= Solution
Use <suffix notation> with indices ranging from $1$ to $3$, and sum over repeated indices. By the component definitions of the <dot product> and <cross product>,
$$
\mathbf a\cdot(\mathbf b\times\mathbf c)
=a_i\epsilon_{ijk}b_jc_k.
$$
The <Levi-Civita symbol> is unchanged by a cyclic permutation of its three indices: the cycle is two transpositions and hence has positive sign. Thus $\epsilon_{ijk}=\epsilon_{kij}$. Commuting the scalar components gives
$$
a_i\epsilon_{ijk}b_jc_k
=c_k\epsilon_{kij}a_ib_j
=c_k(\mathbf a\times\mathbf b)_k.
$$
Therefore the <scalar triple product> obeys
$$
\boxed{\mathbf a\cdot(\mathbf b\times\mathbf c)
=\mathbf c\cdot(\mathbf a\times\mathbf b).}
$$
\b[Cyclically moving the vectors preserves the scalar triple product; interchanging just two reverses its sign.]
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