SunnyLin/FormalGeo7k_v1 huggingface dataset:
Problem: As shown in the diagram, AC=8/5, BC=1/2, EF=15, triangle ABC is similar to triangle DEF.. Find the length of line DF.
问题：如图所示，AC=8/5，BC=1/2，EF=15，三角形ABC与三角形DEF是相似三角形。求直线DF的长度。
Construction (CDL):
Shape(AB,BC,CA)
Shape(DE,EF,FD)
Conditions (CDL):
Equal(LengthOfLine(AC),8/5)
Equal(LengthOfLine(BC),1/2)
Equal(LengthOfLine(EF),15)
SimilarBetweenTriangle(ABC,DEF)
Goal: Value(LengthOfLine(DF))
Answer: 48
Theorem sequence:
similar_triangle_property_line_ratio(1,ABC,DEF)
similar_triangle_property_line_ratio(1,BCA,EFD)
Problem: As shown in the diagram, BC=4, CA=1, DC=2, AB is perpendicular to EB, AC is perpendicular to DC. Find the length of line EB.
问题：如图所示，BC=4，CA=1，DC=2，AB⊥EB，AC⊥DC。求直线EB的长度。
Construction (CDL):
Shape(DA,AC,CD)
Shape(ED,DC,CB,BE)
Collinear(ADE)
Collinear(ACB)
Conditions (CDL):
Equal(LengthOfLine(BC),4)
Equal(LengthOfLine(CA),1)
Equal(LengthOfLine(DC),2)
PerpendicularBetweenLine(AB,EB)
PerpendicularBetweenLine(AC,DC)
Goal: Value(LengthOfLine(EB))
Answer: 10
Theorem sequence:
similar_triangle_judgment_aa(1,DAC,EAB)
similar_triangle_property_line_ratio(1,DAC,EAB)
similar_triangle_property_line_ratio(1,ACD,ABE)
line_addition(1,AC,CB)
Problem: As shown in the diagram, BD=8, DC=4, ED=1, AB⊥CB, ED is perpendicular to CD. Find the length of line AB.
问题：如图所示，BD=8，DC=4，ED=1，AB⊥CB，ED垂直于CD。求直线AB的长度。
Construction (CDL):
Shape(AB,BD,DE,EA)
Shape(ED,DC,CE)
Collinear(AEC)
Collinear(BDC)
Conditions (CDL):
Equal(LengthOfLine(BD),8)
Equal(LengthOfLine(DC),4)
Equal(LengthOfLine(ED),1)
PerpendicularBetweenLine(AB,CB)
PerpendicularBetweenLine(ED,CD)
Goal: Value(LengthOfLine(AB))
Answer: 3
Theorem sequence:
similar_triangle_judgment_aa(1,EDC,ABC)
similar_triangle_property_line_ratio(1,EDC,ABC)
similar_triangle_property_line_ratio(1,CED,CAB)
line_addition(1,BD,DC)
Problem: As shown in the diagram, AB=80, BC=60, EF=900, triangle ABC is similar to triangle DEF.. Find the length of line DE.
问题：如图所示，AB=80，BC=60，EF=900，△ABC与△DEF是相似三角形。求直线DE的长度。
Construction (CDL):
Shape(AB,BC,CA)
Shape(DE,EF,FD)
Conditions (CDL):
Equal(LengthOfLine(AB),80)
Equal(LengthOfLine(BC),60)
Equal(LengthOfLine(EF),900)
SimilarBetweenTriangle(ABC,DEF)
Goal: Value(LengthOfLine(DE))
Answer: 1200
Theorem sequence:
similar_triangle_property_line_ratio(1,ABC,DEF)
similar_triangle_property_line_ratio(1,CAB,FDE)
Problem: As shown in the diagram, AB=80, AC=100, EC=60, AB⊥CB, AD is perpendicular to ED. Find the length of line DB.
问题：如图所示，AB=80，AC=100，EC=60，AB垂直于CB，AD⊥ED。求直线DB的长度。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Collinear(AEC)
Collinear(ADB)
Conditions (CDL):
Equal(LengthOfLine(AB),80)
Equal(LengthOfLine(AC),100)
Equal(LengthOfLine(EC),60)
PerpendicularBetweenLine(AB,CB)
PerpendicularBetweenLine(AD,ED)
Goal: Value(LengthOfLine(DB))
Answer: 48
Theorem sequence:
similar_triangle_judgment_aa(1,EAD,CAB)
similar_triangle_property_line_ratio(1,EAD,CAB)
similar_triangle_property_line_ratio(1,DEA,BCA)
line_addition(1,AE,EC)
line_addition(1,AD,DB)
Problem: As shown in the diagram, ∠ABD=50°, the diameter of circle O is AB. Find the measure of ∠DCB.
