If the vectors $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ form the sides $BC$, $CA$and $AB$, respectively, of triangle $ABC$, then 

Given $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{0}$ (by triangle law).

Therefore, $\overrightarrow{a}\times (\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c})=\overrightarrow{a}\times \overrightarrow{0}$

$$\begin{gathered} \therefore \overrightarrow{a}\times \overrightarrow{a}+\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{a}\times \overrightarrow{c}=\overrightarrow{0} \\ \therefore \overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{c}\times \overrightarrow{a} \end{gathered}$$

Similarly, by taking cross product with $\overrightarrow{b}$ we get $\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{b}\times \overrightarrow{c}$

$\therefore \overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{b}\times \overrightarrow{c}=\overrightarrow{c}\times \overrightarrow{a}$