Maximize distance among three points along a circle

Question

Find three points along a circle with radius r such that the sum of distance between any pair of the three points is maximised.

Intuition is pointing to a equilateral triangle. How do I prove that?

Answer 1

HINT Consider three points on a circle $(1,0)$, $(\cos\theta,\sin\theta)$ and $(\cos\phi,\sin\phi)$ with $\phi>\theta$

The expression to be maximized using calculus is $$S=2\sin(\frac{\theta}2)+2\sin(\frac{\phi-\theta}2)+2\sin(\frac{\phi}2)$$