问题：如图所示，∠ABD=50°，⊙O的直径为AB。求∠DCB的大小。
Construction (CDL):
Shape(ODA,AD)
Shape(DA,AE,ED)
Shape(DE,EB,BD)
Shape(OBD,DB)
Shape(EA,OAC,CE)
Shape(EC,CB,BE)
Shape(OCB,BC)
Collinear(AEB)
Collinear(DEC)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(ABD),50)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(DCB))
Answer: 40
Theorem sequence:
diameter_of_circle_property_right_angle(1,BDA,O)
triangle_property_angle_sum(1,ABD)
arc_property_circumference_angle_external(1,OBD,A)
arc_property_circumference_angle_external(1,OBD,C)
Problem: As shown in the diagram, ∠CAB=42°, AB is the diameter of ⊙O. Find the measure of ∠ABC.
问题：如图所示，∠CAB=42°，AB是圆O的直径。求∠ABC的大小。
Construction (CDL):
Shape(OCA,AC)
Shape(OBC,CB)
Shape(CA,AB,BC)
Shape(OAB,BA)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),42)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ABC))
Answer: 48
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,ABC)
Problem: As shown in the diagram, AH=8, BC=12, BH⊥AH, DGFE is a square. Find the length of line DE.
问题：如图所示，AH=8，BC=12，BH⊥AH，DGFE是正方形。求直线DE的长度。
Construction (CDL):
Shape(AD,DM,MA)
Shape(AM,ME,EA)
Shape(DB,BG,GD)
Shape(DG,GH,HM,MD)
Shape(MH,HF,FE,EM)
Shape(EF,FC,CE)
Collinear(ADB)
Collinear(AEC)
Collinear(BGHFC)
Collinear(DME)
Collinear(AMH)
Conditions (CDL):
Equal(LengthOfLine(AH),8)
Equal(LengthOfLine(BC),12)
PerpendicularBetweenLine(BH,AH)
Square(DGFE)
Goal: Value(LengthOfLine(DE))
Answer: 24/5
Theorem sequence:
parallel_property_collinear_extend(1,FG,ED,C)
parallel_property_collinear_extend(2,CG,ED,B)
parallel_property_corresponding_angle(1,DE,BC,A)
similar_triangle_judgment_aa(1,EAD,CAB)
similar_triangle_judgment_aa(1,MAD,HAB)
parallel_property_collinear_extend(3,DE,GF,M)
parallel_property_collinear_extend(3,FG,MD,H)
parallel_judgment_ipsilateral_internal_angle(1,GD,HM)
parallelogram_judgment_parallel_and_parallel(1,DGHM)
parallelogram_property_opposite_line_equal(1,DGHM)
similar_triangle_property_line_ratio(1,EAD,CAB)
similar_triangle_property_line_ratio(1,ADE,ABC)
similar_triangle_property_line_ratio(1,MAD,HAB)
similar_triangle_property_line_ratio(1,DMA,BHA)
line_addition(1,AM,MH)
Problem: As shown in the diagram, ∠BAO=60°, ∠CDA=85°, the center of ⊙O is O. Find the measure of ∠BCO.
问题：如图所示，∠BAO=60°，∠CDA=85°，O是⊙O的圆心。求∠BCO的大小。
Construction (CDL):
Shape(OCB,BD,DO,OC)
Shape(DB,BA,AD)
Shape(CO,OD,DC)
Shape(DA,OAC,CD)
Shape(OBA,AB)
Collinear(BDC)
Collinear(ODA)
Cocircular(O,BAC)
Conditions (CDL):
Equal(MeasureOfAngle(BAO),60)
Equal(MeasureOfAngle(CDA),85)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(BCO))
Answer: 35
Theorem sequence:
adjacent_complementary_angle(1,CDA,ADB)
adjacent_complementary_angle(1,ODC,CDA)
triangle_property_angle_sum(1,DBA)
arc_property_circumference_angle_external(1,OAC,B)
arc_property_center_angle(1,OAC,O)
triangle_property_angle_sum(1,ODC)
Problem: As shown in the diagram, ∠ABO=40°, ∠OCA=30°, the center of circle O is O. Find the measure of ∠COB.
