The open mouth theorem in higher dimensions

Propositions 24 and 25 of Book I of Euclid’s Elements state the fairly obvious fact that if an angle in a triangle is increased without changing the lengths of its arms, then the length of the opposite side increases, and conversely. A satisfactory analogue that holds for orthocentric tetrahedra is established by S. Abu-Saymeh, M. Hajja, M. Hayajneh in a yet unpublished paper, where it is also shown that no reasonable analogue holds for general tetrahedra. In this paper, the result is shown to hold for orthocentric d-simplices for all d⩾3. The ingredients of the proof consist in finding a suitable parametrization (by a single real number) of the family of orthocentric d-simplices whose edges emanating from a certain vertex have fixed lengths, and in making use of properties of certain polynomials and of Gram and positive definite matrices and their determinants.