The isogonal conjugate
of a point in the plane of the triangle is constructed by reflecting
the lines ,
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).
trilinear coordinates of the isogonal
conjugate of the point with coordinates
The following labels lists a number of isogonal conjugate pairs of points.
In the above figure with
and isogonal conjugates,
(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
meets the circumcircle ,
does not intersect , the isogonal transform
is an ellipse;
is tangent to ,
the transform is a parabola;
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).
See also Antipedal Triangle
Line at Infinity
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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 Dublin: Hodges,
Figgis, & Co., pp. 165-173, 1888. 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. Casey, J. Dublin: Hodges, Figgis, & Co., 1893. 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. Coolidge, J. L.
New York: Chelsea, p. 49,
Treatise on the Geometry of the Circle and Sphere. Coxeter, H. S. M. and Greitzer, S. L. Washington, DC: Math. Assoc. Amer., p. 93, 1967. Geometry
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 London: Hodgson, pp. 57 and
74-85, 1913. The
Modern Geometry of the Triangle, 2nd ed. Honsberger, R. Washington, DC: Math.
Assoc. Amer., pp. 53-57, 1995. Episodes
in Nineteenth and Twentieth Century Euclidean Geometry. Johnson, R. A.
Boston, MA: Houghton Mifflin, pp. 153-158, 1929. Modern
Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Kimberling, C.
"Central Points and Central Lines in the Plane of a Triangle." Math.
Mag. 67, 163-187, 1994. Lachlan, R. §10 in London: Macmillian, pp. 55-57,
Elementary Treatise on Modern Pure Geometry. Morley, F. and Morley, F. V. New York: Ginn, 1954. Inversive
Geometry. 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. Referenced
on Wolfram|Alpha Isogonal Conjugate
Cite this as:
Weisstein, Eric W. "Isogonal Conjugate."
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