Problem (2014EGMOp2): Let $D$ and $E$ be points in the interiors of sides $AB$ and $AC$, respectively, of a triangle $ABC$, such that $DB = BC = CE$. Let the lines $CD$ and $BE$ meet at $F$. Prove that the incentre $I$ of triangle $ABC$, the orthocentre $H$ of triangle $DEF$ and the midpoint $M$ of the arc $BAC$ of the circumcircle of triangle $ABC$ are collinear.
AlphaGeometry formalization: a b c = triangle; d = on_line a b, on_circle b c; e = on_line a c, on_circle c b; f = on_line c d, on_line b e; i = incenter a b c; h = orthocenter d e f; m = on_bline b c, on_circum a b c ? coll i h m
HAGeo formalization: A B C = triangle; AB = line A B; (B,C) = circle_center_point B C; D = intersection AB (B,C); AC = line A C; (C,B) = circle_center_point C B; E = intersection AC (C,B); CD = line C D; BE = line B E; F = intersection CD BE; I = incenter A B C; H = orthocenter D E F; M = midarc B A C; Prove: collinear I H M
Difficulty score: 3.4