AUSTRIAN-POLISH 2002 Problem 6

The diagonals of a convex quadrilateral $ABCD$ intersect in the point $E$ . Let $U$ be the circumcenter of the triangle $ABE$ and $H$ be its orthocenter. Similarly, let $V$ be the circumcenter of the triangle $CDE$ and $K$ be its orthocenter. Prove that $E$ lies on the line $UK$ if and only if it lies on the line $VH$ .