Problem

Source: Romania JBMO TST 2015 Day 1 Problem 5

Tags: geometry, quadrilateral, orthocenter, parallel



Let ABCD be a convex quadrilateral with non perpendicular diagonals and with the sides AB and CD non parallel . Denote by O the intersection of the diagonals , H1 the orthocenter of the triangle AOB and H2 the orthocenter of the triangle COD . Also denote with M the midpoint of the side AB and with N the midpoint of the side CD . Prove that H1H2 and MN are parallel if and only if AC=BD