mg.metric geometry - Routh's theorem in three dimensions

The following paper establishes a generalization of Routh's theorem to 3 dimensions:

Semyon Litvinov,
František Marko, Routh’s theorem for tetrahedra.Geom. Dedicata 174 (2015), 155–167

First, a new tetrahedron is determined by its vertices, which are points on 4 edges of the original one. The authors start with a tetrahedron $ABCD$, choose points $M, N, K, L$ on the edges $AB$, $BC$, $CD$, $DA$ which cut these edges in ratios $x, y, z, t$ respectively, and give an explicit formula for the ratio of the volumes of $KLMN$ and $ABCD$ in terms of $x, y, z, t$. Another tetrahedron is enclosed by the planes $ABK, BCL, CDM, DAN$ and the authors find explicitly the corresponding volume ratio.

A follow-up paper is available on the ArXiv and contains $n$-dimensional generalizations and interesting bibliographical remarks.