= Solution
Take two edge vectors in the face, $\mathbf b-\mathbf a$ and $\mathbf c-\mathbf a$. Their <cross product> is the face <normal vector>
$$
\mathbf n=(\mathbf b-\mathbf a)\times(\mathbf c-\mathbf a)
=\mathbf a\times\mathbf b+\mathbf b\times\mathbf c+\mathbf c\times\mathbf a.
$$
The face <plane> is $(\mathbf r-\mathbf a)\cdot\mathbf n=0$. Every term in $\mathbf a\cdot\mathbf n$ except $\mathbf a\cdot(\mathbf b\times\mathbf c)$ vanishes, giving
$$
\boxed{\mathbf r\cdot(\mathbf a\times\mathbf b+\mathbf b\times\mathbf c+\mathbf c\times\mathbf a)
=\mathbf a\cdot(\mathbf b\times\mathbf c)}.
$$
The <scalar triple product> is nonzero, so this normal cannot vanish.
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