FKMO 2013 Problem 4

For a triangle $ABC$ , let $B_1 ,C_1$ be the excenters of $B, C$ . Line $B_1 C_1$ meets with the circumcircle of $\triangle ABC$ at point $D (\ne A)$ . $E$ is the point which satisfies $B_1 E \bot CA$ and $C_1 E \bot AB$ . Let $w$ be the circumcircle of $\triangle ADE$ . The tangent to the circle $w$ at $D$ meets $AE$ at $F$ . $G , H$ are the points on $AE, w$ such that $DGH \bot AE$ . The circumcircle of $\triangle HGF$ meets $w$ at point $I ( \ne H )$ , and $J$ be the foot of perpendicular from $D$ to $AH$ . Prove that $AI$ passes the midpoint of $DJ$ .