NL-NWO 1967 Problem 1

In this exercise we only consider convex quadrilaterals.

(a) For such a quadrilateral $ABCD$ , determine the set of points $P$ contained within that quadrilateral for which $PA$ and $PC$ divide the quadrilateral into two pieces of equal areas.

(b) Prove that there is a point $P$ inside such a quadrilateral, such that the triangles $PAB$ and $PCD$ have equal areas, as well as the triangles $PBC$ and $PAD$ .

(c) Find out which quadrilaterals $ABCD$ contains a point $P$ , so that the triangles $PAB$ , $PBC$ , $PCD$ and $PDA$ have equal areas.