ZA-SAMO 2007 Problem 4

Let $ABC$ be a triangle and $PQRS$ a square with $P$ on $AB$ , $Q$ on $AC$ , and $R$ and $S$ on $BC$ . Let $H$ on $BC$ such that $AH$ is the altitude of the triangle from $A$ to base $BC$ . Prove that:

(a) $\frac{1}{AH} +\frac{1}{BC}=\frac{1}{PQ}$ (b) the area of $ABC$ is twice the area of $PQRS$ iff $AH=BC$