GULF 2012 Problem 1

Let $X,\ Y$ and $Z$ be the midpoints of sides $BC,\ CA$ , and $AB$ of the triangle $ABC$ , respectively. Let $P$ be a point inside the triangle. Prove that the quadrilaterals $AZPY,\ BXPZ$ , and $CYPX$ have equal areas if, and only if, $P$ is the centroid of $ABC$ .