NO-ABEL 1998 Problem 4

Let $A,B,P$ be points on a line $\ell$ , with $P$ outside the segment $AB$ . Lines $a$ and $b$ pass through $A$ and $B$ and are perpendicular to $\ell$ . A line $m$ through $P$ , which is neither parallel nor perpendicular to $\ell$ , intersects $a$ and $b$ at $Q$ and $R$ , respectively. The perpendicular from $B$ to $AR$ meets $a$ and $AR$ at $S$ and $U$ , and the perpendicular from $A$ to $BQ$ meets $b$ and $BQ$ at $T$ and $V$ , respectively. (a) Prove that $P,S,T$ are collinear. (b) Prove that $P,U,V$ are collinear.