Let $ABC$ be a scalene triangle. The inscribed circle of $ABC$ touches the sides $BC$, $CA$, and $AB$ at the points $A_1$, $B_1$, $C_1$ respectively. Let $I$ be the incenter, $O$ be the circumcenter, and lines $OI$ and $BC$ meet at point $D$. The perpendicular line from $A_1$ to $B_1 C_1$ intersects $AD$ at point $E$. Prove that $B_1 C_1$ passes through the midpoint of $EA_1$.