AT-OEMO-REG 2011 Problem 3

Let $k$ be a circle centered at $M$ and let $t$ be a tangentline to $k$ through some point $T\in k$ . Let $P$ be a point on $t$ and let $g\neq t$ be a line through $P$ intersecting $k$ at $U$ and $V$ . Let $S$ be the point on $k$ bisecting the arc $UV$ not containing $T$ and let $Q$ be the the image of $P$ under a reflection over $ST$ . Prove that $Q$ , $T$ , $U$ and $V$ are vertices of a trapezoid.