2013 OIM Problems/Problem 4

Problem

Let $\Gamma$ be a circle with center $O$ , $AE$ be a diameter of $\Gamma$ and $B$ be the midpoint of one of the arcs $AE$ of $\Gamma$ . Point $D \ne E$ is on segment $OE$ . 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 $CD$ intersect at point $F$ . The line $OF$ cuts the minor arc $EB$ of $\Gamma$ at point $I$ . Prove that the line $EI$ is the bisector of the angle $BEC$ .