Routh's theorem for simplices

TL;DR

This paper extends Routh's theorem from triangles to tetrahedra and higher-dimensional simplices using elementary geometry and inclusion-exclusion, providing new proofs and algebraic identities.

Contribution

It offers a new proof of Routh's theorem for tetrahedra combining elementary geometry with inclusion-exclusion and generalizes the approach to higher-dimensional simplices.

Findings

Extended Routh's theorem to tetrahedra and simplices

Provided a new geometric proof using inclusion-exclusion

Derived an interesting algebraic identity

Abstract

It is shown in our earlier paper that, using only tools of elementary geometry, the classical Routh's theorem for triangles can be fully extended to tetrahedra. In this article we first give another proof of Routh's theorem for tetrahedra where methods of elementary geometry are combined with the inclusion-exclusion principle. Then we generalize this approach to $(n-1)-$ dimensional simplices. A comparison with the formula obtained using vector analysis yields an interesting algebraic identity.