Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Vol. 51, No. 2, pp. 477491 (2010) 

Generalized Gergonne and Nagel pointsBoris OdehnalInstitut für Diskrete Mathematik und Geometrie, TU Wien, Wiedner Hauptstraße 810, A1040 Wien, Austria, email: boris@geometrie.tuwien.ac.atAbstract: 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 Full text of the article (for subscribers):
Electronic version published on: 24 Jun 2010. This page was last modified: 8 Sep 2010.
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