RIOPLATENSE 1990 Problem 3

Let $ABCD$ be a trapezium with bases $AB$ and $CD$ such that $AB = 2 CD$ . From $A$ the line $r$ is drawn perpendicular to $BC$ and from $B$ the line $t$ is drawn perpendicular to $AD$ . Let $P$ be the intersection point of $r$ and $t$ . From $C$ the line $s$ is drawn perpendicular to $BC$ and from $D$ the line $u$ perpendicular to $AD$ . Let $Q$ be the intersection point of $s$ and $u$ . If $R$ is the intersection point of the diagonals of the trapezium, prove that points $P, Q$ and $R$ are collinear.