Routh's theorem for simplices

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.