construction (CDL):
Shape(EA,AD,DE)
Shape(CE,ED,DB,BC)
Collinear(AEC)
Collinear(ADB)
Conditions (CDL):
Equal(LengthOfLine(AB),5)
Equal(LengthOfLine(AC),10)
Equal(LengthOfLine(DB),LengthOfLine(AE))
ParallelBetweenLine(DE,BC)
Goal: Value(LengthOfLine(AE))
Answer: 10/3
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_line_ratio(1,DEA,BCA)
Problem: As shown in the diagram, Div(LengthOfLine(BF)=LengthOfLine(CF)), AC=1, AF bisects ∠CAB, ED∥FC, EF is parallel to AD, FE is parallel to CD, FE is parallel to DA. Find the length of line AB.
问题：如图所示，Div(LengthOfLine(BF)=LengthOfLine(CF))，AC=1，AF是∠CAB的角平分线，ED∥FC，EF平行于AD，FE∥CD，FE平行于DA。求直线AB的长度。
Construction (CDL):
Shape(AE,EG,GA)
Shape(GE,EF,FG)
Shape(EB,BF,FE)
Shape(DG,GF,FC,CD)
Shape(AG,GD,DA)
Collinear(AEB)
Collinear(AGF)
Collinear(ADC)
Collinear(EGD)
Collinear(BFC)
Conditions (CDL):
Equal(Div(LengthOfLine(BF),LengthOfLine(CF)),2)
Equal(LengthOfLine(AC),1)
IsBisectorOfAngle(AF,CAB)
ParallelBetweenLine(ED,FC)
ParallelBetweenLine(EF,AD)
ParallelBetweenLine(FE,CD)
ParallelBetweenLine(FE,DA)
Goal: Value(LengthOfLine(AB))
Answer: 2
Theorem sequence:
line_addition(1,AE,EB)
line_addition(1,AD,DC)
line_addition(1,BF,FC)
parallel_property_corresponding_angle(1,FE,CD,B)
parallel_property_corresponding_angle(2,CF,DE,A)
parallel_property_corresponding_angle(2,AD,EF,B)
parallel_property_alternate_interior_angle(1,FE,DA)
isosceles_triangle_judgment_angle_equal(1,EFA)
parallelogram_judgment_parallel_and_parallel(1,EFCD)
parallelogram_property_opposite_line_equal(1,EFCD)
similar_triangle_judgment_aa(1,EBF,ABC)
similar_triangle_judgment_aa(1,BFE,EDA)
similar_triangle_property_line_ratio(1,EBF,ABC)
similar_triangle_property_line_ratio(1,FEB,CAB)
similar_triangle_property_line_ratio(1,BFE,EDA)
similar_triangle_property_line_ratio(1,FEB,DAE)
Problem: As shown in the diagram, AO=OD, DF=OD, OE=6, AB is parallel to CD, EF∥AB, EF is parallel to CD, FE is parallel to BA. Find the length of line BE.
问题：如图所示，AO=OD，DF=OD，OE=6，AB∥CD，EF平行于AB，EF平行于CD，FE∥BA。求直线BE的长度。
Construction (CDL):
Shape(BA,AO,OB)
Shape(OC,CD,DO)
Shape(CE,EF,FD,DC)
Collinear(AODF)
Collinear(BOCE)
Conditions (CDL):
Equal(LengthOfLine(AO),LengthOfLine(OD))
Equal(LengthOfLine(DF),LengthOfLine(OD))
Equal(LengthOfLine(OE),6)
ParallelBetweenLine(AB,CD)
ParallelBetweenLine(EF,AB)
ParallelBetweenLine(EF,CD)
ParallelBetweenLine(FE,BA)
Goal: Value(LengthOfLine(BE))
Answer: 9
Theorem sequence:
vertical_angle(1,AOB,DOC)
line_addition(1,OD,DF)
line_addition(1,BO,OE)
parallel_property_alternate_interior_angle(1,FE,BA)
similar_triangle_judgment_aa(1,BAO,EFO)
similar_triangle_property_line_ratio(1,BAO,EFO)
similar_triangle_property_line_ratio(1,AOB,FOE)
Problem: As shown in the diagram, Div(LengthOfLine(DE)=LengthOfLine(BC)), AD=6, DE is parallel to BC. Find the length of line AB.
