KöMaL 2021 B5193

In an acute-angled triangle $ABC$, $\angle BCA =45^{\circ}$, the feet of the altitudes on sides $BC$, $CA$, $AB$ are $D$, $E$, $F$, respectively, and the orthocentre is $M$. Point $F$ divides line segment $AB$ in a ratio ${AF}:{FB}={2}:{3}$. $G$ is the point on side $AC$ for which $CG=BM$. Show that the centroid of triangle $ABG$ is $M$.