For any three vectors `a ,\ b ,\ c` prove that `[ vec a+ vec b\ vec b+ vec c\ vec c+ vec a]=2\ [ vec a\ vec b\ vec c]dot`

Explanation

$$ [\vec{a}+\vec{b} \vec{b}+\vec{c} \vec{c}+\vec{a}]=(\vec{a}+\vec{b}) \cdot\{(\vec{b}+\vec{c}) \times(\vec{c}+\vec{a})\} $$
$$
\begin{aligned}
&=(\vec{a}+\vec{b}) .\{\vec{b} \times \vec{c}+\vec{b} \times \vec{a}+\vec{c} \times \vec{c}+\vec{c} \times \vec{a}\}\\
&=(\vec{a}+\vec{b}) .\{\vec{b} \times \vec{c}+\vec{b} \times \vec{a}+\vec{c} \times \vec{a}\}\\
&=\vec{a} \cdot(\vec{b} \times \vec{c})+\vec{a} \cdot(\vec{b} \times \vec{a})+\vec{a} \cdot(\vec{c} \times \vec{a})\\
...