问题：如图所示，Div(LengthOfLine(DE)=LengthOfLine(BC))，AD=6，DE∥BC。求直线AB的长度。
Construction (CDL):
Shape(EA,AD,DE)
Shape(CE,ED,DB,BC)
Collinear(AEC)
Collinear(ADB)
Conditions (CDL):
Equal(Div(LengthOfLine(DE),LengthOfLine(BC)),3/5)
Equal(LengthOfLine(AD),6)
ParallelBetweenLine(DE,BC)
Goal: Value(LengthOfLine(AB))
Answer: 10
Theorem sequence:
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_line_ratio(1,ADE,ABC)
Problem: As shown in the diagram, AC=4, AD=2, Triangle ACD is mirror similar to triangle ABD.. Find the length of line AB.
问题：如图所示，AC=4，AD=2，△ACD与△ABD是镜像相似三角形。求直线AB的长度。
Construction (CDL):
Shape(AB,BD,DA)
Shape(BC,CD,DB)
Collinear(ABC)
Conditions (CDL):
Equal(LengthOfLine(AC),4)
Equal(LengthOfLine(AD),2)
MirrorSimilarBetweenTriangle(ACD,ABD)
Goal: Value(LengthOfLine(AB))
Answer: 1
Theorem sequence:
mirror_similar_triangle_property_line_ratio(1,DAC,BDA)
mirror_similar_triangle_property_line_ratio(1,CDA,DAB)
Problem: As shown in the diagram, AB=10, AC=8, AD=4, Triangle ABC is mirror similar to triangle ADE.. Find the length of line AE.
问题：如图所示，AB=10，AC=8，AD=4，△ABC与△ADE是镜像相似三角形。求直线AE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(AC),8)
Equal(LengthOfLine(AD),4)
MirrorSimilarBetweenTriangle(ABC,ADE)
Goal: Value(LengthOfLine(AE))
Answer: 5
Theorem sequence:
mirror_similar_triangle_property_line_ratio(1,BCA,EAD)
mirror_similar_triangle_property_line_ratio(1,CAB,DEA)
Problem: As shown in the diagram, AB=4, AP=5, CD=6, AB is parallel to CD. Find the length of line AD.
问题：如图所示，AB=4，AP=5，CD=6，AB平行于CD。求直线AD的长度。
Construction (CDL):
Shape(AP,PB,BA)
Shape(PC,CD,DP)
Collinear(APD)
Collinear(BPC)
Conditions (CDL):
Equal(LengthOfLine(AB),4)
Equal(LengthOfLine(AP),5)
Equal(LengthOfLine(CD),6)
ParallelBetweenLine(AB,CD)
Goal: Value(LengthOfLine(AD))
Answer: 25/2
Theorem sequence:
vertical_angle(1,APB,DPC)
parallel_property_alternate_interior_angle(2,AB,CD)
similar_triangle_judgment_aa(1,APB,DPC)
similar_triangle_property_line_ratio(1,PBA,PCD)
similar_triangle_property_line_ratio(1,BAP,CDP)
line_addition(1,AP,PD)
Problem: As shown in the diagram, AB=4, CD=7, PD=10, AB∥CD. Find the length of line AP.
问题：如图所示，AB=4，CD=7，PD=10，AB∥CD。求直线AP的长度。
Construction (CDL):
Shape(AP,PB,BA)
Shape(PC,CD,DP)
Collinear(APD)
Collinear(BPC)
Conditions (CDL):
Equal(LengthOfLine(AB),4)
Equal(LengthOfLine(CD),7)
Equal(LengthOfLine(PD),10)
ParallelBetweenLine(AB,CD)
Goal: Value(LengthOfLine(AP))
Answer: 40/7
Theorem sequence:
vertical_angle(1,APB,DPC)
parallel_property_alternate_interior_angle(2,AB,CD)
similar_triangle_judgment_aa(1,APB,DPC)
similar_triangle_property_line_ratio(1,PBA,PCD)
similar_triangle_property_line_ratio(1,BAP,CDP)
Problem: As shown in the diagram, Div(LengthOfLine(DE)=LengthOfLine(DA)), CD=8, EF is parallel to AB, AD and BC are opposite sides of the parallelogram ABCD. Find the length of line EF.
