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)
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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)
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(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)
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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)
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and a rectangular hyperbola if
(5)
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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 |