${\text{If z}}_1,z_2 \text{and} z_3\text{ are three points lying on the circle} \left|z\right| =2 t\text{nimum value of }{\left|z_1+z_2\right|}^2\text{+}{\left|z_2+z_3\right|}^2\text{+}{\left|z_3+z_1\right|}^2\text{is}$

$\begin{array}{l}\left|z_1\right|=\left|z_2\right|=\left|z_3\right|=2\\ \text{ Now },{\left|z_1+z_2+z_3\right|}^2\ge 0\\ \Rightarrow \left(z_1+z_2+z_3\right)\left({\overline{z}}_1+{\overline{z}}_2+{\overline{z}}_3\right)\ge 0\end{array}$

$\begin{array}{l}\Rightarrow \sum {\left|z_1\right|}^2+\sum \left(z_1{\overline{z}}_2+{\overline{z}}_1{z}_2\right)\ge 0\\ \Rightarrow 2\sum {\left|z_1\right|}^2+\sum \left(z_1{\overline{z}}_2+{\overline{z}}_1{z}_2\right)\ge \sum {\left|z_1\right|}^2\\ \Rightarrow {\left|z_1+z_2\right|}^2+{\left|z_2+z_3\right|}^2+{\left|z_3+z_1\right|}^2\ge 12\end{array}$