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A hexagon theorem

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  • xpolakis@otenet.gr
    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
    Message 1 of 3 , Jul 1, 2000
      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)


      Antreas
    • Richard Guy
      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
      Message 2 of 3 , 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)
      • Lambrou Michael
        ... I don t know of an earlier reference but I am glad you raise the point. I do, however, have two references different from yours. I will write about this
        Message 3 of 3 , Jul 3, 2000
          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)
          >
          >

          I don't know of an earlier reference but I am glad you raise the
          point. I do, however, have two references different from yours.
          I will write about this when I check my notes at home. For
          now I just remember that I once saw the theorem as an excersize
          for traing students for a mathematical competition.
          I have seen the article of Kyriazis you quote, and I knew the
          result years now.
          Unfortunately Kyriazis presents it as if he discovered THE theorem
          in geometry. His proof is too elaborate but here is a trivial one:
          By the trigonometric form of Ceva's theorem applied to triangle
          ACE we have

          sin(EAD)
          -------- .--------.-------- = 1
          sin(DAC)

          Use know ED=2Rsin(EAD) etc and you are done.The converse similar.

          Michael
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