Školjka

Let $ABC$ be a triangle with circumcircle $\Omega$ and incentre $I$. Let the line passing through $I$ and perpendicular to $CI$ intersect the segment $BC$ and the arc $BC$ (not containing $A$) of $\Omega$ at points $U$ and $V$, respectively. Let the line passing through $U$ and parallel to $AI$ intersect $AV$ at $X$, and let the line passing through $V$ and parallel to $AI$ intersect $AB$ at $Y$. Let $W$ and $Z$ be the midpoints of $AX$ and $BC$, respectively. Prove that if the points $I$, $X$, and $Y$ are collinear, then the points $I$, $W$, and $Z$ are also collinear. \begin{flushright}\emph{(U.S.A.)}\end{flushright}