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.
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?