MK-MO 2022 Problem 5

An acute $\triangle ABC$ with circumcircle $\Gamma$ is given. $I$ and $I_a$ are the incenter and $A-$ excenter of $\triangle ABC$ respectively. The line $AI$ intersects $\Gamma$ again at $D$ and $A'$ is the antipode of $A$ with respect to $\Gamma$ . $X$ and $Y$ are point on $\Gamma$ such that $\angle IXD = \angle I_aYD = 90^\circ$ . The tangents to $\Gamma$ at $X$ and $Y$ intersect in point $Z$ . Prove that $A', D$ and $Z$ are collinear.

Proposed by Nikola Velov