问题：如图所示，Div(LengthOfLine(DE)=LengthOfLine(DA))，CD=8，EF∥AB，ABCD是平行四边形。求直线EF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(DA)),2/5)
Equal(LengthOfLine(CD),8)
ParallelBetweenLine(EF,AB)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(EF))
Answer: 16/5
Theorem sequence:
parallelogram_property_opposite_line_equal(1,ABCD)
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,FDE,BDA)
similar_triangle_property_line_ratio(1,DEF,DAB)
Problem: As shown in the diagram, AB=80, CD=20, AB is perpendicular to FB, BF is perpendicular to EF, EF is perpendicular to CF, FC is perpendicular to DC. Find the length of line EF.
问题：如图所示，AB=80，CD=20，AB⊥FB，BF垂直于EF，EF⊥CF，FC⊥DC。求直线EF的长度。
Construction (CDL):
Shape(AB,BE,EA)
Shape(EB,BF,FE)
Shape(DE,EC,CD)
Shape(CE,EF,FC)
Collinear(AEC)
Collinear(BED)
Collinear(BFC)
Conditions (CDL):
Equal(LengthOfLine(AB),80)
Equal(LengthOfLine(CD),20)
PerpendicularBetweenLine(AB,FB)
PerpendicularBetweenLine(BF,EF)
PerpendicularBetweenLine(EF,CF)
PerpendicularBetweenLine(FC,DC)
Goal: Value(LengthOfLine(EF))
Answer: 16
Theorem sequence:
parallel_judgment_per_per(1,BA,FE)
parallel_property_corresponding_angle(1,EF,AB,C)
similar_triangle_judgment_aa(1,CEF,CAB)
vertical_angle(1,BEA,DEC)
parallel_judgment_per_per(1,FE,CD)
parallel_judgment_par_par(1,BA,FE,CD)
parallel_property_alternate_interior_angle(2,BA,CD)
similar_triangle_judgment_aa(1,BEA,DEC)
similar_triangle_property_line_ratio(1,EAB,ECD)
similar_triangle_property_line_ratio(1,BEA,DEC)
line_addition(1,AE,EC)
similar_triangle_property_line_ratio(1,FCE,BCA)
similar_triangle_property_line_ratio(1,CEF,CAB)
Problem: As shown in the diagram, AC=3, CE=4, AB is parallel to CD, CD is parallel to EF, ACDB is similar to AEFB. Find the ratio of the length of line BD to the length of line BF.
问题：如图所示，AC=3，CE=4，AB平行于CD，CD平行于EF，ACDB相似于AEFB。求直线BD的长度与直线BF的长度之比。
Construction (CDL):
Shape(AC,CD,DB,BA)
Shape(CE,EF,FD,DC)
Collinear(ACE)
Collinear(BDF)
Conditions (CDL):
Equal(LengthOfLine(AC),3)
Equal(LengthOfLine(CE),4)
ParallelBetweenLine(AB,CD)
ParallelBetweenLine(CD,EF)
SimilarBetweenQuadrilateral(ACDB,AEFB)
Goal: Value(Div(LengthOfLine(BD),LengthOfLine(BF)))
Answer: 3/7
Theorem sequence:
line_addition(1,AC,CE)
similar_quadrilateral_property_line_ratio(1,ACDB,AEFB)
similar_quadrilateral_property_line_ratio(1,DBAC,FBAE)
Problem: As shown in the diagram, ∠DBC=72°, ∠EAD=35°, △ABC is similar to △ADE.. Find the measure of ∠DEA.
问题：如图所示，∠DBC=72°，∠EAD=35°，三角形ABC与三角形ADE是相似三角形。求∠DEA的大小。
Construction (CDL):
Shape(BC,CE,ED,DB)
Shape(DE,EA,AD)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(MeasureOfAngle(DBC),72)
Equal(MeasureOfAngle(EAD),35)
SimilarBetweenTriangle(ABC,ADE)
Goal: Value(MeasureOfAngle(DEA))
Answer: 73
Theorem sequence:
triangle_property_angle_sum(1,ABC)
similar_triangle_property_angle_equal(1,CAB,EAD)
Problem: As shown in the diagram, ∠ADE=80°, ∠EAD=60°, Triangle ABC is mirror similar to triangle ADE.. Find the measure of ∠DBC.
