Question 4
In the acute-angled triangle $ABC$, with $AB \neq AC$, $D$ is the foot of the angle bisector of angle $A$, and $E$, $F$ are the feet of the altitudes from $B$ and $C$, respectively. The circumcircles of triangles $DBF$ and $DCE$ intersect for the second time at $M$. Prove that $ME = MF$.