1993 IMO Problems/Problem 2

Let $D$ be a point inside acute triangle $ABC$ such that $\angle ADB = \angle ACB+\frac{\pi}{2}$ and $AC\cdot BD=AD\cdot BC$ . \renewcommand{\labelenumi}{\alph{enumi}} \begin{enumerate} \item Calculate the ratio $\frac{AC\cdot CD}{AC\cdot BD}$ \item Prove that the tangents at $C$ to the circumcircles of $\triangle ACD$ and $\triangle BCD$ are perpendicular. \end{enumerate}