KöMaL problem A, 2023

Geometry Difficulty 7.0 National Olympiad, round 2 Prove it Hungary

Let $ABC$ be a given acute triangle, in which $BC$ is the longest side. Let $H$ be the orthocenter of the triangle, and let $D$ and $E$ be the feet of the altitudes from $B$ and $C$ , respectively. Let $F$ and $G$ be the midpoints of sides $AB$ and $AC$ , respectively. $X$ is the point of intersection of lines $DF$ and $EG$ . Let $O_1$ and $O_2$ be the circumcenters of triangles $EFX$ and $DGX$ , respectively. Finally, $M$ is the midpoint of line segment $O_1O_2$ . Prove that points $X$ , $H$ and $M$ are collinear.

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