问题：如图所示，∠ABO=40°，∠OCA=30°，O是⊙O的圆心。求∠COB的大小。
Construction (CDL):
Shape(BA,OAB)
Shape(AB,BO,OA)
Shape(AO,OC,CA)
Shape(AC,OCA)
Shape(OB,OBC,CO)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(ABO),40)
Equal(MeasureOfAngle(OCA),30)
IsCentreOfCircle(O,O)
Goal: Value(MeasureOfAngle(COB))
Answer: 140
Theorem sequence:
radius_of_circle_property_length_equal(1,OA,O)
radius_of_circle_property_length_equal(1,OB,O)
radius_of_circle_property_length_equal(1,OC,O)
isosceles_triangle_judgment_line_equal(1,OAB)
isosceles_triangle_judgment_line_equal(1,OCA)
isosceles_triangle_property_angle_equal(1,OAB)
isosceles_triangle_property_angle_equal(1,OCA)
angle_addition(1,CAO,OAB)
arc_property_circumference_angle_external(1,OBC,A)
arc_property_center_angle(1,OBC,O)
Problem: As shown in the diagram, AB=10, CD=15, triangle ABD is similar to triangle MHD., triangle MAB is similar to triangle MDC.. Find the length of line MH.
问题：如图所示，AB=10，CD=15，△ABD与△MHD是相似三角形，△MAB相似于△MDC。求直线MH的长度。
Construction (CDL):
Shape(AB,BM,MA)
Shape(MB,BH,HM)
Shape(MH,HD,DM)
Shape(MD,DC,CM)
Collinear(AMD)
Collinear(BMC)
Collinear(BHD)
Conditions (CDL):
Equal(LengthOfLine(AB),10)
Equal(LengthOfLine(CD),15)
SimilarBetweenTriangle(ABD,MHD)
SimilarBetweenTriangle(MAB,MDC)
Goal: Value(LengthOfLine(MH))
Answer: 6
Theorem sequence:
similar_triangle_property_line_ratio(1,MAB,MDC)
similar_triangle_property_line_ratio(1,BMA,CMD)
similar_triangle_property_line_ratio(1,BDA,HDM)
similar_triangle_property_line_ratio(1,DAB,DMH)
line_addition(1,AM,MD)
Problem: As shown in the diagram, BD=1, BE=3, DC=2, triangle ADC is similar to triangle ABE.. Find the length of line AB.
问题：如图所示，BD=1，BE=3，DC=2，△ADC与△ABE是相似三角形。求直线AB的长度。
Construction (CDL):
Shape(AD,DC,CA)
Shape(DB,BE,EC,CD)
Collinear(ADB)
Collinear(ACE)
Conditions (CDL):
Equal(LengthOfLine(BD),1)
Equal(LengthOfLine(BE),3)
Equal(LengthOfLine(DC),2)
SimilarBetweenTriangle(ADC,ABE)
Goal: Value(LengthOfLine(AB))
Answer: 3
Theorem sequence:
similar_triangle_property_line_ratio(1,ADC,ABE)
similar_triangle_property_line_ratio(1,CAD,EAB)
line_addition(1,AD,DB)
Problem: As shown in the diagram, BC=6, DB=1/2, EB=1, triangle DEB is similar to triangle ABC.. Find the length of line AC.
问题：如图所示，BC=6，DB=1/2，EB=1，△DEB相似于△ABC。求直线AC的长度。
Construction (CDL):
Shape(DE,EB,BD)
Shape(AB,BC,CA)
Shape(DB,BA)
Collinear(EBC)
Conditions (CDL):
Equal(LengthOfLine(BC),6)
Equal(LengthOfLine(DB),1/2)
Equal(LengthOfLine(EB),1)
SimilarBetweenTriangle(DEB,ABC)
Goal: Value(LengthOfLine(AC))
Answer: 3
Theorem sequence:
similar_triangle_property_line_ratio(1,DEB,ABC)
similar_triangle_property_line_ratio(1,EBD,BCA)
Problem: As shown in the diagram, AB=1/2, BC=1, CD=12, △CAB is similar to △CED.. Find the length of line DE.
问题：如图所示，AB=1/2，BC=1，CD=12，三角形CAB与三角形CED是相似三角形。求直线DE的长度。
Construction (CDL):
Shape(AB,BC,CA)
Shape(CE,ED,DC)
Collinear(ACE)
Collinear(BCD)
Conditions (CDL):
Equal(LengthOfLine(AB),1/2)
Equal(LengthOfLine(BC),1)
Equal(LengthOfLine(CD),12)
SimilarBetweenTriangle(CAB,CED)
Goal: Value(LengthOfLine(DE))
Answer: 6
Theorem sequence:
similar_triangle_property_line_ratio(1,CAB,CED)
similar_triangle_property_line_ratio(1,ABC,EDC)
Problem: As shown in the diagram, AB=8, BC=16/5, BD=22, triangle ABC is similar to triangle ADE.. Find the length of line DE.
