Prove the identities: `|[a, b, c],[ a-b,b-c,c-a],[ b+c,c+a, a+b]|=a^3+b^3+c^3-3a b c`

Explanation

To prove the identity

\[
\left| \begin{array}{ccc}
a & b & c \\
a-b & b-c & c-a \\
b+c & c+a & a+b
\end{array} \right| = a^3 + b^3 + c^3 - 3abc,
...