Question 15
For a triangle $ABC$ let $B_1$, $C_1$ be the excenters corresponding to $B$, $C$, respectively. The line $B_1C_1$ intersects the circumcircle of $ABC$ at $D$ ($\neq A$). Let $E$ be the intersection of the perpendiculars from $B_1$ to $CA$ and from $C_1$ to $AB$. Let $\omega$ be the circumcircle of the triangle $ADE$. The tangent to $\omega$ at $D$ intersects the line $AE$ at $F$. The perpendicular from $D$ to $AE$ intersects $AE$ at $G$ and this line intersects $\omega$ at $H$ ($\neq D$). The circumcircle of the triangle $HGF$ intersects $\omega$ at $I$ ($\neq H$). Let $J$ be the perpendicular foot at $AH$ from $D$. Show that $AI$ passes through the midpoint of $DJ$.