International Mathematical Olympiad 1993 Problem 2

Let $D$ be a point inside acute triangle $ABC$ such that $\angle ADB = \angle ACB + \pi/2$ and $AC \cdot BD = AD \cdot BC$.

(a) Calculate the ratio $(AB \cdot CD)/(AC \cdot BD)$.

(b) Prove that the tangents at $C$ to the circumcircles of $\triangle ACD$ and $\triangle BCD$ are perpendicular.