Question 72
Let $D$ and $E$ be two points on the sides $AB$ and $AC$, respectively, of a triangle $ABC$, such that $DB = BC = CE$, and let $F$ be the point of intersection of the lines $CD$ and $BE$. Prove that the incenter $I$ of the triangle $ABC$, the orthocenter $H$ of the triangle $DEF$ and the midpoint $M$ of the $\operatorname{arc} BAC$ of the circumcircle of the triangle $ABC$ are collinear.