<s>[INST] Equilateral triangle and another triangle with same perimeter. Which has larger area? There is an equilateral triangle with sides $a$ and another triangle with sides $p,q,r$, both having the same perimeter $S$.
How can we mathematically show which of them has a larger area?
 [/INST] We have M=√(A(A-a)(A-b)(A-c),A=(1/2)(a+b+c)=(1/2).circumference. Thus M=√(A(A-a)(A-b)(A-{2A-a-b})=√(A(A-a)(A-b)(-A+a+b)). Let N=((M²)/A)=(A-a)(A-b)(-A+a+b). We find ∂N/a=0=∂N/b. These translates to (A-b)(-1)(-A+a+b)+(A-b)(A-a)(1)=0,
(A-a)(-1)(-A+a+b)+(A-a)(A-b)(1)=0. Thus A-a=-A+a+b,A-b=-A+a+b. Thus 2A=2a+b=2b+a,a=b=c. Thus when the sides are equal we get an extremum. In fact a maximum.

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