Question 113
Let $ABC$ be a right triangle with hypotenuse $AB$. A circle centred at $C$ meets the segment $AB$ twice, at the points $P$ and $Q$, where $P$ lies between $A$ and $Q$. Let $R$ be the point on the segment $BC$ with $\angle RAC = \frac{1}{2} \angle PCQ$ and let $S$ be the point on the segment $AC$ with $\angle CBS = \frac{1}{2} \angle PCQ$. Further, let $T$ be the intersection of the segments $CP$ and $AR$, and $U$ the intersection of the segments $CQ$ and $BS$. Show that $RSTU$ is a cyclic quadrilateral.