Question

一道数学问题,问题在问题补充里一个任意三角形,在一个边上取两个三等分点.在另一条边上取一个二等分点,分别将这三个点与对应

Answer 1

提示一下

设三角形ABC,AB边上有三等分点D、E,BC边上二等分点F,AF交DC、EC于G、N,

过D作DM平行BC交AF于M、过E作EN平行BC交AF于N

设三角形ABC面积为S

AD/AB=DM/BF=DM/CF=DG/GC=1/3,DG/CD=1/(1+3)=1/4

则S(ADC)=S/3,S(ADG)=S(ADC)/4=S/12,S(AGC)=S/3-S/12=S/4

AE/AB=EN/BF=EN/CF=EG/GC=2/3,EG/EC=2/(2+3)=2/5

则S(AEC)=2S/3,S(AEH)=S(AEC)*2/5=4S/15,S(AHC)=2S/3-4S/15=2S/5

S(EDGH)=S(AEH)-S(ADG)=4S/15-S/12=11/60

S(GHC)=S(AHC)-S(AGC)=2S/5-S/4=3S/20

S(ABF)=S/2,S(AFC)=S/2

S(BEHF)=S(ABF)-S(AEH)=S/2-4S/15=7S/30

S(GFC)=S(AFC)-S(AHC)=S/2-2S/5=S/10

面积最大的三角形为AGC,则S/4=60,S=240

则S(ADG)=20

S(EDGH)=44

S(BEHF)=56

S(AGC)=60

S(GHC)=36

S(GFC)=24