I discovered empirically what to me is an amazing lemma
concerning face angles of a tetrahedron.
Let $\triangle abc$ be a triangle in the $xy$-plane, and $d$ the
apex of a tetrahedron with positive $z$ coordinate.
The lemma is this:

> <b>Lemma</b>. The locus of points $d$ for which the sum
of the nonbase tetrahedron face angles incident to $b$ sum to $\pi$,
$$\angle dba + \angle dbc = \pi$$
is a vertical (parallel to $z$) <em>bisector plane</em> which meets the $xy$-plane
in a line $L$ that has the property
that the angle of incidence $\beta$ between $ab$ and $L$ is equal
to the angle of reflection $\beta$ between $bc$ and $L$.

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<img src="http://cs.smith.edu/~orourke/MathOverflow/TetraSumPi.jpg" alt="TetraSumPi" />
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If I express this relationship in terms of `ArcCos( )`'s of the relevant
angles, it all works out algebraically/trigonometrically.  So I have a "proof" in this
(limited) sense.
But surely for such a simple *angle of incidence = angle of reflection*
relationship there is a concise geometrical explanation&mdash;maybe involving reflecting light rays...?

Amidst a much longer proof, this was at one point a critical lemma, but now I have circumvented
its need (I think?!).  Nevertheless, it would be illuminating to see a more revealing proof.  Thanks for ideas and/or insights!