VMO 2005 — Division A Question 2

Let $(O)$ be a fixed circle of radius $R$. Fixed points $A,B$ on $(O)$ satisfy that $A,B,O$ are not collinear. Let $C$ vary on $(O)$, with $C\ne A,B$. Construct circles $(O_1),(O_2)$ passing through $A,B$ and tangent at $C$ to $BC,AC$, respectively. The circles meet again at $D\ne C$. Prove that:
(a) $CD\le R$;
(b) line $CD$ always passes through a point independent of $C$.