satisfy that: 
,![\[
\sqrt{a + b} + \sqrt{b + c} + \sqrt{c + a} \geq \frac{4(ab + bc + ca)}{\sqrt{(a + b)(b + c)(c + a)}}
\]](http://latex.artofproblemsolving.com/e/7/2/e72fd0f134289c359bc8766440babc52ae131961.png)
Y by
Consider square 
with side length of 
. The point 
is selected on the diagonal 
such that 
. Denote the circumcenters of 
as 
. Find the length of 
with side length of 
. The point 
is selected on the diagonal 
such that 
. Denote the circumcenters of 
as 
. Find the length of 
,
.
and both triangles are isosceles, so 
.
by SAS congruence, and hence 
.
