Circumconic

A circumconic is a conic section that passes through the vertices of a triangle (Kimberling 1998, p. 235). Every circumconic has a trilinear equation of the form

(1)

where , , and are functions of the side lengths , , and and , and conversely every circumconic has such an equation.

The center of a circumconic is given by

(2)

(Kimberling 1998, p. 235).

Isogonal conjugation maps the interior of a triangle onto itself. This mapping transforms lines into circumconics. The type of conic section is determined by whether the line meets the circumcircle ,

1. If does not intersect , the isogonal transform is an ellipse;

2. If is tangent to , the transform is a parabola;

3. If cuts , the transform is a hyperbola, which is a rectangular hyperbola if the line passes through the circumcenter

(Casey 1893, Vandeghen 1965).

The line

(3)

meets the circumcircle of a circumconic's triangle on 0, 1, or 2 points if the conic is an ellipse, parabola, or hyperbola (Kimberling 1998, p. 235).

A circumconic is a parabola if

(4)

and a rectangular hyperbola if

(5)

In the latter case, the hyperbola passes through the orthocenter and has center on the nine-point circle (Kimberling 1998, p. 236), a result known as the Feuerbach's conic theorem (Coolidge 1959, p. 198).

The following table summarizes some circumconics.

circumconic	Kimberling	center
circumcircle		circumcenter
excentral-hexyl ellipse		circumcenter
Feuerbach hyperbola		Feuerbach point
Jerabek hyperbola		center of the Jerabek hyperbola
Johnson circumconic		nine-point center
Kiepert hyperbola		center of the Kiepert hyperbola
MacBeath circumconic		symmedian point
Steiner circumellipse		triangle centroid