AUSTRIAN-POLISH 1997 Problem 4

In a trapezoid $ABCD$ with $AB // CD$ , the diagonals $AC$ and $BD$ intersect at point $E$ . Let $F$ and $G$ be the orthocenters of the triangles $EBC$ and $EAD$ . Prove that the midpoint of $GF$ lies on the perpendicular from $E$ to $AB$ .