Smallest Perimeter of Quadrilateral With Equal Area

[tarek like math]

Prove perimeter of square smallest of all Quadrilateral's perimeter when their areas are equal. Quadrilateral means a polygon of four sides.
Mathematically, $ab=c^2$ now prove $2(a+b)>4c$

Mod edit: Edited the post and subject replacing with "Quadrilateral"

[tarek like math]

Here from u can say, if multiple of some pair integers equal then smaller sum is of that pair's which has equal integers.

[Tahmid Hasan]

AM-GM

[tarek like math]

Please post ur proof.

[Tahmid Hasan]

it can be easily proved that a quad can have least perimeter if all the angles are right angled.just divide it into 2 parts with a diagonal and apply the $\Delta=.5.ab.sin C$ formula of area of the triangles.then the sine would become 1 and the angle will be right.then apply AM-GM inequality to prove that the perimeter of square is lesser than rectangle.

[tarek like math]

i have a different and very easy proof, i hope i will post it.

[tarek like math]

$ab=c^2$. let $a=c+x, b=c-y$. then $ab=c^2-cy+cx-xy=c^2$ or, $c(x-y)=xy$ or, $(x-y)=xy/c$ it means $(x-y)$ is positive.
Now perimeter of tetrahedren, $2(a+b)=4c+2(x-y)$
since $(x-y)>0$ So $2(a+b)>4c$
(proved)

[Tahmid Hasan]

the definition of tetrahedron was wrong in your post,it should be quadrilateral.If is is you have to consider the cases of trapeziums(not trapezoids)

[tarek like math]

i mean a geometric figure with four sides by tetrahedren if any error with that please don't think something different. for example triangle is a geometric figure with three sides.
@Tahmid, u can comment about my proof which can inspire me.

[Tahmid Hasan]

i didn't means to hurt your feelings Tarek.It is after all a math forum ,we are here to help each other become better mathematicians.so I'm not holding anything personal against you.well the proof was nice.keep moving forward