# Difference between revisions of "Euler's inequality"

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− | + | Euler's Inequality states that <cmath>R \ge 2r</cmath> where R is the circumradius and r is the inradius of a non-degenerate triangle | |

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− | Euler's Inequality states that <cmath>R \ge 2r</cmath> | ||

==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> | + | 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, <math> R-2r \ge 0 </math> <math> R \ge 2r </math> as desired. |

## Latest revision as of 23:44, 24 June 2015

Euler's Inequality states that 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 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.