Question 177
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 $E, F \ne A$). Show that
$$
[BEGF] + [CFGE] \ge \frac{4}{9}[ABC].
$$
When does the equality hold? (Here $[WXYZ]$ is the area of the polygon $WXYZ$, etc.)
$$
[BEGF] + [CFGE] \ge \frac{4}{9}[ABC].
$$
When does the equality hold? (Here $[WXYZ]$ is the area of the polygon $WXYZ$, etc.)