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
Geometry
Easy
Suggested Book
An Excursion in Mathematics

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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