NO-ABEL 2006 Problem 4

Let $\gamma$ be the circumscribed circle about a right-angled triangle $ABC$ with right angle $C$ . Let $\delta$ be the circle tangent to the sides $AC$ and $BC$ and tangent to the circle $\gamma$ internally. (a) Find the radius $i$ of $\delta$ in terms of $a$ when $AC$ and $BC$ both have length $a$ . (b) Show that the radius $i$ is twice the radius of the inscribed circle of $ABC$ .