Question 110
Let $ABC$ be a triangle with $m(\angle B) = 90^\circ$. The incircle of $\triangle ABC$ is tangent to the side $BC$ at point $D$. Let the intersection of lines $XZ$ and $AD$ be $K$, where $X$ and $Z$ are the centers of incircles of $\triangle ABD$ and $\triangle ADC$, respectively. $XZ$ intersects the circumscribed circle of $\triangle ABC$ at $U$ and $V$ and the midpoint of segment $[UV]$ is $M$. Let $Y$ be the point of intersection of the line $AD$ with the circumscribed circle of $\triangle ABC$ different from $A$. Prove that $|CY| = 2|MK|$. (Cafer Tayyar Yıldırım).