Does this proof work to prove that the greatest area of a triangle inside a circle is when the triangle is equilateral? I gather it doesn't because most of the proofs I've seen use derivatives etc. If so why doesn't it work?
Consider a triangle $ABC$ inscribed inside a circle $\Gamma$. Assume WLOG one of the sided is fixed then one can easily see that the other two sides being equal maximizes the area of the triangle (it has the greatest height). Now consider one of the two equal sides - using that side as a base we can see from the same argument as before that the triangle must in fact be equilateral to maximize the area (this triangle has the greatest height as before).
Thanks in advance.