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

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

$\begin{array}{l}\overline{\mathrm{GA}}+\overline{\mathrm{GB}}+\overline{\mathrm{GC}}=\overline{0} \\ \Rightarrow \overline{\mathrm{GA}}+\overline{\mathrm{GC}}=-\overline{\mathrm{GB}} ...\left(\mathrm{ii}\right)\\ \text{ Now. }\overline{\mathrm{GA}}+\overline{\mathrm{BG}}+\overline{\mathrm{GC}}\\ \begin{array}{l}=-\overline{\mathrm{GB}}+\overline{\mathrm{BG}}\text{ [From Eq.(ii)] }\\ =2\overline{\mathrm{BG}}\end{array}\end{array}$