Question 53
The acute-angled triangle $ABC$ with circumcenter $O$ is given. The midpoints of the sides $BC$, $CA$ and $AB$ are $D$, $E$ and $F$ respectively. An arbitrary point $M$ on the side $BC$, different from $D$, is chosen. The straight lines $AM$ and $EF$ intersect at the point $N$ and the straight line $ON$ cuts again the circumscribed circle of the triangle $ODM$ at the point $P$. Prove that the reflection of the point $M$ with respect to the midpoint of the segment $DP$ belongs on the nine points circle of the triangle $ABC$.