Question 5
Let $ABC$ be an acute triangle. Denote by $D, E, F$ the midpoints of sides $BC, CA, AB$ respectively. The circle with diameter $AD$ intersects lines $AB$ and $AC$ again at $P$ and $Q$ respectively. The line through $P$ parallel to $BC$ meets line $DE$ at $R$, the line through $Q$ parallel to $BC$ meets line $DF$ at $S$. The circumcircle of $DPR$ meets $AB$ again at $X$, the circumcircle of $DQS$ meets $AC$ again at $Y$, and those two circles meet again at $Z$. Prove that $Z$ is the midpoint of $XY$.