Re: a claim on closed curves
- R. Nandakumar <nandakumarr@...> wrote (in geometry-puzzles):
>Here is a claim for which I have only a tentative proof. Is it valid?In the American Mathematical Monthly appeared the following problem:
>'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.
Give appropriate conditions under which a simple closed curve in the
plane contains three points which form the vertices of an equilateral
(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.