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Strengthened
Euler's inequality in spherical and hyperbolic geometries
Ren Guo, Estonia Black and Caleb Smith |
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Vol. 18 (2025), No. 5, 909–926
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Abstract |
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Euler’s inequality is a well-known inequality relating the inradius and circumradius of a triangle. In Euclidean geometry, this inequality takes the form , where is the circumradius and is the inradius. In spherical geometry, the inequality takes the form , as proved by Mitrinović, Pečarić and Volenec (Math. Appl. East Eur. Ser. 28 (1989)); similarly, we have 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. |
Keywords
Euler's inequality, Euclidean geometry, spherical geometry,
hyperbolic geometry
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Mathematical Subject Classification
Primary: 51M04, 51M09
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Milestones
Received: 16 April 2024
Revised: 11 July 2024
Accepted: 13 July 2024
Published: 13 November 2025
Communicated by Michael Dorff
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Authors |
| Ren Guo | |
| Department of Mathematics Oregon State University Corvallis, OR United States |
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| Estonia Black | |
| Department of Mathematics University of Tennessee Knoxville, TN United States |
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| Caleb Smith | |
| Department of Mathematics Oregon State Univeristy Corvallis, OR United States |
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| © 2025 MSP (Mathematical Sciences Publishers). |