问题：如图所示，∠ADE=80°，∠EAD=60°，三角形ABC镜像相似于三角形ADE。求∠DBC的大小。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(MeasureOfAngle(ADE),80)
Equal(MeasureOfAngle(EAD),60)
MirrorSimilarBetweenTriangle(ABC,ADE)
Goal: Value(MeasureOfAngle(DBC))
Answer: 40
Theorem sequence:
triangle_property_angle_sum(1,ADE)
mirror_similar_triangle_property_angle_equal(1,BCA,EAD)
Problem: As shown in the diagram, AB=5, AC=4, the ratio of mirror similar triangles ABC and BCD is 4/3, △ABC is mirror similar to △BCD., ∠BCA is the right angle of the right triangle BCA, triangle BCD is a right triangle. Find the length of line CD.
问题：如图所示，AB=5，AC=4，三角形ABC和三角形BCD的镜像相似比为4/3，△ABC与△BCD是镜像相似三角形，BC和CA是直角三角形BCA的两个直角边，∠BCD是直角三角形BCD的直角。求直线CD的长度。
Construction (CDL):
Shape(CD,DB,BC)
Shape(DA,AB,BD)
Collinear(ADC)
Conditions (CDL):
Equal(LengthOfLine(AB),5)
Equal(LengthOfLine(AC),4)
Equal(RatioOfMirrorSimilarTriangle(ABC,BCD),4/3)
MirrorSimilarBetweenTriangle(ABC,BCD)
RightTriangle(BCA)
RightTriangle(BCD)
Goal: Value(LengthOfLine(CD))
Answer: 9/4
Theorem sequence:
right_triangle_property_pythagorean(1,BCA)
mirror_similar_triangle_property_line_ratio(1,ABC,BCD)
Problem: As shown in the diagram, AB=3, DB=2, EC=6, △ADB is similar to △AEC.. Find the length of line BC.
问题：如图所示，AB=3，DB=2，EC=6，△ADB相似于△AEC。求直线BC的长度。
Construction (CDL):
Shape(EC,CB,BD,DE)
Shape(DB,BA,AD)
Collinear(EDA)
Collinear(CBA)
Conditions (CDL):
Equal(LengthOfLine(AB),3)
Equal(LengthOfLine(DB),2)
Equal(LengthOfLine(EC),6)
SimilarBetweenTriangle(ADB,AEC)
Goal: Value(LengthOfLine(BC))
Answer: 6
Theorem sequence:
similar_triangle_property_line_ratio(1,ADB,AEC)
similar_triangle_property_line_ratio(1,DBA,ECA)
line_addition(1,CB,BA)
Problem: As shown in the diagram, AE=4, EC=5, triangle EDC is similar to triangle ABC.. Find the ratio of the length of line CD to the length of line CB.
问题：如图所示，AE=4，EC=5，△EDC与△ABC是相似三角形。求直线CD的长度与直线CB的长度之比。
Construction (CDL):
Shape(AB,BD,DE,EA)
Shape(ED,DC,CE)
Collinear(AEC)
Collinear(BDC)
Conditions (CDL):
Equal(LengthOfLine(AE),4)
Equal(LengthOfLine(EC),5)
SimilarBetweenTriangle(EDC,ABC)
Goal: Value(Div(LengthOfLine(CD),LengthOfLine(CB)))
Answer: 5/9
Theorem sequence:
line_addition(1,AE,EC)
similar_triangle_property_line_ratio(1,DCE,BCA)
similar_triangle_property_line_ratio(1,EDC,ABC)
Problem: As shown in the diagram, ∠AED=40°, ∠EBC=80°, △ABC is mirror similar to △AED.. Find the measure of ∠DAE.
问题：如图所示，∠AED=40°，∠EBC=80°，△ABC与△AED是镜像相似三角形。求∠DAE的大小。
Construction (CDL):
Shape(AE,ED,DA)
Shape(EB,BC,CD,DE)
Collinear(AEB)
Collinear(ADC)
Conditions (CDL):
Equal(MeasureOfAngle(AED),40)
Equal(MeasureOfAngle(EBC),80)
MirrorSimilarBetweenTriangle(ABC,AED)
Goal: Value(MeasureOfAngle(DAE))
Answer: 60
Theorem sequence:
mirror_similar_triangle_property_angle_equal(1,BCA,DAE)
triangle_property_angle_sum(1,DAE)
Problem: As shown in the diagram, BC=1, the ratio of similar triangles ABC and DEF is 1/2, triangle ABC is similar to triangle DEF.. Find the length of line EF.
