VMO 2005 Problem 2

Let $(O)$ be a fixed circle with the radius $R$ . Let $A$ and $B$ be fixed points in $(O)$ such that $A,B,O$ are not collinear. Consider a variable point $C$ lying on $(O)$ ( $C\neq A,B$ ). Construct two circles $(O_1),(O_2)$ passing through $A,B$ and tangent to $BC,AC$ at $C$ , respectively. The circle $(O_1)$ intersects the circle $(O_2)$ in $D$ ( $D\neq C$ ). Prove that: a) $$CD\leq R$$ b) The line $CD$ passes through a point independent of $C$ (i.e. there exists a fixed point on the line $CD$ when $C$ lies on $(O)$ ).