ISL 2000 G5

The tangents at $B$ and $A$ to the circumcircle of an acute angled triangle $ABC$ meet the tangent at $C$ at $T$ and $U$ respectively. $AT$ meets $BC$ at $P$ , and $Q$ is the midpoint of $AP$ ; $BU$ meets $CA$ at $R$ , and $S$ is the midpoint of $BR$ . Prove that $\angle ABQ=\angle BAS$ . Determine, in terms of ratios of side lengths, the triangles for which this angle is a maximum.