问题：如图所示，AB=8，BC=16/5，BD=22，三角形ABC相似于三角形ADE。求直线DE的长度。
Construction (CDL):
Shape(AB,BC,CA)
Shape(CB,BD,DE,EC)
Collinear(ACE)
Collinear(ABD)
Conditions (CDL):
Equal(LengthOfLine(AB),8)
Equal(LengthOfLine(BC),16/5)
Equal(LengthOfLine(BD),22)
SimilarBetweenTriangle(ABC,ADE)
Goal: Value(LengthOfLine(DE))
Answer: 12
Theorem sequence:
line_addition(1,AB,BD)
similar_triangle_property_line_ratio(1,ABC,ADE)
similar_triangle_property_line_ratio(1,CAB,EAD)
Problem: As shown in the diagram, AB=1, BP=3/2, PD=12, △PAB is mirror similar to △PDC.. Find the length of line CD.
问题：如图所示，AB=1，BP=3/2，PD=12，△PAB与△PDC是镜像相似三角形。求直线CD的长度。
Construction (CDL):
Shape(AB,BP,PA)
Shape(PD,DC,CP)
Collinear(BPD)
Conditions (CDL):
Equal(LengthOfLine(AB),1)
Equal(LengthOfLine(BP),3/2)
Equal(LengthOfLine(PD),12)
MirrorSimilarBetweenTriangle(PAB,PDC)
Goal: Value(LengthOfLine(CD))
Answer: 8
Theorem sequence:
mirror_similar_triangle_property_line_ratio(1,PAB,PDC)
mirror_similar_triangle_property_line_ratio(1,ABP,CPD)
Problem: As shown in the diagram, ∠CAB=40°, the diameter of ⊙O is AB. Find the measure of ∠ABC.
问题：如图所示，∠CAB=40°，AB是⊙O的直径。求∠ABC的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(AC,OCA)
Shape(BA,OAB)
Shape(CB,OBC)
Collinear(AOB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),40)
IsDiameterOfCircle(AB,O)
Goal: Value(MeasureOfAngle(ABC))
Answer: 50
Theorem sequence:
diameter_of_circle_property_right_angle(1,BCA,O)
triangle_property_angle_sum(1,ABC)
Problem: As shown in the diagram, AC=3, BD=6, CD=10, △EAC is mirror similar to △EDB.. Find the length of line ED.
问题：如图所示，AC=3，BD=6，CD=10，△EAC与△EDB是镜像相似三角形。求直线ED的长度。
Construction (CDL):
Shape(AC,CE,EA)
Shape(ED,DB,BE)
Collinear(CED)
Conditions (CDL):
Equal(LengthOfLine(AC),3)
Equal(LengthOfLine(BD),6)
Equal(LengthOfLine(CD),10)
MirrorSimilarBetweenTriangle(EAC,EDB)
Goal: Value(LengthOfLine(ED))
Answer: 20/3
Theorem sequence:
mirror_similar_triangle_property_line_ratio(1,EAC,EDB)
mirror_similar_triangle_property_line_ratio(1,ACE,BED)
line_addition(1,CE,ED)
Problem: As shown in the diagram, AB=2, BD=7/3, CE=9, triangle ECA is similar to triangle DBA.. Find the length of line BC.
问题：如图所示，AB=2，BD=7/3，CE=9，△ECA相似于△DBA。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),2)
Equal(LengthOfLine(BD),7/3)
Equal(LengthOfLine(CE),9)
SimilarBetweenTriangle(ECA,DBA)
Goal: Value(LengthOfLine(BC))
Answer: 40/7
Theorem sequence:
line_addition(1,CB,BA)
similar_triangle_property_line_ratio(1,ECA,DBA)
similar_triangle_property_line_ratio(1,AEC,ADB)
Problem: As shown in the diagram, AD=3, BC=3/2, BD=1, triangle ABC is similar to triangle ADE.. Find the length of line DE.
问题：如图所示，AD=3，BC=3/2，BD=1，三角形ABC与三角形ADE是相似三角形。求直线DE的长度。
Construction (CDL):
Shape(AB,BC,CA)
Shape(BD,DE,EC,CB)
Collinear(ABD)
Collinear(ACE)
Conditions (CDL):
Equal(LengthOfLine(AD),3)
Equal(LengthOfLine(BC),3/2)
Equal(LengthOfLine(BD),1)
SimilarBetweenTriangle(ABC,ADE)
Goal: Value(LengthOfLine(DE))
Answer: 9/4
Theorem sequence:
line_addition(1,AB,BD)
similar_triangle_property_line_ratio(1,ABC,ADE)
similar_triangle_property_line_ratio(1,CAB,EAD)
Problem: As shown in the diagram, BD=5, CD=3, △ABD is similar to △ECD.. Find the ratio of the length of line AB to the length of line EC.
