Euler’s inequality in absolute geometry
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Two results are proved synthetically in Hilbert’s absolute geometry: (i) of all triangles inscribed in a circle, the equilateral one has the greatest area; (ii) of all triangles inscribed in a circle, the equilateral one has the greatest radius of the inscribed circle (which amounts, in the Euclidean case, to Euler’s inequality \(R\ge 2r\)).
KeywordsAbsolute plane geometry Euler’s inequality Area Optimization
Mathematics Subject ClassificationPrimary 51F05 Secondary 51M16
- 1.Chapple, W.: An essay on the properties of triangles inscribed in and circumscribed about two given circles. Miscellaneous Curiosa Mathematica 4, 117–124 (1746)Google Scholar
- 2.Euler, L.: Solutio facilis problematum quorumdam geometricorum difficillimorum. Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae 11, 103–123 (1765/printed 1767)Google Scholar
- 6.Kazarinoff, N.D.: Geometric inequalities. New Mathematical Library 4. Random House, New York (1961)Google Scholar