# Difference between revisions of "Euler's inequality"

## Euler's Inequality

Euler's Inequality states that $$R \ge 2r$$

## Proof

Let the circumradius be $R$ and inradius $r$. Let $d$ be the distance between the circumcenter and the incenter. Then $$d=\sqrt{R(R-2r)}$$ From this formula, Euler's Inequality follows as $$d^2=R(R-2r)$$ By the Trivial Inequality, $R(R-2r)$ is positive. Since $R$ has to be positive as it is the circumradius, $$R-2r \ge 0\\R \ge 2r$$ as desired