问题：如图所示，BD=5，CD=3，三角形ABD与三角形ECD是相似三角形。求直线AB的长度与直线EC的长度之比。
Construction (CDL):
Shape(AB,BC,CE,EA)
Shape(EC,CD,DE)
Collinear(AED)
Collinear(BCD)
Conditions (CDL):
Equal(LengthOfLine(BD),5)
Equal(LengthOfLine(CD),3)
SimilarBetweenTriangle(ABD,ECD)
Goal: Value(Div(LengthOfLine(AB),LengthOfLine(EC)))
Answer: 5/3
Theorem sequence:
similar_triangle_property_line_ratio(1,ABD,ECD)
similar_triangle_property_line_ratio(1,DAB,DEC)
Problem: As shown in the diagram, AB=3, AD=6, DE=16, triangle ADB is similar to triangle EDC.. Find the length of line EC.
问题：如图所示，AB=3，AD=6，DE=16，△ADB相似于△EDC。求直线EC的长度。
Construction (CDL):
Shape(AD,DB,BA)
Shape(AB,BC,CE,EA)
Collinear(DAE)
Collinear(DBC)
Conditions (CDL):
Equal(LengthOfLine(AB),3)
Equal(LengthOfLine(AD),6)
Equal(LengthOfLine(DE),16)
SimilarBetweenTriangle(ADB,EDC)
Goal: Value(LengthOfLine(EC))
Answer: 8
Theorem sequence:
similar_triangle_property_line_ratio(1,DBA,DCE)
similar_triangle_property_line_ratio(1,BAD,CED)
Problem: As shown in the diagram, AC=2, BC=2, DF=3/2, triangle ABC is similar to triangle DEF.. Find the length of line EF.
问题：如图所示，AC=2，BC=2，DF=3/2，三角形ABC与三角形DEF是相似三角形。求直线EF的长度。
Construction (CDL):
Shape(AB,BC,CA)
Shape(DE,EF,FD)
Conditions (CDL):
Equal(LengthOfLine(AC),2)
Equal(LengthOfLine(BC),2)
Equal(LengthOfLine(DF),3/2)
SimilarBetweenTriangle(ABC,DEF)
Goal: Value(LengthOfLine(EF))
Answer: 3/2
Theorem sequence:
similar_triangle_property_line_ratio(1,ABC,DEF)
similar_triangle_property_line_ratio(1,BCA,EFD)
Problem: As shown in the diagram, ∠BAO=20°, the center of circle O is O, OE⊥AE. Find the measure of ∠ABC.
问题：如图所示，∠BAO=20°，⊙O的圆心为O，OE垂直于AE。求∠ABC的大小。
Construction (CDL):
Shape(OA,OAB,BE,EO)
Shape(OE,EA,AO)
Shape(EB,BC,CE)
Shape(CB,OBC)
Shape(EC,OCA,AE)
Collinear(OEC)
Collinear(AEB)
Cocircular(O,ABC)
Conditions (CDL):
Equal(MeasureOfAngle(BAO),20)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(OE,AE)
Goal: Value(MeasureOfAngle(ABC))
Answer: 35
Theorem sequence:
triangle_property_angle_sum(1,OEA)
arc_property_center_angle(1,OCA,O)
arc_property_circumference_angle_external(1,OCA,B)
Problem: As shown in the diagram, BC=1, DE=3/2, EC=1, △ADC is similar to △BEC.. Find the length of line AB.
问题：如图所示，BC=1，DE=3/2，EC=1，△ADC与△BEC是相似三角形。求直线AB的长度。
Construction (CDL):
Shape(AD,DE,EB,BA)
Shape(BE,EC,CB)
Collinear(CED)
Collinear(ABC)
Conditions (CDL):
Equal(LengthOfLine(BC),1)
Equal(LengthOfLine(DE),3/2)
Equal(LengthOfLine(EC),1)
SimilarBetweenTriangle(ADC,BEC)
Goal: Value(LengthOfLine(AB))
Answer: 3/2
Theorem sequence:
line_addition(1,CE,ED)
line_addition(1,AB,BC)
similar_triangle_property_line_ratio(1,ADC,BEC)
similar_triangle_property_line_ratio(1,DCA,ECB)
Problem: As shown in the diagram, AB=6, AC=5, CD=30, ED∥AB. Find the length of line DE.
