# Difference between revisions of "Euler's inequality"

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− | 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> | + | 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 as <cmath>d^2=R(R-2r)</cmath> By the [[Trivial Inequality]], <math>R(R-2r)</math> is positive. Since <math>R</math> has to be positive as it is the circumradius, <cmath>R-2r \ge 0\\R \ge 2r</cmath> as desired |

## Revision as of 11:19, 4 June 2013

## Euler's Inequality

Euler's Inequality states that

## Proof

Let the circumradius be and inradius . Let be the distance between the circumcenter and the incenter. Then From this formula, Euler's Inequality follows as By the Trivial Inequality, is positive. Since has to be positive as it is the circumradius, as desired