Area S of triangle with sides a, b and c and with perimeter a+b+c=2*p is

S=sqrt(p*(p-a)(p-b)(p-c))

Therefore, (1/p)S²=(p-a)(p-b)(p-c)

The area of the triangle will be the largest when (p-a)(p-b)(p-c)  is maximum.

Notice, that the sum of each term in that product is constant.

(p-a)(p-b)(p-c)  => p-a + p-b + p-c = 3*p -(a+b+c)= 3p-2p=p

We conclude what sum will be largest when a=b=c. In other words, equilateral triangle has the largest area.

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