No CrossRef data available.
Article contents
The three Steiner-Lehmus theorems
Published online by Cambridge University Press: 06 June 2019
Extract
Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.
Steiner’s proof of what is now called the Steiner-Lehmus theorem was published in 1844, the same year as the book The three musketeers, written by the French author Alexandre Dumas. The motto One for all, all for one (Einer für alle, alle für einen; Un pour tous, tous pour un; Uno per tutti, tutti per uno) of the three musketeers came into widespread use in Europe in the 19th century, and its essence is that the three musketeers are inseparable; each member pledges to support the group, and the group supports each member. Now there are three classical geometries of constant curvature, namely Euclidean, spherical and hyperbolic geometries, and one can argue that, like the three musketeers, these geometries should be considered as being inseparable; that is, an idea, theorem or proof in any one of them should automatically be considered in the other two. The issue here should be not only to decide whether a particular result is true, or false, in a given geometry, but to understand which particular properties of the geometries make it so.
Information
- Type
- Articles
- Information
- Copyright
- © Mathematical Association 2019
References
Steiner, J., Elementare Lösung einer Aufgabe über das ebene und sphärische Reiecke, J. Reine Angew. Math. 28 (1844) pp. 375-379.10.1515/crll.1844.28.375CrossRefGoogle Scholar
McBride, J. A., The equal internal bisectors theorem, 1840-1940 Many solutions or none? Edinburgh Math. Notes 33 (1943) pp. 1-12.10.1017/S0950184300000021CrossRefGoogle Scholar
Sylvester, J. J., On a simple geometric problem illustrating a conjectured principle in the theory of geometrical method, Philosophical Mag. 4 (1852) pp. 366-369.10.1080/14786445208647142CrossRefGoogle Scholar
Coxeter, H. S. M. and Greitzer, S. L., Geometry revisited, Random House of Canada (1967).Google Scholar
MacKay, J. S., History of a theorem in elementary geometry, Proc. Edinburgh Math. Soc. XX (1901-02) pp. 18-22.CrossRefGoogle Scholar
Hajja, M., The hyperbolic version of the Steiner-Lehmus theorem, Math. Gaz. 101 (July 2017) pp. 306-307.10.1017/mag.2017.76CrossRefGoogle Scholar
Kiyota, K., A trigonometric proof of the Steiner-Lehmus theorem in hyperbolic geometry, accessed August 2015, available at https://arxiv.org/pdf/1508.03248.pdfGoogle Scholar
Ungar, A. A., A gyrovector space approach to hyperbolic geometry, Synthesis Lectures on Mathematics and Statistics 4, Morgan and Claypool (2009).Google Scholar
Barbu, C., Trigonometric proof of Steiner-Lehmus theorem in hyperbolic geometry, Acta Univ. Apulensis, Math. Inform. 23 (2010) pp. 63-67.Google Scholar
Sönmez, N., Trigonometric proof of Steiner-Lehmus theorem in hyperbolic geometry, KoG 12 (2008) p. 35-36 available at: https://hrcak.srce.hr/31445Google Scholar
Anderson, J. W., Hyperbolic geometry (2nd edn.), Springer Undergraduate Math. Ser., Springer-Verlag (2005).Google Scholar
Berger, M., Geometry I, Springer Universitext, Springer-Verlag (1987).10.1007/978-3-540-93815-6CrossRefGoogle Scholar
Ratcliffe, J. G., Foundations of hyperbolic manifolds, Graduate Texts in Math., Vol. 149, Springer-Verlag (1994).CrossRefGoogle Scholar
Hajja, M., A short trigonometric proof of the Steiner-Lehmus theorem, Forum Geom. 8 (2008) p. 39-42.Google Scholar
You have Access