OSN 2007 Problem 7

Points $A,B,C,D$ are on circle $S$ , such that $AB$ is the diameter of $S$ , but $CD$ is not the diameter. Given also that $C$ and $D$ are on different sides of $AB$ . The tangents of $S$ at $C$ and $D$ intersect at $P$ . Points $Q$ and $R$ are the intersections of line $AC$ with line $BD$ and line $AD$ with line $BC$ , respectively.

(a) Prove that $P$ , $Q$ , and $R$ are collinear.

(b) Prove that $QR$ is perpendicular to line $AB$ .