Let $ABC$ be a triangle. Let $M$ be the midpoint of $BC,$ and let $G$ be the centroid of $\triangle ABC.$ Let $D$ be a point on the segment $GM.$ A straight line passing through $D$ meets the sides $AB$ and $AC$ at $E$ and $F$ respectively (with $E, F \neq A$). Show that $$[BEGF]+[CFGD]\geq \frac{4}{9} [ABC].$$When does the equality hold? (Here $[WXYZ]$ is the area of the polygon $WXYZ,$ etc.)