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Re: a claim on closed curves

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  • xpolakis@otenet.gr
    ... In the American Mathematical Monthly appeared the following problem: Give appropriate conditions under which a simple closed curve in the plane contains
    Message 1 of 1 , Oct 6, 2000
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      R. Nandakumar <nandakumarr@...> wrote (in geometry-puzzles):

      >Here is a claim for which I have only a tentative proof. Is it valid?
      >
      >'Given any closed curve, not necessarily smooth, not necessarily
      >confined to 2 dimensions, one can have an INFINITE number of
      >EQUILATERAL triangles such that all 3 vertices of each triangle lie
      >on the given curve'
      >
      >Even if the claim holds for closed 1-d curves which 'live in' lower
      >dimensional spaces, I am not sure how it would turn out if the curve
      >lies in a space of dimensionality above 3.


      In the American Mathematical Monthly appeared the following problem:

      Give appropriate conditions under which a simple closed curve in the
      plane contains three points which form the vertices of an equilateral
      triangle.
      (AMM 95(1988) p. 555, #E3273 by Orrin Frink)

      There is an interesting discussion (Editorial Comment) on the problem
      and its generalizations (squares), in the solution which appeared in
      AMM 97(1990), p. 159.

      Note: As the solvers proved, no extra conditions are necessary.

      Antreas
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