Maximise perimeter of triangle in a circle

[twinkle21]

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

I want to show that for a triangle inscribedin a circle an equilateral traingle gives the maximal perimeter. I've tried a few things and just get bogged down in algebra and I am 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 2sin(C/2) + 2sin(B/2) + sqrt(2-2cos(C-B)) which is very messy... can anyone give me some pointers?

Thank you!

 

[AlephZero]

Try joining the vertices to the center of the circle, and find the perimeter in terms of the angles at the center. That won't involve any square roots.

Or start from the sine rule: $a / \sin A = b / \sin B = c / \sin C = 2R$ where $R$ is the radius of the circle.