问题：如图所示，AB=6，AC=5，CD=30，ED∥AB。求直线DE的长度。
Construction (CDL):
Shape(EC,CD,DE)
Shape(CA,AB,BC)
Collinear(ACD)
Collinear(ECB)
Conditions (CDL):
Equal(LengthOfLine(AB),6)
Equal(LengthOfLine(AC),5)
Equal(LengthOfLine(CD),30)
ParallelBetweenLine(ED,AB)
Goal: Value(LengthOfLine(DE))
Answer: 36
Theorem sequence:
parallel_property_alternate_interior_angle(1,ED,AB)
parallel_property_alternate_interior_angle(2,ED,AB)
similar_triangle_judgment_aa(1,CDE,CAB)
similar_triangle_property_line_ratio(1,CDE,CAB)
similar_triangle_property_line_ratio(1,ECD,BCA)
Problem: As shown in the diagram, AB=CD, BC=DE, ∠CDE=128°. Find the measure of ∠ABC.
问题：如图所示，AB=CD，BC=DE，∠CDE=128°。求∠ABC的大小。
Construction (CDL):
Shape(CB,OBC)
Shape(DC,OCD)
Shape(ED,ODE)
Shape(AE,OEA)
Shape(BA,OAB)
Shape(AB,BC,CD,DE,EA)
Cocircular(O,ABCDE)
Conditions (CDL):
Equal(LengthOfLine(AB),LengthOfLine(CD))
Equal(LengthOfLine(BC),LengthOfLine(DE))
Equal(MeasureOfAngle(CDE),128)
Goal: Value(MeasureOfAngle(ABC))
Answer: 128
Theorem sequence:
congruent_arc_judgment_chord_equal(1,OAB,OCD)
congruent_arc_judgment_chord_equal(1,OBC,ODE)
congruent_arc_property_length_equal(1,OAB,OCD)
congruent_arc_property_length_equal(1,OBC,ODE)
arc_addition_length(1,OEA,OAB)
arc_addition_length(1,OEB,OBC)
arc_addition_length(1,OCD,ODE)
arc_addition_length(1,OCE,OEA)
congruent_arc_judgment_length_equal(1,OCA,OEC)
congruent_arc_property_measure_equal(1,OCA,OEC)
arc_property_circumference_angle_external(1,OCA,B)
arc_property_circumference_angle_external(1,OEC,D)
Problem: As shown in the diagram, BD=3, CD=2, HM=1, MN=1/2, triangle HMN is similar to triangle ABE., △HMN is similar to △CDE.. Find the length of line AB.
问题：如图所示，BD=3，CD=2，HM=1，MN=1/2，△HMN相似于△ABE，三角形HMN与三角形CDE是相似三角形。求直线AB的长度。
Construction (CDL):
Shape(HM,MN,NH)
Shape(AB,BD,DC,CA)
Shape(CD,DE,EC)
Collinear(BDE)
Collinear(ACE)
Conditions (CDL):
Equal(LengthOfLine(BD),3)
Equal(LengthOfLine(CD),2)
Equal(LengthOfLine(HM),1)
Equal(LengthOfLine(MN),1/2)
SimilarBetweenTriangle(HMN,ABE)
SimilarBetweenTriangle(HMN,CDE)
Goal: Value(LengthOfLine(AB))
Answer: 8
Theorem sequence:
similar_triangle_property_line_ratio(1,HMN,CDE)
similar_triangle_property_line_ratio(1,NHM,ECD)
similar_triangle_property_line_ratio(1,HMN,ABE)
similar_triangle_property_line_ratio(1,NHM,EAB)
line_addition(1,BD,DE)
Problem: As shown in the diagram, CD=8, DE=4, EF=2, △DFE is mirror similar to △DCB., ∠FED is the right angle of the right △ FED. Find the length of line BC.
问题：如图所示，CD=8，DE=4，EF=2，三角形DFE镜像相似于三角形DCB，FE和ED是直角三角形FED的两个直角边。求直线BC的长度。
Construction (CDL):
Shape(DF,FE,ED)
Shape(EF,FC,CB,BE)
Collinear(DEB)
Collinear(DFC)
Conditions (CDL):
Equal(LengthOfLine(CD),8)
Equal(LengthOfLine(DE),4)
Equal(LengthOfLine(EF),2)
MirrorSimilarBetweenTriangle(DFE,DCB)
RightTriangle(FED)
Goal: Value(LengthOfLine(BC))
Answer: 4
Theorem sequence:
mirror_similar_triangle_property_line_ratio(1,DFE,DCB)
mirror_similar_triangle_property_line_ratio(1,FED,BDC)
Problem: As shown in the diagram, OA=2, the ratio of similar △OBA and △OEF is 1/2, triangle OBA is similar to triangle OEF.. Find the length of line AF.
