CMO 1969 Problem 10

Let $ABC$ be the right-angled isosceles triangle whose equal sides have length 1. $P$ is a point on the hypotenuse, and the feet of the perpendiculars from $P$ to the other sides are $Q$ and $R$ . Consider the areas of the triangles $APQ$ and $PBR$ , and the area of the rectangle $QCRP$ . Prove that regardless of how $P$ is chosen, the largest of these three areas is at least $2/9$ .