Let $ABCD$ be a convex quadrilateral. Is it possible to find a point $P$ such that the segments drawn between $P$ and the midpoints of the sides of $ABCD$ divide the quadrilateral into four sections of equal area? If $P$ exists, is it unique?
Problem
Source: Taiwan 1st TST 2005, 3rd independent study, problem 2
Tags: geometry, geometry proposed