RMO 2016 A1

Let $ABC$ be a right-angled triangle with $\angle B=90^{\circ}$ . Let $I$ be the incenter of $ABC$ . Draw a line perpendicular to $AI$ at $I$ . Let it intersect the line $CB$ at $D$ . Prove that $CI$ is perpendicular to $AD$ and prove that $ID=\sqrt{b(b-a)}$ where $BC=a$ and $CA=b$ .