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| In [[geometry]], '''Napoleon points''' are a pair of special points associated with a [[Plane (geometry)|plane]] [[triangle]]. It is generally believed that the existence of these points was discovered by [[Napoleon Bonaparte]], the [[Emperor of the French]] from 1804 to 1815, but many have questioned this belief.<ref name=Coexeter/> The Napoleon points are [[triangle center]]s and they are listed as the points X(17) and X(18) in [[Clark Kimberling]]'s [[Encyclopedia of Triangle Centers]].
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| The name "Napoleon points" has also been applied to a different pair of triangle centers, better known as the [[isodynamic point]]s.<ref>{{cite journal
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| | last = Rigby | first = J. F.
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| | doi = 10.1007/BF01230612
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| | issue = 1–2
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| | journal = Journal of Geometry
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| | mr = 963992
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| | pages = 129–146
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| | title = Napoleon revisited
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| | volume = 33
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| | year = 1988}}</ref>
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| ==Definition of the points==
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| ===First Napoleon point===
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| [[File:First Napoleon Point.svg|300px|right|thumb|First Napoleon point]]
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| Let ''ABC'' be any given [[Plane (geometry)|plane]] [[triangle]]. On the sides ''BC'', ''CA'', ''AB'' of the triangle, construct outwardly drawn [[equilateral triangle]]s ''DBC'', ''ECA'' and ''FAB'' respectively. Let the [[centroid]]s of these triangles be ''X'', ''Y'' and ''Z'' respectively. Then the lines ''AX'', ''BY'' and ''CZ'' are [[Concurrent lines|concurrent]]. The point of concurrence ''N1'' is the first Napoleon point, or the outer Napoleon point, of the triangle ''ABC''.
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| The triangle ''XYZ'' is called the outer Napoleon triangle of the triangle ''ABC''. [[Napoleon's theorem]] asserts that this triangle is an [[equilateral triangle]].
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| In [[Clark Kimberling]]'s [[Encyclopedia of Triangle Centers]], the first Napoleon point is denoted by X(17).<ref name=ETC>{{cite web|last=Kimberling|first=Clark|title=Encyclopedia of Triangle Centers|url=http://faculty.evansville.edu/ck6/encyclopedia/ETC.html|accessdate=2 May 2012}}</ref>
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| * The [[trilinear coordinates]] of N1:
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| :: <math>
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| \begin{align}
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| & \left(\csc\left(A + \frac{\pi}{6}\right), \csc\left(B + \frac{\pi}{6}\right), \csc\left(C + \frac{\pi}{6}\right)\right) \\
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| & = \left( \sec\left(A -\frac{\pi}{3}\right), \sec\left(B -\frac{\pi}{3}\right), \sec\left(C - \frac{\pi}{3}\right)\right)
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| \end{align}
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| </math> | |
| * The [[Barycentric coordinate system (mathematics)|barycentric coordinates]] of N1:
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| :: <math>\left(a \csc\left(A + \frac{\pi}{6}\right), b \csc\left(B +\frac{\pi}{6}\right), c \csc\left(C + \frac{\pi}{6}\right)\right)</math>
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| ===Second Napoleon point===
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| [[File:Second Napoleon Point.svg|300px|right|thumb| Second Napoleon point]]
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| Let ''ABC'' be any given [[Plane (geometry)|plane]] [[triangle]]. On the sides ''BC'', ''CA'', ''AB'' of the triangle, construct inwardly drawn equilateral triangles ''DBC'', ''ECA'' and ''FAB'' respectively. Let the [[centroid]]s of these triangles be ''X'', ''Y'' and ''Z'' respectively. Then the lines ''AX'', ''BY'' and ''CZ'' are concurrent. The point of concurrence ''N2'' is the second Napoleon point, or the inner Napoleon point, of the triangle ''ABC''.
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| The triangle ''XYZ'' is called the inner Napoleon triangle of the triangle ''ABC''. [[Napoleon's theorem]] asserts that this triangle is an equilateral triangle.
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| In Clark Kimberling's Encyclopedia of Triangle Centers, the second Napoleon point is denoted by ''X''(18).<ref name=ETC/>
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| * The trilinear coordinates of N2:
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| :: <math>
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| \begin{align}
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| & \left(\csc\left(A - \frac{\pi}{6}\right), \csc\left(B - \frac{\pi}{6}\right), \csc\left(C - \frac{\pi}{6}\right)\right) \\
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| & = \left(\sec\left(A + \frac{\pi}{3}\right), \sec\left(B +\frac{\pi}{3}\right), \sec\left(C + \frac{\pi}{3}\right)\right)
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| \end{align}
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| </math>
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| * The barycentric coordinates of N2:
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| :: <math>\left(a \csc\left(A - \frac{\pi}{6}\right), b \csc\left(B -\frac{\pi}{6}\right), c \csc \left(C - \frac{\pi}{6}\right)\right)</math>
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| ==Generalizations==
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| The results regarding the existence of the Napoleon points can be [[generalization|generalized]] in different ways. In defining the Napoleon points we begin with equilateral triangles drawn on the sides of the triangle ABC and then consider the centers X, Y, and Z of these triangles. These centers can be thought as the vertices of [[isosceles triangle]]s erected on the sides of triangle ABC with the base angles equal to π/6 (30 degrees). The generalizations seek to determine triangles with less restrictive conditions which, when erected over the sides of the triangle ABC, are such that the lines joining their external vertices and the vertices of triangle ABC are concurrent.{{citation needed|date = May 2012}}
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| ===Generalization to isosceles triangles===
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| [[File:KiepertPoint.svg|300px|right|thumb|A point on the Kiepert hyperbola.]]
