Problem

Source: Bosnia and Herzegovina Junior Balkan Mathematical Olympiad TST 2016

Tags: geometry, perpendicular, quadrilateral, perimeter



Let $O$ be a center of circle which passes through vertices of quadrilateral $ABCD$, which has perpendicular diagonals. Prove that sum of distances of point $O$ to sides of quadrilateral $ABCD$ is equal to half of perimeter of $ABCD$.