2014 IMO Shortlist G7

Let $A B C$ be a triangle with circumcircle $\Omega$ and incentre $I$. Let the line passing through $I$ and perpendicular to $C I$ intersect the segment $B C$ and the $\operatorname{arc} B C($ not containing $A)$ of $\Omega$ at points $U$ and $V$, respectively. Let the line passing through $U$ and parallel to $A I$ intersect $A V$ at $X$, and let the line passing through $V$ and parallel to $A I$ intersect $A B$ at $Y$. Let $W$ and $Z$ be the midpoints of $A X$ and $B C$, respectively. Prove that if the points $I, X$, and $Y$ are collinear, then the points $I, W$, and $Z$ are also collinear.

设$A B C$ 为外接圆$\Omega$ 和心$I$ 的三角形。设穿过$I$并垂直于$C I$的直线分别与线段$B C$和$\Omega$的$\operatorname{arc} B C($不包含$A)$相交于点$U$和$V$。让穿过$U$并平行于$A I$的直线与$A V$相交于$X$，让穿过$V$并平行于$A I$的直线与$A B$相交于$Y$。设$W$和$Z$分别为$A X$和$B C$的中点。证明如果点 $I、X$ 和 $Y$ 共线，则点 $I、W$ 和 $Z$ 也共线。