Journal of Geometry

, Volume 83, Issue 1, pp 46–56

The Gergonne and Nagel centers of an n-dimensional simplex

Original Paper

DOI: 10.1007/s00022-005-0011-3

Cite this article as:
Hajja, M. J. geom. (2005) 83: 46. doi:10.1007/s00022-005-0011-3

Abstract.

In contrast with the analogous situation for a triangle, the cevians that join the vertices of a tetrahedron to the points where the faces touch the insphere (or the exspheres) are not concurrent in general. This observation led the present author and P. Walker in [4] to devise alternative definitions of the Gergonne and Nagel centers of a tetrahedron that do not assume the concurrence of such cevians and that coincide with the ordinary definitions in the case of a triangle. They then proved that the Gergonne center exists and is unique for all tetrahedra and that the Nagel center, though unique, exists only for tetrahedra that satisfy certain conditions. In this article, we extend these definitions to simplices of any dimension. By keeping the requirement that the Gergonne center be interior and relaxing such a condition for the Nagel center, we prove that both centers exist and are unique for all simplices, thus polishing the definitions and generalizing the results of the above-mentioned article.

Mathematics Subject Classification (2000).

51M0451M2052B10

Key words.

Gergonne centerNagel centern–simplicesn-dimensional Ceva’s theorembarycentric coordinates

Copyright information

© Birkhäuser Verlag, Basel 2005

Authors and Affiliations

  1. 1.Mathematics DepartmentYarmouk UniversityIrbidJordan