EGMO 2014 Problem 2

Let $D$ and $E$ be two points on the sides $A B$ and $A C$, respectively, of a triangle $A B C$, such that $D B=B C=C E$, and let $F$ be the point of intersection of the lines $C D$ and $B E$. Prove that the incenter $I$ of the triangle $A B C$, the orthocenter $H$ of the triangle $D E F$ and the midpoint $M$ of the $\operatorname{arc} B A C$ of the circumcircle of the triangle $A B C$ are collinear.

Proposed by Danylo Khilko, UKR