The orthocentric simplex as an extreme simplex

Abstract

Let s4 be a variable n -simplex containing a fixed point Q and having vertices A, and corresponding opposite faces .^/,, / = 0, 1, , n. We use the properties of orthocentric simplexes to present brief solutions to the following problems and obtain several Erds-Mordell type inequalities as a by-product, some of which are stronger than known inequalities.