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)