# Finding the maximum area of a triangle

A triangle has integer side lengths and sum of its side lengths is 7.What is the maximum possible area of this triangle?

Please give me a hint on starting with this problem.

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The triangles have sides $3,3,1$, or $3,2,2$. You can compute the area of each using Heron's Formula.
Or else note these are both isosceles. The height of the $3,3,1$, with respect to base $1$, is $\sqrt{3^2-\frac{1}{4}}$ (Pythagorean Theorem). You can now find its area. Do a similar calculation for the other triangle.
To see that the area is maximum when sides are close fix the sum of the sides $2s=a+b+c$ and note that the product $s(s-a)(s-b)(s-c)$ which appears in Heron's formula is greatest when $(s-a)(s-b)(s-c)$ is greatest. The sum of these three terms is $s$. The product of terms having a fixed sum is greatest when the terms are equal -$AM\ge GM$ –  Mark Bennet Aug 31 '13 at 7:45