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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

 x/alpha+y/beta+z/gamma=0
(1)

where x, y, and z are functions of the side lengths a, b, and c and xyz!=0, and conversely every circumconic has such an equation.

The center of a circumconic is given by

 x(-ax+by+cz):y(ax-by+cz):z(ax+by-cz)
(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 d meets the circumcircle C^',

1. If d does not intersect C^', the isogonal transform is an ellipse;

2. If d is tangent to C^', the transform is a parabola;

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

(Casey 1893, Vandeghen 1965).

The line

 xalpha+ybeta+zgamma=0
(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

 x^2a^2+y^2b^2+z^2c^2-2yzbc-2zxca-2xyab=0
(4)

and a rectangular hyperbola if

 xcosA+ycosB+zcosC=0.
(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.


See also

Circumcircle, Circumhyperbola, Conic Section, de Longchamps Ellipse, Feuerbach Hyperbola, Feuerbach's Conic Theorem, Inconic, Isogonal Conjugate, Jerabek Hyperbola, Johnson Circumconic, Kiepert Hyperbola, MacBeath Circumconic, Steiner Circumellipse

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References

Brianchon, C.-J. and Poncelet, J.-V. "Recherches sur la détermination d'une hyperbole équilatère, au moyen de quatre conditions données." Ann. des Math. 11, 205-220, 1821.Casey, J. A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions with Numerous Examples, 2nd rev. enl. ed. Dublin: Hodges, Figgis, & Co., 1893.Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 198, 1959.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.Eves, H.; Kimberling, C.; Lossers, O. P.; and Yff, P. "Problem E2990 and Solution." Amer. Math. Monthly 93, 132-133, 1983.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Vandeghen, A. "Some Remarks on the Isogonal and Cevian Transforms. Alignments of Remarkable Points of a Triangle." Amer. Math. Monthly 72, 1091-1094, 1965.

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Circumconic

Cite this as:

Weisstein, Eric W. "Circumconic." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Circumconic.html

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