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[[Image:Napoleon's theorem.svg|200px|right|]]
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


In [[mathematics]], '''[[Napoleon]]'s theorem''' states that if [[equilateral triangle]]s are constructed on the sides of any [[triangle]], either all outward, or all inward, the [[centre (geometry)|centre]]s of those [[equilateral]] triangles themselves form an equilateral triangle.
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The triangle thus formed is called the ''Napoleon triangle'' (inner and outer). The difference in area of these two triangles equals the area of the original triangle.
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The theorem is often attributed to [[Napoleon|Napoleon Bonaparte]] (1769–1821). However, it may just date back to W. Rutherford's 1825 publication ''[[The Ladies' Diary]]'', four years after the French emperor's death.<ref>http://mathworld.wolfram.com/NapoleonsTheorem.html</ref>
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


==Proofs==
<!--'''PNG'''  (currently default in production)
A quick way to see that the triangle LMN is equilateral is to observe that MN  becomes CZ under a [[clockwise]] rotation of 30° around A and an [[Homothetic transformation|homothety]] of ratio √<span style = "text-decoration:overline">''3''</span> with the same center, and that LN also becomes CZ after a counterclockwise rotation of 30° around B and an homothety of ratio √<span style = "text-decoration:overline">''3''</span> with the same center. The respective spiral [[similarity (geometry)|similarities]]<ref>{{MathWorld |title=Spiral Similarity |urlname=SpiralSimilarity}}</ref> are A(√<span style = "text-decoration:overline">''3''</span>,-30°) and B(√<span style = "text-decoration:overline">''3''</span>,30°). That implies MN = LN and the angle between them must be 60°.<ref>For a visual demonstration see ''[http://www.cut-the-knot.org/Curriculum/Geometry/NapoleonSmyth.shtml Napoleon's Theorem via Two Rotations]'' at [[Cut-the-Knot]].</ref>
:<math forcemathmode="png">E=mc^2</math>


[[Analytic geometry|Analytically]], it can be determined<ref name="MathPages270">{{MathPages|id=home/kmath270/kmath270|title=Napoleon's Theorem}}</ref> that each of the three segments of the LMN triangle has a length of:
'''source'''
:<math>\sqrt{{a^2+b^2+c^2  \over  6} + {\sqrt{(a+b+c)(a+b-c)(a-b+c)(-a+b+c)}  \over  {2\sqrt{3}}}}</math>
:<math forcemathmode="source">E=mc^2</math> -->


There are in fact many proofs of the theorem's statement, including a [[trigonometry|trigonometric]] one,<ref name="MathPages270"/> a [[symmetry]]-based approach,<ref>[http://www.cut-the-knot.org/proofs/napoleon.shtml#second Proof #2 (an argument by symmetrization)]</ref> and proofs using [[complex number]]s.<ref name="MathPages270"/>
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


==Background==
==Demos==
[[File:LadiesDiary 1826 p38.jpg|thumb|right|480px|Extract from the 1826 Ladies' Diary giving geometric and analytic proofs]]


The following entry appeared on page 47 in the Ladies' Diary of 1825. As the earliest known reference it may fairly be regarded as the official birth certificate of Napoléon's theorem.  
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


:VII. Quest.(1439); ''by Mr. W. Rutherford, Woodburn.''


&nbsp;&nbsp;Describe equilateral triangles (the vertices being either all outward or all inward) upon the three sides of any triangle ABC: then the lines which join the centres of gravity of those three equilateral triangles will constitute an equilateral triangle. Required a demonstration.  
* accessibility:
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
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** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


Since [[s:Rutherford, William (DNB00)|William Rutherford]] was clearly a very able mathematician his motive for requesting a proof of a theorem that he could certainly have proved himself is unknown. Maybe he posed the question as a challenge to his peers, or perhaps he hoped that the responses would yield a more elegant solution.
==Test pages ==


Plainly there is no reference to Napoléon in either the question or the published responses, though the Editor evidently omitted some submissions. Also Rutherford himself does not appear amongst the named solvers.
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
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Several intriguing mysteries survive to this day :-
==Bug reporting==
* did Rutherford discover the theorem or was it communicated to him by someone else?
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
* when and by whom was the theorem first attributed to Napoléon?
* did Napoléon have anything to do with the initial discovery or proof of the theorem, and if not why does it bear his name?
 
==See also==
*[[Napoleon's problem]]
*[[Napoleon points]]
*[[Lemoine's problem]]
 
==References==
<references/>
 
==External links==
*[http://www.cut-the-knot.org/proofs/napoleon_intro.shtml Napoleon's Theorem and Generalizations], at [[Cut-the-Knot]]
*[http://instrumenpoche.sesamath.net/IMG/lecteur_iep.php?anim=xml_triangle_napoleon2.xml To see the construction], at [[instrumenpoche]]
*[http://demonstrations.wolfram.com/NapoleonsTheorem/ Napoleon's Theorem] by Jay Warendorff, The [[Wolfram Demonstrations Project]].
* {{MathWorld |title=Napoleon's Theorem |urlname=NapoleonsTheorem}}
* [http://dynamicmathematicslearning.com/napole1.html Napoleon's Theorem and some generalizations, variations & converses] at [http://dynamicmathematicslearning.com/JavaGSPLinks.htm Dynamic Geometry Sketches]
* [http://www.cut-the-knot.org/proofs/napoleon.shtml Napoleon's Theorem, Two Simple Proofs]
* [http://alvyray.com/Papers/PapersCG.htm#HexagonSequences Infinite Regular Hexagon Sequences on a Triangle (generalization of Napoleon's Theorem)] by [[Alvy Ray Smith]].
 
{{PlanetMath attribution|id=4538|title=Napoleon's theorem}}
 
[[Category:Triangle geometry]]
[[Category:Napoleon|Theorem]]
[[Category:Theorems in plane geometry]]
 
[[ar:مبرهنة نابليون]]
[[ca:Teorema de Napoleó]]
[[de:Napoleon-Dreieck]]
[[es:Teorema de Napoleón]]
[[fr:Théorème de Napoléon]]
[[ko:나폴레옹의 정리]]
[[it:Teorema di Napoleone]]
[[he:משפט נפוליאון]]
[[nl:Stelling van Napoleon]]
[[ja:ナポレオンの定理]]
[[pms:Teorema ëd Napoleon]]
[[pl:Twierdzenie Napoleona]]
[[pt:Teorema de Napoleão]]
[[ro:Teorema lui Napoleon]]
[[ru:Теорема Наполеона]]
[[fi:Napoleonin lause]]
[[ta:நெப்போலியன் தேற்றம்]]
[[uk:Теорема Наполеона]]
[[zh:拿破侖定理]]

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

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