AUSTRIAN-POLISH 1999 Problem 8

Let $P,Q,R$ be points on the same side of a line $g$ in the plane. Let $M$ and $N$ be the feet of the perpendiculars from $P$ and $Q$ to $g$ respectively. Point $S$ lies between the lines $PM$ and $QN$ and satisfies and satisfies $PM = PS$ and $QN = QS$ . The perpendicular bisectors of $SM$ and $SN$ meet in a point $R$ . If the line $RS$ intersects the circumcircle of triangle $PQR$ again at $T$ , prove that $S$ is the midpoint of $RT$ .