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Conjugation 1 – Conjugate points in a triangle
Published online by Cambridge University Press: 01 August 2016
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The concept of isogonal conjugate points and isotomic conjugate points relative to a given triangle appears to have been introduced by Casey in 1888. Since then these topics have been treated in many important books, such as Johnson Coxeter and Greitzer and more recently Honsberger. The idea originated from the fact that the isogonal conjugates of the points on a fixed line form a circumconic of the triangle involved. That certain pairs of points are isogonal conjugates is of considerable theoretical significance. For example, that the symmedian point K in a triangle is the isogonal conjugate of its centroid G is an important result in understanding the significance of the symmedian point. The concepts involved, though well known to enthusiasts of triangle geometry, deserve additional publicity.
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References
1.
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. Dublin: Hodges, Figgis & Co., (1888) pp. 165–173.Google Scholar
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Johnson, R.A., Modern geometry: an elementary treatise on the geometry of the triangle and the circle, Boston, Mass.: Houghton Mifflin, (1929) pp. 153–158.Google Scholar
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Coxeter, H.S.M and Greitzer, S.L., Geometry revisited, Washington DC: Math. Assoc. Amer., (1967) p. 93.Google Scholar
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Honsberger, R., Episodes in nineteenth and twentieth century Euclidean geometry, Washington DC: Math. Assoc. Amer., (1995) p. 53–57.CrossRefGoogle Scholar
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