engleski

Gergonne conic

An elementary notion of geometry, the concept of Gergonne point is generalized in this paper. Given a triangle $V_1V_2V_3$, 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 $V_iQ_i$ 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.

Gergonne point; conics; projectivity; pencil of conics

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