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A generalisation of the arbelos theorem of Archimedes

Published online by Cambridge University Press:  23 January 2015

Shailesh A Shirali
Shailesh A Shirali*
Affiliation:
Rishi Valley School, Rishi Valley 517 352, Andhra Pradesh, Indiae-mail:[email protected]
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Extract

Over two thousand years ago, Archimedes discovered a remarkable result concerning two circles drawn with reference to a configuration of three circles and a straight line. Figure 1 displays this result.

In the figure, A, B, C are three collinear points, with B between A and C; circles ω1, ω2, ω3 are drawn on AB, BC, AC as diameters, respectively; a line l is drawn through B, perpendicular to AC; a circle ω4 is inscribed in the region bounded by {ωl, ω3, l}; and a circle ω5 is inscribed in the region bounded by {ω2, ω3, l}. The Archimedean property is that ω4 and ω5 have equal radii.

Type
Articles
Information
The Mathematical Gazette , Volume 95 , Issue 533 , July 2011 , pp. 197 - 205
Copyright
Copyright © The Mathematical Association 2012

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References

1. Bankoff, Leon, A mere coincidence, College Math. Journal (MAA) 23 (1992) p. 106.Google Scholar
2. Bankoff, Leon, Are the twin circles of Archimedes really twins?, Mathematics Magazine (MAA) 47 (1974) pp. 214–218.CrossRefGoogle Scholar
3. Bogomolny, A., Arbelos—the shoemaker's knife, from Interactive mathematics miscellany and puzzles, http://www.cut-the-knot.org/proofs/arbelos.shtml, accessed 27 February 2009.Google Scholar
4. Bogomolny, A., The book of lemmas: Proposition 5, from Interactive mathematics miscellany and puzzles, http://www.cut-the-knot.org/Curriculum/Geometry/BookOfLemmas/BOL5.shtml, accessed 27 February 2009.Google Scholar
5. Boas, H. P., Reflections on the arbelos, http://www.math.tamu.edu/~harold.boas/preprints/arbelos.pdf Google Scholar
6. van Lamoen, Floor, Archimedean adventures, Forum geometricorum 6, 2006, http://forumgeom.fau.edu/FG2006volume6/FG200609index.html Google Scholar
7. van Lamoen, Floor, Online catalogue of Archimedean circles, http://home.planet.nl/~lamoen/wiskunde/Arbelos/Catalogue.htm Google Scholar
8. Schoch, Thomas, Arbelos: amazing properties, http://www.retas.de/thomas/arbelos/arbelos.html Google Scholar
9. Schoch, Thomas, My arbelos story, http://www.retas.de/thomas/arbelos/references.html#ref1 Google Scholar
10. Weisstein, Eric W., Arbelos, From MathWorld—A Wolfram Web Resource, http://mathworld.wolfram.com/Arbelos.html Google Scholar
11. http://mathworld.wolfram.com/search/index.html?query=arbelos Google Scholar