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.
Problem
Source: Morrocan TST 2005 (pb 4)
Tags: geometry, cyclic quadrilateral, perpendicular bisector, geometry proposed