If G is the centroid of the $\Delta ABC$ , then $\overline{GA}+\overline{BG}+\overline{GC}$ is equal to

Given G is the centroid of a triangle. We know that

$\overline{GA}+\overline{GB}+\overline{GC}=\overline{0}$

$\Rightarrow \overline{GA}+\overline{GC}=-\overline{GB}\mathrm{....}\left(\text{ii}\right)$

Now, $\overline{GA}+\overline{BG}+\overline{GC}$

$=-\overline{GB}+\overline{BG}$ [From Eq.(ii)]

$=2\overline{BG}$