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A colleague of mine just told me this amusing optimization problem:

Given an equilateral triangle, cut it into two pieces of the same size with a curve of minimal length.

I only want some insight or hint that might help me solve this. I am thinking that this must be related to circle arcs, for instance I computed that the length of the circle arc (centered at one vertex) is shorter than just cutting in a straight line through that vertex.

Any reference or hint, or clever solution will be appreciated.

Tal-Botvinnik
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1 Answers1

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Here are some hints to push you in the direction of a clever solution:

Suppose you have such a curve, and that it goes from one edge to another. Arrange six copies equilateral triangles intersecting at a point, and draw this curve on one triangle, reflected on the next and so on. This gives a closed curve enclosing a fixed area (namely three times the area of a single triangle). What can we conclude about the length of this closed curve?

Similarly, what if the curve starts on one edge and exits the same edge?

Marcus M
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  • good point to check the second case for possible solutions – orangeskid Apr 10 '18 at 16:13
  • Wow, nice solution, I see it. One has still to check if those two solutions are shorter than cutting off a circle inside the triangle, right? Since the statement doesn't say that we have to cut it through the boundary – Tal-Botvinnik Apr 10 '18 at 17:01
  • The good thing about this solution is that it allows to solve for a whole class of triangles (for instance isosceles triangles whose single angle is a fraction of $2\pi$) – Tal-Botvinnik Apr 10 '18 at 20:08