The Inspherical Gergonne Center of a Tetrahedron

M. Hajja ; P. Walker

Journal for geometry and graphics, Tome 8 (2004) no. 1, pp. 23-32.

Résumé

With the traditional definition of the Gergonne center of a triangle in mind, it is natural to consider, for a given tetrahedron, the intersection of the cevians that join the vertices to the points where the insphere touches the faces. However, these cevians are concurrent for the limited class of what we call inspherical tetrahedra. Another approach is to note that the cevians through any point inside a triangle divide the sides into 6 segments, and that the Gergonne center is characterized by the requirement that every two segments sharing a vertex are equal. Similarly, the cevians through any point inside a tetrahedron divide the faces into 12 subtriangles, and one may define the Gergonne center as the point for which every two subtriangles that share an edge are equal in area. It turns out that tetrahedra whose Gergonne center has the stronger property that every two subtriangles that share an edge are congruent (resp. skew-congruent) are precisely the inspherical (resp. equifacial) ones. Furthermore, we give a characterization of the inspherical tetrahedra and we outline a method for constructing them.

Détail

Classification : 51M04, 51M20, 52B10
Mots-clés : Barycentric coordinates, Ceva's theorem, Gergonne point, Fermat-Torricelli point, isosceles or equifacial tetrahedron, isogonal tetrahedron