Arrangement of Central Points on the Faces of a Tetrahedron

TL;DR

This paper systematically studies the positions of various triangle centers on the faces of tetrahedra, identifying conditions for special geometric configurations using computational methods.

Contribution

It provides a comprehensive analysis of triangle centers on tetrahedral faces across different tetrahedron types, revealing new geometric concurrency results.

Findings

Lines from vertices to Gergonne points are concurrent in certain tetrahedra.

Identifies conditions for centers to be coplanar or collinear.

Uses computational methods to analyze over 600 configurations.

Abstract

We systematically investigate properties of various triangle centers (such as orthocenter or incenter) located on the four faces of a tetrahedron. For each of six types of tetrahedra, we examine over 100 centers located on the four faces of the tetrahedron. Using a computer, we determine when any of 16 conditions occur (such as the four centers being coplanar). A typical result is: The lines from each vertex of a circumscriptible tetrahedron to the Gergonne points of the opposite face are concurrent.