Altitudes of a Tetrahedron and Traceless Quadratic Forms

TL;DR

It is shown that the altitudes of a general tetrahedron are mutually skew, for they are generators of an equilateral hyperboloid.

Abstract

It is well known that the three altitudes of a triangle are concurrent at the so-called orthocenter of the triangle. So one might expect that the altitudes of a tetrahedron also meet at a point. However, it was already pointed out in 1827 by the Swiss geometer Jakob Steiner (1796--1863) that the altitudes of a general tetrahedron are mutually skew, for they are generators of an equilateral hyperboloid.

Theorems & Definitions (8)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8