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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

Miklós Hoffmann
Institute of Mathematics and Computer Science, Károly Eszterházy College, Leányka str. 4, 3300 Eger, Hungary

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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