Question 52
Given that $ABC$ is a triangle where $AB \lt AC$. On the half-lines $BA$ and $CA$ we take points $F$ and $E$ respectively such that $BF = CE = BC$. Let $M, N$ and $H$ be the mid-points of the segments $BF, CE$ and $BC$ respectively and $K$ and $O$ be the circumcenters of the triangles $ABC$ and $MNH$ respectively. We assume that $OK$ cuts $BE$ and $HN$ at the points $A_1$ and $B_1$ respectively and that $C_1$ is the point of intersection of $HN$ and $FE$. If the parallel line from $A_1$ to $OC_1$ cuts the line $FE$ at $D$ and the perpendicular from $A_1$ to the line $DB_1$ cuts $FE$ at the point $M_1$, prove that $E$ is the orthocenter of the triangle $A_1OM_1$.