OIM 2013 Problem 4

Let $\Gamma$ be a circunference and $O$ its center. $AE$ is a diameter of $\Gamma$ and $B$ the midpoint of one of the arcs $AE$ of $\Gamma$ . The point $D \ne E$ in on the segment $OE$ . The point $C$ is such that the quadrilateral $ABCD$ is a parallelogram, with $AB$ parallel to $CD$ and $BC$ parallel to $AD$ . The lines $EB$ and $CD$ meets at point $F$ . The line $OF$ cuts the minor arc $EB$ of $\Gamma$ at $I$ .

Prove that the line $EI$ is the angle bissector of $\angle BEC$ .