Gergonne and Nagel points for simplices in the \(n\)-dimensional space (Q2718701)

Many propositions referring to triangles and corresponding special-objects can be easily transferred to a tetrahedral. In search of analogies very interesting problems arise. Let \(A_1A_2A_3A_4\) be a tetrahedron and let \(S\) be the sphere inscribed into this tetrahedron. Besides let \(B_i\) be the point in which the face opposite to corner \(A_i\) \((i=1,2,3,4)\) touches the sphere \(S\). The authors give a condition necessary and sufficient for the tetrahedron in which the four straight lines \(A_iB_i\) meet at one common point (Gergonne point). For that polar transformations are used. This resultat is generalized to simplices in \(n\)-dimensional geometry.NEWLINENEWLINENEWLINEIn an analogous way a necessary and sufficient condition for the existence of the Nagel point for an \(n\)-simplex is given. Here this point is related to the points of tangency of the escribed spheres.NEWLINENEWLINENEWLINEIn the references a few papers are missing, e.g., \textit{D. Brânzei}, Notes on geometry, Pitesti: Editura Paralela 45 (1999; Zbl 0939.51021), \textit{V. Thébault}, Sur des points de Gergonne et de Nagel d'un tétraèdre, Math. Gaz. 33, 270-272 (1949; Zbl 0035.22002).