Understand the problem

Show that among all quadrilaterals of a given perimeter the square has the largest area.

Source of the problem
Indian National Mathematical Olympiad 1986
Topic
Geometry
Difficulty Level
Easy
Suggested Book
An Excursion in Mathematics

Start with hints

Do you really need a hint? Try it first!

Start with a quadrilateral with sides a,b,c and d. Divide it into two triangles and write its area as the sum of the areas of the triangles.
Show that the area A satisfies A\le\frac{ab+cd}{2} and A\le\frac{ad+bc}{2}.
Using hint 2, derive that A\le \frac{(a+c)(b+d)}{2}.
From the AM-GM inequality, we can write that (a+c)(b+d)\le\frac{(a+b+c+d)^2}{4}. Hence \text{Area}\le\frac{(\text{Perimetre)^2}{8}. Equality is achieved iff all the angles are right angles (this follows from hint 1) and a+c=b+d. If all the angles are right angles then the quadrilateral is a rectangle and hence a=c and b=d. Finally, a=b=c=d. Thus the area is maximised for a square.

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