
Journal for Geometry and Graphics 15 (2011), No. 1, 019028 Copyright Heldermann Verlag 2011 The Gergonne Conic Sonja Gorjanc Dept. of Mathematics, Faculty of Civil Engineering, University of Zagreb, Kaciceva 26, 10000 Zagreb, Croatia sgorjanc@grad.hr Miklós Hoffmann Institute of Mathematics and Computer Science, Károly Eszterházy College, Leányka str. 4, 3300 Eger, Hungary hofi@ektf.hu The notion of Gergonne point was generalized in several ways during the last decades. Given a triangle V_{1}V_{2}V_{3}, a point I and three arbitrary directions q_{i}, we find a distance x = IQ_{1} = IQ_{2} = IQ_{3} along these directions, for which the three cevians V_{i}Q_{i} are concurrent. If I is the incenter, q_{i} are the direction of the altitudes, and x is the radius of the incenter, the point of concurrency is the Gergonne point. For arbitrary directions q_{i}, it is shown that each point I generally yields two solutions, and points of concurrency lie on a conic, which can be called the Gergonne conic. Keywords: Gergonne point, conics, projectivity, pencil of conics. MSC: 51M04; 51N35 [ Fulltextpdf (374 KB)] for subscribers only. 