Geodesic
Gergonne and Nagel Points for Simplices in the n-Dimensional Space
Journal for geometry and graphics, Tome 4 (2000) no. 2, pp. 119-128.

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Properties of triangles related to so called Gergonne and Nagel points are known in elementary geometry. 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. 
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     author = {E. Kozniewski and R. A. Gorska},
     title = {Gergonne and {Nagel} {Points} for {Simplices} in the {n-Dimensional} {Space}},
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E. Kozniewski; R. A. Gorska. Gergonne and Nagel Points for Simplices in the n-Dimensional Space. Journal for geometry and graphics, Tome 4 (2000) no. 2, pp. 119-128. http://geodesic.mathdoc.fr/item/JGG_2000_4_2_a2/