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1105Re: [EMHL] A hexagon theorem

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  • Richard Guy
    Jul 3, 2000
      Is it possible that there is any connexion with Martin Gardner, The
      Asymmetric Propeller, Coll. Math. J., 30(1999) 18-22, and the Bankoff,
      Erd"os, Klamkin Math Mag article (1977?) referred to therein? R.

      On Sat, 1 Jul 2000 xpolakis@... wrote:

      > In the Greek mathematical periodical Diastasis [=dimension] #3-4, 1998,
      > pp. 24-37, I read the article by Nikos Kyriazis:
      > A New Geometry Theorem and its more Significant Applications.
      >
      > The author uses the theorem as a lemma to prove well-known theorems:
      > The three angle-bisectors/medians/altitudes of a triangle concur, etc.
      >
      > Actually the theorem is not new. The earliest reference I found is the
      > following:
      >
      > Soit A_1A_2A_3A_4A_5A_6 un hexagone convexe inscrit dans un cercle.
      > Si le produit A_1A_2 X A_3A_4 X A_5A_6 des cotes de rang pair egale
      > le produit des cotes de rang impair A_2A_3 X A_4A_5 X A_6A_1, les
      > diagonales A_1A_4, A_2A_5, A_3A_6 sont concourantes, et reciprocuement.
      > (L'education mathematique 49(1946-1947) 150, #8941)
      >
      > Does anyone know an earlier one?
      >
      > Note that the theorem was once proposed as a problem by HSMC:
      > Given six consecutive points A, B, C, D, E, and F on a circle, prove
      > that if (AB)(CD)(EF) = (BC)(DE)(FA), then AD, BE, and CF are concurrent.
      > (The Mathematics Student Journal, v. 27, #5, 1980, p. 3, # 527,
      > by H. S. M. Coxeter)
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