Maximise perimeter of triangle in a circle

Hey guys, I hope someone can give me some pointers with this because it should be really easy but im just not getting it!

I want to show that for a triangle inscribed in a circle an equilateral traingle gives the maximal perimeter. Ive tried a few things and just get bogged down in algebra and im sure there should be a clean geometric proof!

For example if you take a unit circle on the origin then I can set one of my points at the north pole (0,1), then in polars assign the other 2 points at B and C. But this gives me the problem of maximising $\displaystyle 2sin(\frac{C}{2}) + 2sin(\frac{B}{2}) + \sqrt{2-2cos(C-B)}$ which gets very messy... can anyone give me some pointers?

Thank you!

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Re: Maximise perimeter of triangle in a circle

This is not a pure geometric proof, but if you know about Lagrange multipliers, it is simple.

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