Question 167
Let $H$ be the orthocenter of triangle $ABC$. Let $E, F$ be the feet of the $B, C$-altitudes. Let $D, M, N$ be the midpoints of segments $AH, BD, CD$ respectively, and $T$ be the intersection of line $FM$ and $EN$. Suppose $D, E, T$, and $F$ are concyclic. Prove that $DT$ passes through the circumcentre of $ABC$.