= Solution
{c}
Put $V=\mathbf a\cdot(\mathbf b\times\mathbf c)$. The <vector triple product identity> gives
$$
(\mathbf b\times\mathbf c)\times(\mathbf c\times\mathbf a)
=\mathbf c[(\mathbf b\times\mathbf c)\cdot\mathbf a]-\mathbf a[(\mathbf b\times\mathbf c)\cdot\mathbf c]
=V\mathbf c.
$$
Taking its <dot product> with $\mathbf a\times\mathbf b$ and using cyclic invariance of the <scalar triple product> yields
$$
\boxed{(\mathbf a\times\mathbf b)\cdot[(\mathbf b\times\mathbf c)\times(\mathbf c\times\mathbf a)]=V^2.}
$$
There was no division by $V$, so this also proves the identity for coplanar <vectors>.
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