问题：如图所示，BC=1，三角形ABC和三角形DEF的相似比为1/2，三角形ABC与三角形DEF是相似三角形。求直线EF的长度。
Construction (CDL):
Shape(CA,AB,BC)
Shape(FD,DE,EF)
Conditions (CDL):
Equal(LengthOfLine(BC),1)
Equal(RatioOfSimilarTriangle(ABC,DEF),1/2)
SimilarBetweenTriangle(ABC,DEF)
Goal: Value(LengthOfLine(EF))
Answer: 2
Theorem sequence:
similar_triangle_property_line_ratio(1,ABC,DEF)
Problem: As shown in the diagram, ∠DCB=30°, △BCA is similar to △DCF.. Find the measure of ∠FCA.
问题：如图所示，∠DCB=30°，三角形BCA相似于三角形DCF。求∠FCA的大小。
Construction (CDL):
Shape(BE,EF,FB)
Shape(FE,EC,CF)
Shape(AF,FC,CA)
Shape(BD,DE,EB)
Shape(DC,CE,ED)
Collinear(BFA)
Collinear(BEC)
Collinear(DEF)
Conditions (CDL):
Equal(MeasureOfAngle(DCB),30)
SimilarBetweenTriangle(BCA,DCF)
Goal: Value(MeasureOfAngle(FCA))
Answer: 30
Theorem sequence:
similar_triangle_property_angle_equal(1,CAB,CFD)
angle_addition(1,DCE,ECF)
angle_addition(1,ECF,FCA)
Problem: As shown in the diagram, AC=3, BC=4, CD=2, △ABC is mirror similar to △CDB.. Find the length of line AB.
问题：如图所示，AC=3，BC=4，CD=2，三角形ABC与三角形CDB是镜像相似三角形。求直线AB的长度。
Construction (CDL):
Shape(CA,AD,DC)
Shape(CD,DB,BC)
Collinear(ADB)
Conditions (CDL):
Equal(LengthOfLine(AC),3)
Equal(LengthOfLine(BC),4)
Equal(LengthOfLine(CD),2)
MirrorSimilarBetweenTriangle(ABC,CDB)
Goal: Value(LengthOfLine(AB))
Answer: 6
Theorem sequence:
mirror_similar_triangle_property_line_ratio(1,BCA,BCD)
mirror_similar_triangle_property_line_ratio(1,CAB,DBC)
Problem: As shown in the diagram, AC=4, BC=3, Triangle ABC is mirror similar to triangle BCD.. Find the length of line CD.
问题：如图所示，AC=4，BC=3，△ABC与△BCD是镜像相似三角形。求直线CD的长度。
Construction (CDL):
Shape(CD,DB,BC)
Shape(DA,AB,BD)
Collinear(CDA)
Conditions (CDL):
Equal(LengthOfLine(AC),4)
Equal(LengthOfLine(BC),3)
MirrorSimilarBetweenTriangle(ABC,BCD)
Goal: Value(LengthOfLine(CD))
Answer: 9/4
Theorem sequence:
mirror_similar_triangle_property_line_ratio(1,BCA,DBC)
mirror_similar_triangle_property_line_ratio(1,ABC,BCD)
Problem: As shown in the diagram, ∠ADE=80°, ∠EAD=60°, △ABC is mirror similar to △ADE.. Find the measure of ∠BCE.
问题：如图所示，∠ADE=80°，∠EAD=60°，三角形ABC与三角形ADE是镜像相似三角形。求∠BCE的大小。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(MeasureOfAngle(ADE),80)
Equal(MeasureOfAngle(EAD),60)
MirrorSimilarBetweenTriangle(ABC,ADE)
Goal: Value(MeasureOfAngle(BCE))
Answer: 80
Theorem sequence:
mirror_similar_triangle_property_angle_equal(1,CAB,DEA)
Problem: As shown in the diagram, AB=6, AE=9, DE=2, ABCD is a rectangle, △ABE is similar to △DEF.. Find the length of line EF.
问题：如图所示，AB=6，AE=9，DE=2，ABCD是长方形，三角形ABE与三角形DEF是相似三角形。求直线EF的长度。
Construction (CDL):
Shape(AB,BE,EA)
Shape(BC,CF,FE,EB)
Shape(EF,FD,DE)
Collinear(AED)
Collinear(DFC)
Conditions (CDL):
Equal(LengthOfLine(AB),6)
Equal(LengthOfLine(AE),9)
Equal(LengthOfLine(DE),2)
Rectangle(ABCD)
SimilarBetweenTriangle(ABE,DEF)
Goal: Value(LengthOfLine(EF))
Answer: sqrt(13)
Theorem sequence:
similar_triangle_property_line_ratio(1,EAB,FDE)
similar_triangle_property_line_ratio(1,BEA,EFD)
right_triangle_judgment_angle(1,FDE)
right_triangle_property_pythagorean(1,FDE)
Problem: As shown in the diagram, AC=4, BC=3, Triangle ABC is mirror similar to triangle BCD., BC and CA are the legs of the right △ BCA. Find the length of line CD.
