OBM 2016 Problem 1

Let $ABC$ be a triangle. $r$ and $s$ are the angle bisectors of $\angle ABC$ and $\angle BCA$ , respectively. The points $E$ in $r$ and $D$ in $s$ are such that $AD \| BE$ and $AE \| CD$ . The lines $BD$ and $CE$ cut each other at $F$ . $I$ is the incenter of $ABC$ .

Show that if $A,F,I$ are collinear, then $AB=AC$ .