Problem:
Of all triangles with a given perimeter, find the one with maximum area.
Solution:
Consider an arbitrary triangle with side lengths a, b, c and perimeter . By Heron’s formula, its area F is given by
Now, the arithmetic mean geometric mean inequality gives
Therefore,
where inequality holds if and only if , that is, when
.
Thus, the area of any triangle with perimeter 2s does not exceed and is equal to
only for an equilateral triangle. QED.
More later,
Nalin Pithwa
PS: Ref: Geometric Problems on Maxima and Minima by Titu Andreescu et al.