Is it always possible to inscribe an equilateral triangle in a closed loop?

Question

Suppose you have a continuous, closed planar curve. It is allowed to intersect itself.

• Is it always possible to find three points on the curve that are equidistant (i.e. form an equilateral triangle?)
• If it is always possible, can the curve have a finite number of sets of 3 equidistant points?
• If it can have a finite number of sets of 3 equidistant points, What's the minimum number of sets of three equidistant points that the curve can have?

Edit: Here's an animation (hopefully it'll load for you) https://i.stack.imgur.com/lbW1M.jpg

Edit 2: I know a circle has an infinite number of equidistant triplets of points - I wanted to know if there existed any curve that only had a finite number of equidistant triplets of points.

Answer 1

Yes, it's always possible to find an equilateral triangle on a (sufficiently smooth) closed curve. Here's an illustration of the proof hinted at above:

1. You start with a closed curve S (black). Find a point A on the curve that has a well-defined tangent (red dot).
2. Rotate the curve by 60 degrees about point A. This produces a second curve T (grey) superimposed on the first.

3. I claim that the two curves S and T intersect in at least two places. Theoretically, the options for any closed curves are that the curves are entirely separate from each other (e.g. far away from each other, or where one curve is entirely contained in the other); or touching at exactly one point (osculate); or touching more than once. But the two curves intersect at the pivot point A, of course, so they touch at least once.

4. And at point A, T goes from being outside of S to being inside of S. Because both curves are closed, T must emerge from S at some point in order to reach point A again from the outside. Hence T must intersect S again somewhere. Call that point B (blue).

5. Point B is a point on curve S and curve T. Consider it as a point on curve T. Let's rotate curve T back onto point S and see where the corresponding point is on curve S. Call that corresponding point C (green).

6. The claim is that ABC is an equilateral triangle.

7. Evidently, due to the rotation process that transforms C into B, angle BAC is 60 degrees.

8. And because rotation about point A transforms C into B, the segments AC and AB have the same length.

9. By the side-angle-side theorem, this shows that triangle ABC is equilateral.

There's not always a finite number of equilateral triangles, because for example a circle can have an infinite number of them. (There are other examples, too: consider a rectangle with one short vertical edge and one very very long horizontal edge. There's an equilateral triangle with a vertex on the top of the rectangle, and an edge entirely on the bottom of the rectangle. You can slide it horizontally to produce an infinitude of congruent equilateral triangles along the length of the rectangle.)