Let $\Gamma$ be a circle with center $O$ and $AE$ be a diameter. Point $D$ lies on segment $OE$ and point $B$ is the midpoint of one of the arcs $\widehat{AE}$ of $\Gamma$. Construct point $C$ such that $ABCD$ is a parallelogram. Lines $EB$ and $CD$ meet at $F$. Line $OF$ meets the minor arc $\widehat{EB}$ at $I$. Prove that $EI$ bisects $\angle BEC$.