Hint: In the picture $GH||BI$. You have to show:

1-$HI||BG$

2- Circle $c_1$ is tangent to BC atK and toAB at A.

3-$KI||FH$

In this way:

$\triangle GFH=\triangle ABI \Rightarrow AB=FG$

Hint: In the picture $GH||BI$. You have to show:

1-$HI||BG$

2- Circle $c_1$ is tangent to $BC$ at $K$ and to $AB$ at $A$.

3-$KI||FH$

In this way:

$\triangle GFH=\triangle ABI \Rightarrow AB=FG$