Strengthened Euler’s inequality in spherical and hyperbolic geometries

Guo, Ren; Black, Estonia; Smith, Caleb

doi:10.2140/involve.2025.18.909

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Strengthened Euler's inequality in spherical and hyperbolic geometries

Ren Guo, Estonia Black and Caleb Smith

Vol. 18 (2025), No. 5, 909–926

DOI: 10.2140/involve.2025.18.909

Abstract

Euler’s inequality is a well-known inequality relating the inradius and circumradius of a triangle. In Euclidean geometry, this inequality takes the form $R \geq 2 r$ , 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 by Mitrinović, Pečarić and Volenec (Math. Appl. $($ East Eur. Ser. $)$ 28 (1989)); similarly, we have $\tanh (R) \geq 2 \tanh (r)$ for hyperbolic triangles; see the work of Svrtan and Veljan (Forum Geom. 12 (2012), 197–209) for a proof. In Euclidean geometry, this inequality can be strengthened as discussed by Svrtan and Veljan (2012). 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.

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Keywords

Euler's inequality, Euclidean geometry, spherical geometry, hyperbolic geometry

Mathematical Subject Classification

Primary: 51M04, 51M09

Milestones

Received: 16 April 2024

Revised: 11 July 2024

Accepted: 13 July 2024

Published: 20 November 2025

Communicated by Michael Dorff

Authors

Ren Guo
Department of Mathematics Oregon State University Corvallis, OR United States

Estonia Black
Department of Mathematics University of Tennessee Knoxville, TN United States

Caleb Smith
Department of Mathematics Oregon State Univeristy Corvallis, OR United States

Vol. 18, No. 5, 2025