\(\left\{{}\begin{matrix}\overrightarrow{GA}+\overrightarrow{GB}+\overrightarrow{GC}+\overrightarrow{GD}=0\\\overrightarrow{G_0B}+\overrightarrow{G_0C}+\overrightarrow{G_0D}=0\end{matrix}\right.\)
\(\overrightarrow{GA}+\overrightarrow{GB}+\overrightarrow{GC}+\overrightarrow{GD}=\overrightarrow{0}\)
\(\Leftrightarrow\left(\overrightarrow{GG_0}+\overrightarrow{G_0A}\right)+\left(\overrightarrow{GG_0}+\overrightarrow{G_0B}\right)+\left(\overrightarrow{GG_0}+\overrightarrow{G_0C}\right)+\left(\overrightarrow{GG_0}+\overrightarrow{G_0D}\right)=\overrightarrow{0}\)
\(\Leftrightarrow4\overrightarrow{GG_0}+\overrightarrow{G_0A}+\left(\overrightarrow{G_0B}+\overrightarrow{G_0C}+\overrightarrow{G_0D}\right)=\overrightarrow{0}\)
\(\Rightarrow\overrightarrow{AG_0}=4\overrightarrow{GG_0}\)
\(\Rightarrow\overrightarrow{AG}+\overrightarrow{GG_0}=4\overrightarrow{GG_0}\)
\(\Rightarrow\overrightarrow{AG}=3\overrightarrow{GG_0}\)
\(\Rightarrow\overrightarrow{GA}=-3\overrightarrow{GG_0}\)
