Question 19
Let $D$ and $E$ be points in the interiors of sides $[A B]$ and $[A C]$ of a triangle $A B C$, such that $D B = B C = C E$. Let $F$ be the intersection of lines $C D$ and $B E$, $I$ the incenter of triangle $A B C$, $H$ the orthocenter of triangle $D E F$ and $M$ the midpoint of arc $B A C$ on the circumcircle of triangle $A B C$. Prove that $I, H$ and $M$ are collinear.