Generalized Gergonne and Nagel points

Boris Odehnal

Abstract: In this paper we show that the Gergonne point $G$ of a triangle $\Delta$ in the Euclidean plane can in fact be seen from a more general point of view, i.e., from the viewpoint of projective geometry. So it turns out that there are up to four Gergonne points associated with $\Delta$. The Gergonne and Nagel point are isotomic conjugates of each other, and thus we find up to four Nagel points associated with a generic triangle. We reformulate the problems in a more general setting and illustrate the different appearances of Gergonne points in different affine geometries. Darboux's cubic can also be found in the more general setting, and finally a projective version of Feuerbach's circle appears.

Keywords: Brianchon's theorem, Darboux's cubic, excenters, Feuerbach's nine point circle, Gergonne point, incenter, isotomic conjugate, Nagel point, triangle

Classification (MSC2000): 51M04, 51M05, 51B20

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