A question at Math.StackExchange asked about “Ceva’s Theorem in three dimensions”.

The result (which I believe may be new, although it could very well exist in the hundreds of years of mathematical literature since Ceva) replaces the traditional ratios of segment lengths with “triple-ratios” of triangle areas, to satisfying effect. Whereas the ratios in Ceva’s Theorem reduce to \(1\) (or “\([1:1]\)”), my triple-ratios reduce to \([1:1:1]\).

I like the result enough that I’ve tidied the derivation, added a smidgeon of context, and touched-on the (current) lack of a corresponding Menelaus-like theorem. See my note: “A Ceva-like Theorem for Tetrahedra” (PDF).