If G is centroid of the $∆$ABC then $\overline{GA} + \overline{BG} + \overline{GC} =$

$$\begin{gathered} \overline{\mathrm{OG}} = \frac{\overline{\mathrm{OA}} + \overline{\mathrm{OB}} + \overline{\mathrm{OC}}}{3} . \\ \overline{\mathrm{GA}} + \overline{\mathrm{BG}} + \overline{\mathrm{GC}}= \overline{\mathrm{OA}} - \overline{\cancel{\mathrm{OG}}} + \cancel{\overline{\mathrm{OG}} }- \overline{\mathrm{OB}} + \overline{\mathrm{OC}} -\overline{\mathrm{OG}} \end{gathered}$$$\begin{array}{rcl} & \Rightarrow & \overline{\mathrm{GA}} + \overline{\mathrm{BG}} + \overline{\mathrm{GC}}= \overline{\mathrm{OA}} - \overline{\mathrm{OB}} + \overline{\mathrm{OC}} - \overline{\mathrm{OG}} \\ & =& \overline{\mathrm{OA}} + \overline{\mathrm{OB}} + \overline{\mathrm{OC}} -2\overline{\mathrm{OB}} - \overline{\mathrm{OG}}\\ & =& 2\left(\overline{\mathrm{OG}} - \overline{\mathrm{OB}}\right) \\ & =& 2\overline{\mathrm{BG}}\end{array}$