Journal for Geometry and Graphics 15 (2011), No. 1, 019--028
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 V1V2V3, a point I and three arbitrary directions qi, we find a distance x = IQ1 = IQ2 = IQ3 along these directions, for which the three cevians ViQi are concurrent. If I is the incenter, qi 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 qi, 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
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