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

(1)

where , , and
are functions of the side lengths , ,
and , 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 of the Brianchon point
as

(2)

Furthermore, the center of an inconic with parameters is the point

in which case the focus is the point which lies on the circumcircle
and the directrix passes through the orthocenter
(Smith 1894, p. 70; Eddy and Fritsch 1994; Kimberling 1998, p. 239).

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.