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