A conic section that is tangent to all sides of a triangle is called an inconic. Any trilinear equation of the form


where x, y, and z are functions of the side lengths a, b, and c, is an inconic, and every inconic has such an equation.

The lines connecting the vertices of a triangle and the corresponding contact points of an inconic are concurrent in a point known as the Brianchon point of the inconic (Veblen and Young 1938, p. 111; Eddy and Fritsch 1994). The inconic parameters are given simply in terms of the trilinear coordinates alpha:beta:gamma of the Brianchon point as


Furthermore, the center of an inconic with parameters x:y:z is the point


(Kimberling 1998, p. 238).

An inconic is a parabola iff


in which case the focus is the point a/x^2:b/y^2:c/z^2 which lies on the circumcircle and the directrix passes through the orthocenter (Smith 1894, p. 70; Eddy and Fritsch 1994; Kimberling 1998, p. 239).

Examples of inconics include the Brocard inellipse, incircle, Kiepert parabola, Steiner inellipse, and Yff parabola.

See also

Brianchon Point, Brocard Inellipse, Circumconic, Conic Section, Incircle, Kiepert Parabola, Lemoine Inellipse, MacBeath Inconic, Mandart Inellipse, Orthic Inconic, Steiner Inellipse, Yff Parabola

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Eddy, R. H. and Fritsch, R. "The Conics of Ludwig Kiepert: A Comprehensive Lesson in the Geometry of the Triangle." Math. Mag. 67, 188-205, 1994.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Smith, C. Geometrical Conics. London: Macmillan, 1894.Veblen, O. and Young, J. W. Projective Geometry, 2 vols. Boston, MA: Ginn, 1938.

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Cite this as:

Weisstein, Eric W. "Inconic." From MathWorld--A Wolfram Web Resource.

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