I'm trying to prove that the triangle of largest area for a given perimeter is equilateral, but I'm having some difficulties.
I've done 2 different proofs for a similar problem but for rectangles – one proof uses the AM-GM mean inequality and the other uses algebra and a little calculus.
I can't manage to use a similar method for the triangle problem – with the algebra/calculus method, we only have 2 equations (one for perimeter, one for area) but 3 unknowns (each side length of triangle) so it looked like I was going to have to do it with respect to 2 different variables at the same time.
Can anyone point me in the right direction as to how to approach this?
Thanks
Best Answer
Using Heron's formula, $\triangle^2=s(s-a)(s-b)(s-c) $
Using $AM\ge GM$, $$\frac{s-a+s-b+s-c}3\ge \{(s-a)(s-b)(s-c)\}^\frac13$$
or $(s-a)(s-b)(s-c)\le (\frac s3)^3$ the equality occurs when $s-a=s-b=s-c$ ie when $a=b=c$