Original scientific paper
https://doi.org/10.21857/y6zolb6ldm
Refined Euler's inequalities in plane geometries and spaces
Darko Veljan
orcid.org/0000-0001-5904-6546
; Department of Mathematics, Faculty of Science, University of Zagreb, 10 000 Zagreb, Croatia
Abstract
Refined famous Euler's inequalities R ≥ nr of an n-dimensional simplex for n = 2, 3 and 4 as well as of non-Euclidean triangles in terms of symmetric functions of edge lengths of a triangle or a simplex in question are shown. Here R is the circumradius and r the inradius of the simplex. We also provide an application to geometric probabilities of our results and an example from astrophysics to the position of a planet within the space of four stars. We briefly discuss a recursive algorithm to get similar inequalities in higher dimensions.
Keywords
Triangle and tetrahedron inequalities; Euler's inequality in 2D, 3D and 4D; non-Euclidean Euler's inequality; geometric probability applied to astrophysics; simplex inequalities.
Hrčak ID:
307494
URI
Publication date:
25.8.2023.
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