 TOPICS # Isogonal Conjugate The isogonal conjugate of a point in the plane of the triangle is constructed by reflecting the lines , , and about the angle bisectors at , , and . The three reflected lines then concur at the isogonal conjugate (Honsberger 1995, pp. 55-56). In older literature, isogonal conjugate points are also known as counter points (Gallatly 1913), Gegenpunkte (Gallatly 1913), and focal pairs (Morley 1954).

The trilinear coordinates of the isogonal conjugate of the point with coordinates (1)

are (2)

The following labels lists a number of isogonal conjugate pairs of points.

 triangle center isogonal conjugate circumcenter orthocenter Clawson point de Longchamps point Feuerbach point first Brocard point second Brocard point first Fermat point first isodynamic point first isodynamic point first Fermat point Gergonne point incenter incenter second Napoleon point isogonal mittenpunkt mittenpunkt Kosnita point Ko nine-point center mittenpunkt isogonal mittenpunkt Nagel point Na nine-point center Kosnita Point Ko orthocenter circumcenter orthocenter of the orthic triangle first Napoleon point Schiffler point orthocenter of the contact triangle second Brocard point first Brocard point second Fermat point second isodynamic point second isodynamic point second Fermat point Spieker center Sp symmedian point triangle centroid symmedian point of the orthic triangle third Brocard point third power point triangle centroid symmedian point  In the above figure with and isogonal conjugates, (3)

(Honsberger 1995, pp. 54-55).

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 isogonal conjugate of a point on the circumcircle is a point at infinity (and conversely). The sides of the pedal triangle of a point are perpendicular to the connectors of the corresponding polygon vertices with the isogonal conjugate. The isogonal conjugate of a set of points is the locus of their isogonal conjugate points.

The product of isotomic and isogonal conjugation is a collineation which transforms the sides of a triangle to themselves (Vandeghen 1965).

Antipedal Triangle, Collineation, Isogonal Line, Isogonal Mittenpunkt, Isogonal Transform, Isotomic Conjugate, Line at Infinity, Symmedian

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## References

Barrow, D. F. "A Theorem about Isogonal Conjugates." Amer. Math. Monthly 20, 251-253, 1913.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.Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 49, 1971.Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 93, 1967.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.Gallatly, W. "Counter Points." Ch. 9 in The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, pp. 57 and 74-85, 1913.Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 53-57, 1995.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 153-158, 1929.Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163-187, 1994.Lachlan, R. §10 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 55-57, 1893.Morley, F. and Morley, F. V. Inversive Geometry. New York: Ginn, 1954.Sigur, S. "Where are the Conjugates?" Forum Geom. 5, 1-15, 2005. http://forumgeom.fau.edu/FG2005volume5/FG200501index.html.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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Weisstein, Eric W. "Isogonal Conjugate." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/IsogonalConjugate.html