BALTIC-WAY 2007 Problem 11

In triangle $ABC$ let $AD,BE$ and $CF$ be the altitudes. Let the points $P,Q,R$ and $S$ fulfil the following requirements: i) $P$ is the circumcentre of triangle $ABC$ . ii) All the segments $PQ,QR$ and $RS$ are equal to the circumradius of triangle $ABC$ . iii) The oriented segment $PQ$ has the same direction as the oriented segment $AD$ . Similarly, $QR$ has the same direction as $BE$ , and $Rs$ has the same direction as $CF$ . Prove that $S$ is the incentre of triangle $ABC$ .