## Geometric maxima minima — a little question for IITJEE Mains

**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.

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