TUYMAADA 2006 Problem 6

Let $ABC$ be a triangle, $G$ its centroid, $H$ its orthocenter, and $M$ the midpoint of the arc $\widehat{AC}$ (not containing $B$ ). It is known that $MG=R$ , where $R$ is the radius of the circumcircle. Prove that $BG\geq BH$ .

Proposed by F. Bakharev