Question 78
Let $ABC$ be a scalene triangle. The inscribed circle of $ABC$ touches the sides $BC$, $CA$ and $AB$ at the points $A_1$, $B_1$ and $C_1$ respectively. Let $I$ be the center of the inscribed circle, $O$ be the center of the circumscribed circle and $OI \cap BC = D$. The perpendicular line from $A_1$ to $B_1C_1$ intersects $AD$ at the point $E$. Prove that $B_1C_1$ passes through the midpoint of $EA_1$.