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In an arbitrary triangle whose circumcircle has radius $R$ and center $O$ and whose inscribed circle has radius $r$ and center $I$, we have Euler's inequality $$R\geq 2r$$ This follows from the equality $$|IO|^2=R(R-2r)$$ (There are many examples in Euclidean geometry, I think Ptolemy's inequality follows from an equality but I can't remember at the moment)