Napoleon's Theorem

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In geometry, Napoleon's theorem states that if equilateral triangles are constructed on the sides of any triangle, either all outward or all inward, the lines connecting the centres of those equilateral triangles themselves form an equilateral triangle. The triangle thus formed is called the inner or outer Napoleon triangle. The difference in area of these two triangles equals the area of the original triangle. The theorem is often attributed to Napoleon Bonaparte (1769–1821). Some have suggested that it may date back to W. Rutherford's 1825 question published in The Ladies' Diary, four years after the French emperor's death, but the result is covered in three questions set in an examination for a Gold Medal at the University of Dublin in October, 1820, whereas Napoleon died the following May.

equilateral napoleon geometry

References

Weisstein, Eric W.. "Spiral Similarity". http://mathworld.wolfram.com/SpiralSimilarity.html.
For a visual demonstration see Napoleon's Theorem via Two Rotations at Cut-the-Knot. http://www.cut-the-knot.org/Curriculum/Geometry/NapoleonSmyth.shtml
Coxeter, H.S.M., and Greitzer, Samuel L. 1967. Geometry Revisited, pages 60-63.
"Napoleon's Theorem". MathPages.com. http://www.mathpages.com/home/kmath270/kmath270.htm.
Alexander Bogomolny. "Proof #2 (an argument by symmetrization)". Cut-the-knot.org. http://www.cut-the-knot.org/proofs/napoleon.shtml#second. Retrieved 2013-09-06.
Cavallaro, V.G. (1949), "Per la storia dei teoremi attribuiti a Napoleone Buonaparte e a Frank Morley", Archimede 1: 286–287
Scriba, Christoph J (1981). "Wie kommt 'Napoleons Satz' zu seinem namen?". Historia Mathematica 8 (4): 458–459. doi:10.1016/0315-0860(81)90054-9. https://dx.doi.org/10.1016%2F0315-0860%2881%2990054-9
Faifofer (1911), Elementi di Geometria (17th ed.), Venezia, p. 186 , but the historical record cites various editions in different years. This reference is from (Wetzel 1992)
http://solo.bodleian.ox.ac.uk/primo_library/libweb/action/dlDisplay.do?vid=OXVU1&docId=oxfaleph014134656 http://dbooks.bodleian.ox.ac.uk/books/PDFs/590315941.pdf [22.8MB]
The First Six and the Eleventh and Twelfth Books of Euclid's Elements; with Notes and Illustrations, and an Appendix in Five Books (Adam and Charles B;ack, Edinburgh; Longman, Rees & co, London; John Cumming, Dublin; Simms & McIntyre, Belfast; James Brash & Co, Glasgow, 1834) https://books.google.com/books?id=dQBfAAAAcAAJ
Clarendon Press, Oxford, 1867, pp. 133--135
Macmillan, London, 1873, pp. 42--43
Weisstein, Eric W. "Inner Napoleon Triangle." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/InnerNapoleonTriangle.html
Coxeter, H.S.M., and Greitzer, Samuel L. 1967. Geometry Revisited, page 64.
Weisstein, Eric W. "Outer Napoleon Triangle." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/OuterNapoleonTriangle.html
"Isogonal Prismatoids". Discrete & Computational Geometry 18: 13–52. doi:10.1007/PL00009307. https://dx.doi.org/10.1007%2FPL00009307
A. Barlotti, Intorno ad una generalizzazione di un noto teorema relativo al triangolo, Boll. Un. Mat. Ital. 7 no. 3 (1952) 182–185.
Una proprietà degli n-agoni che si ottengono transformando in una affinità un n-agono regolare, Boll. Un. Mat. Ital. 10 no. 3 (1955) 96–98.

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