Question

Find the curve of minimum length which divides the equilateral triangle into two pieces of equal area (median curve)

Answer 1

This must be fairly easy with variational methods but I'm wondering if there's a more elementary method that only uses calculus. I think I can sort of see that at every point the curve must be perpendicular to the line segment connecting it to the vertex, but I wouldn't know how to prove it.

The image sort of describes my intuition. Since over the small distance r doesn't change, the biggest area we can get is by going perpendicular to r. I guess this would imply that we need to go along a circle centered on the facing vertex.