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.
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