The Zeros of Heron's Formula in Orthocentric Tetrahedra
Timothy Havel, Garrett Sobczyk
The second author has recently shown that one can define and study Lorentzian vector spaces based on the concept of compatible null vectors. Around the same time the first author used the relation $r = t/s$ between the in-radius $r$, area $t$ and semi-perimeter $s$ of a triangle to generalize Heron's classical formula to simplices in all dimensions. A few years earlier, Udo Hertrich-Jeromin, Alastair King and Jun O’Hara used the conformal model of Euclidean geometry to show that the vertices of a triangle are related to its ex-centers via reflection w.r.t. a time-like vector exchanging its in-center with the point-at-infinity, an operation they named the conformal dual. In this Abstract, we will explore the interconnections among these three previously independent lines of study of the Lorentzian geometric algebras, focusing on the simple case of $\mathcal G_{3,1}$.