BALKAN 2011 Problem 1

Let $A B C D$ be a cyclic quadrilateral which is not a trapezoid and whose diagonals meet at $E$. The midpoints of $A B$ and $C D$ are $F$ and $G$ respectively, and $\ell$ is the line through $G$ parallel to $A B$. The feet of the perpendiculars from $E$ onto the lines $\ell$ and $C D$ are $H$ and $K$, respectively. Prove that the lines $E F$ and $H K$ are perpendicular.