问题：如图所示，AC=4，BC=3，△ABC镜像相似于△BCD，BC和CA是直角三角形BCA的两个直角边。求直线CD的长度。
Construction (CDL):
Shape(DA,AB,BD)
Shape(CD,DB,BC)
Collinear(ADC)
Conditions (CDL):
Equal(LengthOfLine(AC),4)
Equal(LengthOfLine(BC),3)
MirrorSimilarBetweenTriangle(ABC,BCD)
RightTriangle(BCA)
Goal: Value(LengthOfLine(CD))
Answer: 9/4
Theorem sequence:
mirror_similar_triangle_property_line_ratio(1,BCA,DBC)
mirror_similar_triangle_property_line_ratio(1,ABC,BCD)
Problem: As shown in the diagram, AB=9, AC=12, BC=18, CD=4, triangle AED is similar to triangle ABC.. Find the length of line DE.
问题：如图所示，AB=9，AC=12，BC=18，CD=4，三角形AED相似于三角形ABC。求直线DE的长度。
Construction (CDL):
Shape(AE,ED,DA)
Shape(EB,BC,CD,DE)
Collinear(AEB)
Collinear(ADC)
Conditions (CDL):
Equal(LengthOfLine(AB),9)
Equal(LengthOfLine(AC),12)
Equal(LengthOfLine(BC),18)
Equal(LengthOfLine(CD),4)
SimilarBetweenTriangle(AED,ABC)
Goal: Value(LengthOfLine(DE))
Answer: 12
Theorem sequence:
line_addition(1,AD,DC)
similar_triangle_property_line_ratio(1,EDA,BCA)
similar_triangle_property_line_ratio(1,AED,ABC)
Problem: As shown in the diagram, the area of triangle ABC is 18, AB=3, DE=4, triangle ABC is similar to triangle DEF.. Find the area of triangle DEF.
问题：如图所示，△ABC的面积为18，AB=3，DE=4，△ABC与△DEF是相似三角形。求△DEF的面积。
Construction (CDL):
Shape(AB,BC,CA)
Shape(DE,EF,FD)
Conditions (CDL):
Equal(AreaOfTriangle(ABC),18)
Equal(LengthOfLine(AB),3)
Equal(LengthOfLine(DE),4)
SimilarBetweenTriangle(ABC,DEF)
Goal: Value(AreaOfTriangle(DEF))
Answer: 32
Theorem sequence:
similar_triangle_property_line_ratio(1,CAB,FDE)
similar_triangle_property_area_square_ratio(1,ABC,DEF)
Problem: As shown in the diagram, AE=4, BF=3, △AED is mirror similar to △BFA., ABCD is a square. Find the area of ABCD.
问题：如图所示，AE=4，BF=3，三角形AED与三角形BFA是镜像相似三角形，四边形ABCD是正方形。求ABCD的面积。
Construction (CDL):
Shape(AB,BF,FA)
Shape(AF,FE,ED,DA)
Shape(FB,BC,CE,EF)
Collinear(AFE)
Collinear(DEC)
Conditions (CDL):
Equal(LengthOfLine(AE),4)
Equal(LengthOfLine(BF),3)
MirrorSimilarBetweenTriangle(AED,BFA)
Square(ABCD)
Goal: Value(AreaOfQuadrilateral(ABCD))
Answer: 12
Theorem sequence:
mirror_similar_triangle_property_line_ratio(1,EDA,ABF)
mirror_similar_triangle_property_line_ratio(1,DAE,FAB)
parallelogram_area_formula_sine(1,ABCD)
Problem: As shown in the diagram, BE=3, CE=2, triangle ABC is similar to triangle DEB.. Find the ratio of the length of line ED to the length of line AB.
