May 24, 2004
There is an easy way proving the main part of this example. Here it is.

Theorem:
Let Area(Rectangle) = Area(Circle) (Denote A(R)=A(C) for convenience). Show that Perimeter(Circle) < Perimeter(Rectangle). (Denote as P(C)

Proof. A(R)=A(C)
ab=ƒÎr2
r= ã[(ab)/ƒÎ]

Need to show that P(C)< 2(a+b) .

To show:
2ƒÎr<2(a+b) ƒÎr<(a+b)
ƒÎã[(ab)/ƒÎ] < (a+b)
ƒÎab<(a+b)2
So, it is enough to show that ƒÎab<(a+b)2.

Lemma : Given a,b, a>b, then ƒÎab<(a+b).
Proof of Lemma : Since (a-b)>0:

(a-b)2 = a2+b2-2ab > 0
a2+b2 > 2ab
a2+b2 > (ƒÎ-2)ab ;
since 2>(ƒÎ-2)
a2+b2 > ƒÎab-2ab
a2+b2+2ab > ƒÎab
(a+b) 2 > ƒÎab
as claimed.

This proves the theorem.