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

Arrangement of Central Points on the Faces of a Tetrahedron