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Journal for Geometry and Graphics 16 (2012), No. 2, 171--183
Copyright Heldermann Verlag 2012

Conic Construction of a Triangle From Its Incenter, Nine-point Center, and a Vertex

Paul Yiu
Dept. of Mathematical Sciences, Florida Atlantic University, 777 Glades Road, Boca Raton, FL 33431, U.S.A.

We construct a triangle given its incenter, nine-point center and a vertex by locating the circumcenter as an intersection of two rectangular hyperbolas. Some special configurations leading to solutions constructible with ruler and compass are studied. The related problem of construction of a triangle given its circumcenter, incenter, and one vertex is revisited, and it is established that such a triangle exists if and only if the incenter lies inside the cardioid relative to the circumcircle.

Keywords: Triangle geometry, construction problems, nine-point center, rectangular hyperbolas.

MSC: 51M15; 51M04

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