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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