Problem

Source: 2009 Balkan Shortlist BMO G3 - difficult

Tags: diagonals, projections, geometry, midpoints, convex quadrilateral, Circumcenter, Isogonal conjugate



Let $ABCD$ be a convex quadrilateral, and $P$ be a point in its interior. The projections of $P$ on the sides of the quadrilateral lie on a circle with center $O$. Show that $O$ lies on the line through the midpoints of $AC$ and $BD$.