Prove that among all triangles with the same perimeter, the equilateral triangle maximizes the enclosed area.

Answer 1

Heron's formula states that
A = √(s(s-a)(s-b)(s-c))
where s = p/2
So,
A^2 = s(s-a)(s-b)(s-c)
2A dA/da = s(s-b)(2s-2a-b)
dA/da=0 when 2s-2a-b=0
Similarly, dA/db=0 when 2s-2b-a = 0
Solving those two equations gives
a = b = 2/3 s
Since p = a+b+c = 2s, we also have c = 2/3 s
So, a=b=c and we have an equilateral triangle.