Euler's inequality

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Euler's Inequality states that $$R \ge 2r$$ where R is the circumradius and r is the inradius of a non-degenerate triangle

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.