BALTIC-WAY 2013 Problem 11

In an acute triangle $A B C$ with $A C>A B$, let $D$ be the projection of $A$ on $B C$, and let $E$ and $F$ be the projections of $D$ on $A B$ and $A C$, respectively. Let $G$ be the intersection point of the lines $A D$ and $E F$. Let $H$ be the second intersection point of the line $A D$ and the circumcircle of triangle $A B C$. Prove that

$$A G \cdot A H=A D^{2}$$