Problem

Source: 2018 Romania JBMO TST 4.2

Tags: geometry, circumcircle, midpoint, parallel



Let $ABC$ be an acute triangle, with $AB \ne AC$. Let $D$ be the midpoint of the line segment $BC$, and let $E$ and $F$ be the projections of $D$ onto the sides $AB$ and $AC$, respectively. If $M$ is the midpoint of the line segment $EF$, and $O$ is the circumcenter of triangle $ABC$, prove that the lines $DM$ and $AO$ are parallel.

HIDE: PS As source was given Caucasus MO, but I was unable to find this problem in the contest collections