KöMaL 2009 C998

The points $A$, $B$, $C$ and $D$ lie on a line in this order, and $AB$=$BC$. Draw perpendiculars to $AD$ at $B$and $C$. The perpendicular drawn at $B$ intersects the circle of diameter $AD$ at $P$ and $Q$. The perpendicular drawn at $C$intersects the circle of diameter $BD$ at $K$ and $L$. Show that the centre of the circle passing through points $P$, $K$, $L$ and $Q$ is point $B$.