Difference between revisions of "Euler's inequality"
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==Euler's Inequality== | ==Euler's Inequality== | ||
| − | Euler's Inequality states that <cmath>R \ | + | Euler's Inequality states that <cmath>R \ge 2r</cmath> |
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==Proof== | ==Proof== | ||
Let the circumradius be <math>R</math> and inradius <math>r</math>. Let <math>d</math> be the distance between the circumcenter and the incenter. Then <cmath>d=\sqrt{R(R-2r)}</cmath>. From this formula, Euler's Inequality follows | Let the circumradius be <math>R</math> and inradius <math>r</math>. Let <math>d</math> be the distance between the circumcenter and the incenter. Then <cmath>d=\sqrt{R(R-2r)}</cmath>. From this formula, Euler's Inequality follows | ||
Revision as of 10:13, 4 June 2013
Euler's Inequality
Euler's Inequality states that ![\[R \ge 2r\]](http://latex.artofproblemsolving.com/b/6/7/b67a21d28c721bc8c4932576a99292686f277554.png)
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
