Isotomic Conjugate

The isotomic conjugate of a point is the point of concurrence Q of the isotomic lines relative to a point P. The isotomic conjugate alpha^':beta^':gamma^' of a point with trilinear coordinates alpha:beta:gamma is


Vandeghen (1965) calls the transformation taking points to their isotomic conjugates the Cevian transform. The product of isotomic and isogonal is a collineation which transforms the sides of a triangle to themselves (Vandeghen 1965).

An isotomic transversal is sometimes referred to as an isotomic conjugate (Ehrmann and van Lamoen 2004).

There are four points which are isotomically self-conjugate: the triangle centroid G and each of the exmedian points. The following table lists some common centers and their isotomic conjugates.

The isotomic conjugate of a line d having trilinear equation


is a conic section circumscribed on the triangle DeltaABC (Casey 1893, Vandeghen 1965). The isotomic conjugate of the line at infinity having trilinear equation


is the Steiner circumellipse


(Vandeghen 1965).

The isotomic conjugate of the Euler line is called the Jerabek hyperbola (Casey 1893, Vandeghen 1965).

The type of conic section to which d is transformed is determined by whether the line d meets the Steiner circumellipse E.

1. If d does not intersect E, the isotomic transform is an ellipse.

2. If d is tangent to E, the transform is a parabola.

3. If d cuts E, the transform is a hyperbola, which is a rectangular hyperbola if the line passes through the isotomic conjugate of the orthocenter

(Casey 1893, Vandeghen 1965).

See also

Cevian Transform, Gergonne Point, Isogonal Conjugate, Isotomic Lines, Isotomic Transform, Isotomic Transversal, Jerabek Hyperbola, Nagel Point, Steiner Circumellipse

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Casey, J. "Theory of Isogonal and Isotomic Points, and of Antiparallel and Symmedian Lines." Supp. Ch.§1 in A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, Co., pp. 165-173, 1888.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.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.Ehrmann, J.-P. and van Lamoen, F. M. "A Projective Generalization of the Droz-Farny Line Theorem." Forum Geom. 4, 225-227, 2004., R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 157-159, 1929.Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163-187, 1994.Sigur, S. "Where are the Conjugates?" Forum Geom. 5, 1-15, 2005., A. "Some Remarks on the Isogonal and Cevian Transforms. Alignments of Remarkable Points of a Triangle." Amer. Math. Monthly 72, 1091-1094, 1965.

Referenced on Wolfram|Alpha

Isotomic Conjugate

Cite this as:

Weisstein, Eric W. "Isotomic Conjugate." From MathWorld--A Wolfram Web Resource.

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