TR-MO 2007 Problem 5

Let $ABC$ be a triangle with $\angle B=90$ . The incircle of $ABC$ touches the side $BC$ at $D$ . The incenters of triangles $ABD$ and $ADC$ are $X$ and $Z$ , respectively. The lines $XZ$ and $AD$ are intersecting at the point $K$ . $XZ$ and circumcircle of $ABC$ are intersecting at $U$ and $V$ . Let $M$ be the midpoint of line segment $[UV]$ . $AD$ intersects the circumcircle of $ABC$ at $Y$ other than $A$ . Prove that $|CY|=2|MK|$ .