Euler's inequality
Euler's Inequality states that ![\[R \ge 2r\]](http://latex.artofproblemsolving.com/b/6/7/b67a21d28c721bc8c4932576a99292686f277554.png)
where R is the circumradius and r is the inradius of a non-degenerate triangle
Proof
Let the circumradius be 
and inradius 
. Let 
be the distance between the circumcenter and the incenter. Then ![\[d=\sqrt{R(R-2r)}\]](http://latex.artofproblemsolving.com/3/5/0/35000a85b7898dabbc4a2f8ccbb8e845d8aff7b5.png)
From this formula, Euler's Inequality follows as ![\[d^2=R(R-2r)\]](http://latex.artofproblemsolving.com/7/e/2/7e2fcd01749b32bd1072381de2ac6589dd730703.png)
By the Trivial Inequality, 
is positive. Since has to be positive as it is the circumradius, 
as desired.
