OMM 2003 Problem 2

$A, B, C$ are collinear with $B$ betweeen $A$ and $C$ . $K_{1}$ is the circle with diameter $AB$ , and $K_{2}$ is the circle with diameter $BC$ . Another circle touches $AC$ at $B$ and meets $K_{1}$ again at $P$ and $K_{2}$ again at $Q$ . The line $PQ$ meets $K_{1}$ again at $R$ and $K_{2}$ again at $S$ . Show that the lines $AR$ and $CS$ meet on the perpendicular to $AC$ at $B$ .