Euler's inequality

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

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