Strengthened Euler's Inequality in Spherical and Hyperbolic Geometries

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