问题：如图所示，BE=3，CE=2，三角形ABC与三角形DEB是相似三角形。求直线ED的长度与直线AB的长度之比。
Construction (CDL):
Shape(AB,BE,EC,CA)
Shape(BD,DE,EB)
Collinear(BEC)
Conditions (CDL):
Equal(LengthOfLine(BE),3)
Equal(LengthOfLine(CE),2)
SimilarBetweenTriangle(ABC,DEB)
Goal: Value(Div(LengthOfLine(ED),LengthOfLine(AB)))
Answer: 3/5
Theorem sequence:
line_addition(1,BE,EC)
similar_triangle_property_line_ratio(1,ABC,DEB)
similar_triangle_property_line_ratio(1,CAB,BDE)
Problem: As shown in the diagram, ∠BDC=45°, ∠CAB=75°, △ABD is mirror similar to △ABC.. Find the measure of ∠CBD.
问题：如图所示，∠BDC=45°，∠CAB=75°，△ABD镜像相似于△ABC。求∠CBD的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(BD,DC,CB)
Collinear(ACD)
Conditions (CDL):
Equal(MeasureOfAngle(BDC),45)
Equal(MeasureOfAngle(CAB),75)
MirrorSimilarBetweenTriangle(ABD,ABC)
Goal: Value(MeasureOfAngle(CBD))
Answer: 15
Theorem sequence:
triangle_property_angle_sum(1,ABD)
mirror_similar_triangle_property_angle_equal(1,DAB,BCA)
angle_addition(1,ABC,CBD)
Problem: As shown in the diagram, EC=21, OE=6, triangle OCB is similar to triangle OED.. Find the ratio of similar △OCB and △OED.
问题：如图所示，EC=21，OE=6，△OCB与△OED是相似三角形。求三角形OCB和三角形OED的相似比。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DC,CO,OD)
Shape(DO,OE,ED)
Shape(OC,CB,BO)
Shape(OB,BE,EO)
Collinear(ADC)
Collinear(AEB)
Collinear(DOB)
Collinear(EOC)
Conditions (CDL):
Equal(LengthOfLine(EC),21)
Equal(LengthOfLine(OE),6)
SimilarBetweenTriangle(OCB,OED)
Goal: Value(RatioOfSimilarTriangle(OCB,OED))
Answer: 5/2
Theorem sequence:
line_addition(1,CO,OE)
similar_triangle_property_line_ratio(1,BOC,DOE)
Problem: As shown in the diagram, AB=10, AC=8, Triangle CBA is mirror similar to triangle DAC., △ACB is a right △, CD and DA are the legs of the right triangle CDA. Find the length of line AD.
问题：如图所示，AB=10，AC=8，三角形CBA镜像相似于三角形DAC，AC和CB是直角三角形ACB的两个直角边，∠CDA是直角三角形CDA的直角。求直线AD的长度。
Construction (CDL):
Shape(CB,BD,DC)
Shape(CD,DA,AC)
Collinear(BDA)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(LengthOfLine(AC),8)
MirrorSimilarBetweenTriangle(CBA,DAC)
RightTriangle(ACB)
RightTriangle(CDA)
Goal: Value(LengthOfLine(AD))
Answer: 32/5
Theorem sequence:
mirror_similar_triangle_property_line_ratio(1,CBA,DAC)
mirror_similar_triangle_property_line_ratio(1,BAC,CDA)
Problem: As shown in the diagram, AB=9, AD=6, ED=3, △BCF is mirror similar to △DEC., quadrilateral ABCD is a ▱. Find the length of line BF.
问题：如图所示，AB=9，AD=6，ED=3，△BCF镜像相似于△DEC，BA和CD是平行四边形ABCD的一组对边。求直线BF的长度。
Construction (CDL):
Shape(AF,FC,CE,EA)
Shape(FB,BC,CF)
Shape(EC,CD,DE)
Collinear(AED)
Collinear(AFB)
Conditions (CDL):
Equal(LengthOfLine(AB),9)
Equal(LengthOfLine(AD),6)
Equal(LengthOfLine(ED),3)
MirrorSimilarBetweenTriangle(BCF,DEC)
Parallelogram(ABCD)
Goal: Value(LengthOfLine(BF))
Answer: 2
Theorem sequence:
parallelogram_property_opposite_line_equal(1,ABCD)
parallelogram_property_opposite_line_equal(1,DABC)
mirror_similar_triangle_property_line_ratio(1,FBC,ECD)
mirror_similar_triangle_property_line_ratio(1,CFB,CDE)