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If $\mathrm{G}(2,-1,2)$ is the centroid of tetrahedron $\mathrm{OABC}$ where $\mathrm{O}=(0,0,0)$ and $\mathrm{G}_1$ is the centroid of $\triangle \mathrm{ABC}$, then $\left|\overline{O G_1}\right|=$
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$4$
$\begin{aligned} & \text { Now } \frac{a}{4}=2 \Rightarrow a=8 \\ & \frac{b}{4}=-1 \Rightarrow b=-4 \\ & \frac{c}{4}=2 \Rightarrow c=8\end{aligned}$
$\begin{aligned} & \therefore \mathrm{G},\left(\frac{8}{3}, \frac{-4}{3}, \frac{8}{3}\right) \\ & \therefore \Rightarrow \mathrm{b}=\frac{\mathrm{a}+\mathrm{c}}{2}\end{aligned}$
$\cot \left(\frac{\mathrm{A}}{2}\right) \times \cot \left(\frac{\mathrm{c}}{2}\right)=\left[\frac{\cos \left(\frac{\mathrm{A}}{2}\right)}{\sin \left(\frac{\mathrm{A}}{2}\right)}\right] \cdot\left[\frac{\cos \left(\frac{\mathrm{c}}{2}\right)}{\sin \left(\frac{\mathrm{c}}{2}\right)}\right]$
$\cot \left(\frac{\mathrm{A}}{2}\right) \times \cot \left(\frac{\mathrm{c}}{2}\right)=\left[\frac{2 \cos \left(\frac{\mathrm{A}}{2}\right) \times \sin \left(\frac{\mathrm{A}}{2}\right)}{2 \sin ^2\left(\frac{\mathrm{A}}{2}\right)}\right]$
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