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:
Lemma. 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 $z$) bisector plane 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 $L$.
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
But surely for such a simple angle of incidence = angle of reflection
relationship there is a concise geometrical explanation—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!