Journal for Geometry and Graphics, Vol. 4, No. 2, pp. 119-127 (2000)

Gergonne and Nagel Points for Simplices in the n-Dimensional Space

Edwin Kozniewski, Renata A. Gorska

Institute of Civil Engineering, Engineering Graphics and
Computer Methods Division, Bialystok University of Technology
Wiejska st. 45E PL 15-351 Bialystok, Poland

Abstract: Properties of triangles related to so called Gergonne and Nagel points are known in elementary geometry. In this paper we present a discussion on some extensions of these theorems. First, we refer to a relation between a tetrahedron and a sphere inscribed into this tetrahedron in the 3-dimensional space. Next, we generalize the obtained results to simplices in n-dimensional geometry. The problem concerning tetrahedra occurs to be no longer as easy to solve as it is for triangles. It has been shown that there are both tetrahedra, which have Gergonne and Nagel points, and tetrahedra with no such a point. We give conditions necessary and sufficient for a simplex to satisfy the Gergonne and Nagel property.

Keywords: 3-dimensional geometry, n-dimensional geometry, polar transformation, Gergonne point, Nagel point

Classification (MSC2000): 51M04

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