Prove that this segment bisects another 
The circle touches the trapezoid $GFEC$ at the points $C$, $D$ and $E$. The point $A$ is the center of the circle. The rest of the information can be seen in the diagrams below. What we have to prove is that $FI=HI$.

I've added some diagrams below. The first one simply shows what is given and the others show what I've tried. I've tried a lot more than that but the other information  is basically useless (the information I've given here isn't quite useful either, though).


 

Let $s$ the line defined by $FE$, $t$ the line defined by $CD$ And $J$ the point of intersection between $s$ and $t$. See the following figure:

Hints:
Note that $\angle GDC = \angle GCD = \angle FDJ = \angle DJF$.
Therefore $FJ=ED$.
But $FD=FE$ (Why?).
Note that $\triangle JFI \sim \triangle JEC$, hence ... 
