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Concentric Circles and the Generalized Gergonne Point

The well-known triangle center, the Gergonne point is generalized for circles concentric to the inscribed circle in this paper. Given a triangle V1V2V3, a point I and three arbitrary directions q1, q2, q3, we consider a circle with radius r and center I, for which finding r=IQ1=IQ2=IQ3 along the given directions, the three cevians ViQi are concurrent. Solutions can be obtained by the common intersection points of three conics. Geometric properties and open problems related to the solutions are discussed in a synthetic way.

Gergonne point; conic; pencil of conic

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