Question 10
Let $ABC$ be an acute triangle, let $D, E, F$ be the feet of the altitudes from $A, B, C$, respectively, and let $M, N, P$ be the midpoints of the sides $BC, CA, AB$, respectively. The circles $BDP$ and $CDN$ cross again at $X$, the circles $CEM$ and $AEP$ cross again at $Y$, and the circles $AFN$ and $BFM$ cross again at $Z$. Prove that the lines $AX, BY, CZ$ are concurrent.