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Euler's inequality is a well known inequality relating the inradius andcircumradius of a triangle. In Euclidean geometry, this inequality takes theform $R \geq 2r$ where $R$ is the circumradius and $r$ is the inradius. Inspherical geometry, the inequality takes the form $\tan(R) \geq 2\tan(r)$ asproved in \cite{MPV}; similary, we have $\tanh(R) \geq 2\tanh(r)$ forhyperbolic triangles (see \cite{SV} for proof). In Euclidean geometry, thisinequality can be strengthened as discussed in \cite{SV}. We prove an analogousversion of this strengthened inequality which holds in spherical geometry, aswell as an additional strengthening of Euler's inequality which holds inEuclidean geometry and can be generalized into both spherical and hyperbolicgeometry.

Strengthened Euler's Inequality in Spherical and Hyperbolic Geometries