"Urban planning" is a brainteaser asking: There are four towns positioned on the corners of a square. The towns are to be joined by a system of roads such that the total road length is minimized. What is the shape of the road?
well, the solution and question is as attached:
A,B,C,D are the 4 towns, dotted lines show the sequare, solid lines are the solution, $\angle AEB=\angle BFD = \frac{2}{3} \pi$.
It sounds convincing, but, how to prove mathematically the existence of the solution and, the valid of the solution?