KöMaL 2006 B3958

The triangle ABC is right angled at $C$ and its incentre is $O$. The perpendiculars drawn at the point $O$ to the line segments $OA$ and $OB$ meet the side $AB$ at $P$ and $Q$, respectively. The foot of the perpendicular from $P$ to the side $BC$ is $P$', and that from $Q$ to the side $AC$ is $Q$'. Prove that the points $P$', $Q$' and $O$ are collinear.