Can someone help me with that proof.
In triangle IEG
H is on GI such that EG+GH=EI+IH
J is on EI such that GI+IJ=EG+EJ
F is on EG such that GI+GF=EI+EF
Prove that EH,GJ,IF are concurrent
Let $\displaystyle EF=a, \ FG=b, \ GH=c, \ HI=d, \ IJ=e, \ JE=f$
We have:
$\displaystyle \left\{\begin{array}{ll}e+f+a=d+c+b\\a+b+f=c+d+e\e nd{array}\right.\Rightarrow b=e$
$\displaystyle \left\{\begin{array}{ll}c+d+e=a+b+f\\d+e+f=a+b+c\e nd{array}\right.\Rightarrow c=f$
$\displaystyle \left\{\begin{array}{ll}a+e+f=b+c+d\\d+e+f=a+b+c\e nd{array}\right.\Rightarrow a=d$
Then, $\displaystyle \frac{FE}{FG}\cdot\frac{HG}{HI}\cdot\frac{JI}{IE}= \frac{a}{b}\cdot\frac{c}{d}\cdot\frac{e}{f}=1$
And, from Ceva's theorem, EH, GJ, IF are concurrent.