Find point $M$ on the circle such that $MA+MB+MC$ is minimum/maximum

Question

I have a basic geometry problem that I came up with while trying to solve a complex number problem in a geometric way. It looks easy but I get stuck on proving:

Let $ABC$ be an equilateral triangle inscribed in a circle with center $O$. Find the point $M$ on the circle for the value $T=MA+MB+MC$ is maximum and minimum.

I already know the result (by using some software) but I want to prove it clearly.

Answer 1

You can also prove this using the answer given here, which suggests using Ptolemy's theorem. From that link we get this picture, and we know that $p+r=q$, i.e. $T=2q$:

So the value is minimal when $q$ is minimal on the short arc $[AB]$ (So $M$ on $A$ or $B$), and maximal when $q$ is maximal, which happens when $M$ is exactly in the middle of $A$ and $B$.