问题：如图所示，OA=2，三角形OBA和三角形OEF的相似比为1/2，三角形OBA与三角形OEF是相似三角形。求直线AF的长度。
Construction (CDL):
Shape(AO,OB,BA)
Shape(OE,EF,FO)
Collinear(AOF)
Collinear(BOE)
Conditions (CDL):
Equal(LengthOfLine(OA),2)
Equal(RatioOfSimilarTriangle(OBA,OEF),1/2)
SimilarBetweenTriangle(OBA,OEF)
Goal: Value(LengthOfLine(AF))
Answer: 6
Theorem sequence:
similar_triangle_property_line_ratio(1,BAO,EFO)
line_addition(1,AO,OF)
Problem: As shown in the diagram, ∠ECB=130°. Find the measure of ∠CBF minus the measure of ∠CAB.
问题：如图所示，∠ECB=130°。求∠CBF的大小减去∠CAB的大小。
Construction (CDL):
Shape(AB,BC,CA)
Shape(EC,CB)
Shape(CB,BF)
Collinear(ABF)
Collinear(ACE)
Conditions (CDL):
Equal(MeasureOfAngle(ECB),130)
Goal: Value(Sub(MeasureOfAngle(CBF),MeasureOfAngle(CAB)))
Answer: 50
Theorem sequence:
adjacent_complementary_angle(1,ABC,CBF)
adjacent_complementary_angle(1,ECB,BCA)
triangle_property_angle_sum(1,ABC)
Problem: As shown in the diagram, ∠ABO=26°, O is the center of circle O, AF is perpendicular to OF. Find the measure of ∠CDA.
问题：如图所示，∠ABO=26°，O是⊙O的圆心，AF垂直于OF。求∠CDA的大小。
Construction (CDL):
Shape(AD,ODA)
Shape(AE,ED,DA)
Shape(EA,OAC,CE)
Shape(EC,CF,FE)
Shape(OF,FB,BO)
Shape(BF,FC,OCB)
Shape(DE,EF,FO,OB,OBD)
Collinear(DEC)
Collinear(OFC)
Collinear(AEFB)
Cocircular(O,ACBD)
Conditions (CDL):
Equal(MeasureOfAngle(ABO),26)
IsCentreOfCircle(O,O)
PerpendicularBetweenLine(AF,OF)
Goal: Value(MeasureOfAngle(CDA))
Answer: 32
Theorem sequence:
triangle_property_angle_sum(1,OFB)
arc_property_center_angle(1,OCB,O)
circle_property_chord_perpendicular_bisect_chord(1,O,OF,AB)
circle_property_chord_perpendicular_bisect_arc(1,OAB,OFC)
congruent_arc_judgment_length_equal(1,OAC,OCB)
congruent_arc_property_measure_equal(1,OAC,OCB)
arc_property_circumference_angle_external(1,OAC,D)
Problem: As shown in the diagram, the perimeter of triangle EFG is 12, the ratio of similar triangles ABC and EFG is 1/3, triangle ABC is similar to triangle EFG.. Find the perimeter of △ABC.
问题：如图所示，三角形EFG的周长为12，三角形ABC和三角形EFG的相似比为1/3，△ABC相似于△EFG。求△ABC的周长。
Construction (CDL):
Shape(AB,BC,CA)
Shape(EF,FG,GE)
Conditions (CDL):
Equal(PerimeterOfTriangle(EFG),12)
Equal(RatioOfSimilarTriangle(ABC,EFG),1/3)
SimilarBetweenTriangle(ABC,EFG)
Goal: Value(PerimeterOfTriangle(ABC))
Answer: 4
Theorem sequence:
similar_triangle_property_perimeter_ratio(1,ABC,EFG)
Problem: As shown in the diagram, ∠CAB=35°, ∠DBA=40°. Find the measure of ∠DPA.
问题：如图所示，∠CAB=35°，∠DBA=40°。求∠DPA的大小。
Construction (CDL):
Shape(AC,OCA)
Shape(CA,AP,PC)
Shape(PA,OAD,DP)
Shape(PD,DB,BP)
Shape(BD,ODB)
Shape(CP,PB,OBC)
Collinear(APB)
Collinear(CPD)
Cocircular(O,ADBC)
Conditions (CDL):
Equal(MeasureOfAngle(CAB),35)
Equal(MeasureOfAngle(DBA),40)
Goal: Value(MeasureOfAngle(DPA))
Answer: 75
Theorem sequence:
arc_property_circumference_angle_external(1,OBC,A)
arc_property_circumference_angle_external(1,OBC,D)
triangle_property_angle_sum(1,PDB)
adjacent_complementary_angle(1,BPD,DPA)
Problem: As shown in the diagram, the area of △ABC is 18, the ratio of similar triangles ADE and ABC is 2/3, triangle ADE is similar to triangle ABC.. Find the area of triangle ADE.
