A circumconic is a nondegenerate conic section that passes through the vertices of a triangle (Kimberling 1998, p. 235). Every circumconic has an equation in trilinear coordinates of the form
 |
(1)
|
where 
, 
, and 
are functions of the side lengths 
, 
, and 
and 
, and conversely such an
equation defines a circumconic after multiplication by 
.
For a circumconic that is a central conic, the center is given by
 |
(2)
|
(Kimberling 1998, p. 235).
Isogonal conjugation maps the triangle interior onto itself. A line 
avoiding the three vertices transforms
into a circumconic. Its type is determined by whether 
meets the circumcircle 
,
1. If 
does not intersect 
, the isogonal conjugate
is an ellipse;
2. If 
is a tangent line 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 the triangle in 0, 1, or 2 points when the circumconic is an ellipse, parabola, or hyperbola, respectively (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 compares some circumconics and their centers. The notation 
denotes the corresponding Kimberling center.
The drawings show particular reference triangles
and are independently scaled. Special triangles can
cause a named construction to become degenerate. The circumconic is blue, the reference triangle gray, and supporting constructions
green. Red markers identify selected named points, including
the center listed in the table; 
denotes the orthocenter. The
three rectangular hyperbolas are shown with
the nine-point circle, on which their centers
lie. In the Johnson circumconic row, 
, 
, and 
label the green Johnson
triangle. The green circle in the Macbeath
circumconic and Steiner circumellipse
rows is the circumcircle. Only finite portions of
the hyperbolas are shown.
