Question 78
In a scalene triangle $ABC$, points $X$ and $Y$ lie on $BC$ and $AC$, respectively, such that $BC \perp XY$. Points $T$ and $Z$ are the reflections of $Y$ and $X$ with respect to the midpoints of sides $AC$ and $BC$, respectively. Point $P$ lies on $ZT$ such that the circumcenter of triangle $XZP$ coincides with the circumcenter of triangle $ABC$. Show that the nine-point circle of triangle $ABC$ passes through the midpoint of segment $XP$.