  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.
