Question 4
Consider an acute triangle $ABC$ with $|AB| \gt |CA| \gt |BC|$. The vertices $D$, $E$, and $F$ are the base points of the altitudes from $A$, $B$, and $C$, respectively. The line through $F$ parallel to $DE$ intersects $BC$ in $M$. The angular bisector of $\angle MFE$ intersects $DE$ in $N$. Prove that $F$ is the circumcentre of $\triangle DMN$ if and only if $B$ is the circumcentre of $\triangle FMN$.