问题：如图所示，三角形ABC的面积为18，三角形ADE和三角形ABC的相似比为2/3，△ADE与△ABC是相似三角形。求△ADE的面积。
Construction (CDL):
Shape(AD,DE,EA)
Shape(DB,BC,CE,ED)
Collinear(ADB)
Collinear(AEC)
Conditions (CDL):
Equal(AreaOfTriangle(ABC),18)
Equal(RatioOfSimilarTriangle(ADE,ABC),2/3)
SimilarBetweenTriangle(ADE,ABC)
Goal: Value(AreaOfTriangle(ADE))
Answer: 8
Theorem sequence:
similar_triangle_property_area_square_ratio(1,ADE,ABC)
Problem: As shown in the diagram, AC=4, BC=3, ∠ACB=90°. Find the value of cos(CB).
问题：如图所示，AC=4，BC=3，∠ACB=90°。求cos(CB)的值。
Construction (CDL):
Shape(BA,AC,CB)
Conditions (CDL):
Equal(LengthOfLine(AC),4)
Equal(LengthOfLine(BC),3)
Equal(MeasureOfAngle(ACB),90)
Goal: Value(Cos(MeasureOfAngle(CBA)))
Answer: 3/5
Theorem sequence:
right_triangle_judgment_angle(1,ACB)
right_triangle_property_pythagorean(1,ACB)
cosine_theorem(1,BAC)
Problem: As shown in the diagram, AC=4, BC=3, ∠ACB=90°. Find the value of sin(BA).
问题：如图所示，AC=4，BC=3，∠ACB=90°。求sin(BA)的值。
Construction (CDL):
Shape(AC,CB,BA)
Conditions (CDL):
Equal(LengthOfLine(AC),4)
Equal(LengthOfLine(BC),3)
Equal(MeasureOfAngle(ACB),90)
Goal: Value(Sin(MeasureOfAngle(BAC)))
Answer: 3/5
Theorem sequence:
right_triangle_judgment_angle(1,ACB)
right_triangle_property_pythagorean(1,ACB)
sine_theorem(1,ACB)
sine_theorem(1,CBA)
Problem: As shown in the diagram, AC=4, BC=3, ∠BCA=90°. Find the value of sin(AB).
问题：如图所示，AC=4，BC=3，∠BCA=90°。求sin(AB)的值。
Construction (CDL):
Shape(AB,BC,CA)
Conditions (CDL):
Equal(LengthOfLine(AC),4)
Equal(LengthOfLine(BC),3)
Equal(MeasureOfAngle(BCA),90)
Goal: Value(Sin(MeasureOfAngle(ABC)))
Answer: 4/5
Theorem sequence:
right_triangle_judgment_angle(1,BCA)
right_triangle_property_pythagorean(1,BCA)
sine_theorem(1,ABC)
Problem: As shown in the diagram, AC=3, BC=4, ∠ACB=90°. Find the value of cos(BA).
问题：如图所示，AC=3，BC=4，∠ACB=90°。求cos(BA)的值。
Construction (CDL):
Shape(AC,CB,BA)
Conditions (CDL):
Equal(LengthOfLine(AC),3)
Equal(LengthOfLine(BC),4)
Equal(MeasureOfAngle(ACB),90)
Goal: Value(Cos(MeasureOfAngle(BAC)))
Answer: 3/5
Theorem sequence:
cosine_theorem(1,ACB)
cosine_theorem(1,CBA)
Problem: As shown in the diagram, AC=4, BC=8, ∠BCA=90°. Find the value of tan(AB).
问题：如图所示，AC=4，BC=8，∠BCA=90°。求tan(AB)的值。
Construction (CDL):
Shape(AB,BC,CA)
Conditions (CDL):
Equal(LengthOfLine(AC),4)
Equal(LengthOfLine(BC),8)
Equal(MeasureOfAngle(BCA),90)
Goal: Value(Tan(MeasureOfAngle(ABC)))
Answer: 1/2
Theorem sequence:
right_triangle_judgment_angle(1,BCA)
right_triangle_property_pythagorean(1,BCA)
sine_theorem(1,ABC)