DE-BUNDESWETTBEWERB 2003 Problem 7

Consider a cyclic quadrilateral $ABCD$ , and let $S$ be the intersection of $AC$ and $BD$ . Let $E$ and $F$ the orthogonal projections of $S$ on $AB$ and $CD$ respectively. Prove that the perpendicular bisector of segment $EF$ meets the segments $AD$ and $BC$ at their midpoints.