BALTIC-WAY 2022 Problem 14

Let $\Gamma$ denote the circumcircle and $O$ the circumcentre of the acute-angled triangle $ABC$ , and let $M$ be the midpoint of the segment $BC$ . Let $T$ be the second intersection point of $\Gamma$ and the line $AM$ , and $D$ the second intersection point of $\Gamma$ and the altitude from $A$ . Let further $X$ be the intersection point of the lines $DT$ and $BC$ . Let $P$ be the circumcentre of the triangle $XDM$ . Prove that the circumcircle of the triangle $OPD$ passes through the midpoint of $XD$ .