Talk:Napoleon's theorem

Other proofs

Hello everybody,

I feel something is missing here,

I mean some other goemetry problems/exercises that have a solution by homothety-rotation or by complex numbers,

thank you, Nicolae-boicu (talk) 01:15, 10 January 2012 (UTC)[reply]

For the "rotation" idea, see http://www.cut-the-knot.org/proofs/napoleon.shtml#second ; for the "complex numbers" idea, see http://www.cut-the-knot.org/proofs/napoleon_complex.shtml . IMHO, I don't think such proofs deserve more than a passing mention and a ref, as i've just added to the article. -- Jokes Free4Me (talk) 03:23, 11 February 2012 (UTC)[reply]

The Ladies' Diary

From the intro it was in 1825, but from the caption it was from 1826. Which is correct? Thanks! VanessaLylithe (talk) 13:22, 11 December 2012 (UTC)[reply]

The question was posed in 1825, but answered in 1826. The illustration is the answer. Bill Cherowitzo (talk) 19:18, 11 December 2012 (UTC)[reply]

I've stuck in a clause to clarify this in the article. Bill Cherowitzo (talk) 03:26, 12 December 2012 (UTC)[reply]

We can visit the list of mathematical contributors to The Ladies' Diary for 1825 to see that both William Rutherford and the first published solver, Thomas Burn, were both already active in solving problems that year (this issue would have appeared in late 1824 ready for 1825; solutions to the problems posed in it would be due in about mid-1825 in order to be considered for possible publication later in the year, in the issue ready for 1826). What is conspicuous from this list is how well Woodburn was represented: there, too, is W. Harrison and also Stephen Fenwick, the latter more precise as to his location: Woodburn School. It is known that Rutherford taught at Woodburn in the years 1822-1825, coinciding with the start of his contributions to the Diary in 1822; later he was to edit The Mathematician with Stephen Fenwick, along initially with T. S. Davies. The Diary served, not only to provide mathematical exercises, but also to advertise the skills of teachers, the merits of schools and the ability of pupils. It seems likely that Rutherford was, in effect, setting an accessible exercise - equilateral triangles placed externally on the edges of a triangle were familiar from the work of Thomas Simpson on maxima and minima by geometrical methods - that would show off the talents of Woodburn. However, publications were competitive also in the problems they published and care was taken to avoid any hint of plagiarism, so the Editor of the Diary would have deemed Rutherford's problem to be sufficiently original, as well as challenging, to justify publication.

Some additional notes

The formula for the side of Nap's triangle includes Heron's formula spelled out, and could be replaced by 2*area/root(3).

According to <https://marveloustriangles.blogspot.com/2014/09/> - verified for a couple of instances! There is a result which combines Napoleon's triangle and Morley's triangle. The result is generalised there, but take the special case noted by Tran Quang Hung, the outer Napolean and inner Morley triangles and join the corresponding vertices in an obvious way they will intersect the near sides of the original triangle (ABC) in points u,v,w so that u is on BC, v on AC, w on AB Then the lines Au, Bv, Cw are concurrent.

A1jrj (talk) 09:47, 26 September 2020 (UTC)[reply]

Well I took another look at this, and the result isn't remarkable. It turns out that you can use a Pseudo Morley triangle (the angles are divided by a fixed ratio but not 1/3), and Pseudo Napoleon triangle (the vertices of the constructed (non equilateral) triangles lie on the perpendicular bisector but the angle with the sides isn't 60), and the result holds. Well, maybe that is remarkable. The proof is almost immediate using ceva's concurrency theorem.

A1jrj (talk) 15:16, 4 October 2020 (UTC)[reply]

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Asymmetric Propellor Theory

The proof below for the general form of the asymmetric propellor theorem proves Napoleon’s theorem. Simply put B1=A2, C1=A3 and C2=B3, then A1B1C1=A1A2A3, A2B2C2=B1B2B3, A3B3C3=C1C2C3 and then for any arbitrary triangle ABC we define A=B1=A2, B=C1=A3, C=C2=B3 (or a permutation of that) and define A1, B2 and C3 so that A1A2A3, B1B2B3, C1C2C3 are equilateral and we have Napoleon’s theorem. We even prove a generalisation if we simply define A1, B2 and C3 so that A1A2A3, B1B2B3, C1C2C3 are similar but not necessarily equilateral. https://www.cut-the-knot.org/m/Geometry/FinalAsymmetricPropeller.shtml Overlordnat1 (talk) 16:10, 17 May 2021 (UTC)[reply]