BALKAN-SL 2017 G8

Given an acute triangle $\triangle A B C(A C \neq A B)$ and let $(C)$ be its circumcircle. The excircle $\left(C_{1}\right)$ corresponding to the vertex $A$, of center $I_{a}$, tangents to the side $B C$ at the point $D$ and to the extensions of the sides $A B, A C$ at the points $E, Z$ respectively. Let $I$ and $L$ are the intersection points of the circles $(C)$ and $\left(C_{1}\right), H$ the orthocenter of the triangle $\triangle E D Z$ and $N$ the midpoint of segment $E Z$. The parallel line through the point $I_{a}$ to the line $H L$ meets the line $H I$ at the point $G$. Prove that the perpendicular line ( $e$ ) through the point $N$ to the to the line $B C$ and the parallel line $(\delta)$ through the point $G$ to the line $I L$ meet each other on the line $\mathrm{HI}_{a}$