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| [[File:KiepertHyperbola.svg|300px|thumb|Kiepert hyperbola of triangle ABC. The hyperbola passes through the vertices (A,B,C), the orthocenter (O) and the centroid (G) of the triangle.]]
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| This generalization asserts the following:<ref name=Eddy/>
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| :''If the three triangles XBC, YCA and ZAB, constructed on the sides of the given triangle ABC as bases, are [[Similar triangle|similar]], [[Isosceles triangle|isosceles]] and similarly situated, then the lines AX, BY, CZ concur at a point N.'' | |
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| If the common base angle is <math>\theta</math>, then the vertices of the three triangles have the following trilinear coordinates.
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| *<math> X ( - \sin \theta, \sin( C + \theta) , \sin( B + \theta) ) </math>
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| *<math> Y ( \sin( C + \theta), - \sin \theta , \sin( A + \theta) ) </math>
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| *<math> Z ( \sin( B + \theta ) , \sin( A + \theta), -\sin \theta ) </math>
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| The trilinear coordinates of N
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| :<math>(\csc(A+\theta),\csc(B+\theta),\csc(C+\theta))</math>
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| A few special cases are interesting.
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| :{| class="wikitable"
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| |-
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| ! Value of <math>\theta</math> !! The point <math> N</math>
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| |-
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| | 0 || G, the centroid of triangle ABC
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| |-
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| | π /2 ( or, – π /2 ) || O, the orthocenter of triangle ABC
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| |-
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| | π /3 || N1, the first Napoleon point
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| |-
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| | – π /3 || N2, the second Napoleon point
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| |-
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| | – A ( if A < π /2 ) <br> π – A ( if A > π /2 ) || The vertex A
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| |-
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| | – B ( if B < π /2 ) <br> π – B ( if B > π /2 ) || The vertex B
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| |-
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| | – C ( if C < π /2 ) <br> π – C ( if C > π /2 ) || The vertex C
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| |}
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| Moreover, the [[Locus (mathematics)|locus]] of N as the base angle <math>\theta</math> varies between -π/2 and π/2 is the [[conic]]
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| :<math>\frac{\sin(B-C)}{x} + \frac{\sin(C-A)}{y} + \frac{\sin(A-B)}{z}=0. </math>
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| This [[conic]] is a [[rectangular hyperbola]] and it is called the ''Kiepert hyperbola'' in honor of Ludwig Kiepert (1846–1934), the mathematician who discovered this result.<ref name=Eddy>{{cite journal|first1=R. H.|last1=Eddy|first2=R.|last2= Fritsch|title=The Conics of Ludwig Kiepert: A Comprehensive Lesson in the Geometry of the Triangle|journal=Mathematics Magazine|date=June 1994|volume=67|issue=3|pages=188–205|url=http://epub.ub.uni-muenchen.de/4550/1/Fritsch_Rudolf_4550.pdf|accessdate=26 April 2012|doi=10.2307/2690610}}</ref> This hyperbola is the unique conic which passes through the five points A, B, C, G and O.
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| ===Further generalizations===
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| ====Generalization to similar (not necessarily isosceles) triangles====
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| [[File:GeneralisationOfNapoleonPointSpecialCase.svg|300px|right|thumb|Generalization of Napoleon point: A special case]]
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| The three triangles XBC, YCA, ZAB erected over the sides of the triangle ABC need not be isosceles for the three lines AX, BY, CZ to be concurrent.<ref name=Michael>{{cite book|first=Michael |last=de Villiers|title=Some Adventures in Euclidean Geometry|year=2009|publisher=Dynamic Mathematics Learning|isbn=9780557102952|pages=138–140}}</ref>
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| :''If similar triangles XBC, AYC, ABZ are constructed outwardly on the sides of any triangle ABC then the lines AX, BY and CZ are concurrent.''
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| ====Generalization to arbitrary (not necessarily similar) triangles====
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| The concurrence of the lines AX, BY, and CZ holds even in much relaxed conditions. The following result states one of the most general conditions for the lines AX, BY, CZ to be concurrent.<ref name=Michael/> | |
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| :''If triangles XBC, YCA, ZAB are constructed outwardly on the sides of any triangle ABC such that
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| ::∠CBX = ∠ABZ, ∠ACY = ∠BCX, ∠BAZ = ∠CAY,
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| :then the lines AX, BY and CZ are concurrent.''
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| [[File:GeneralisationOfNapoleonPoint.svg|300px|right|thumb|A generalization of Napoleon point]] | |
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| ==On the discoverer of Napoleon points==
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| Coxeter and Greitzer state the Napoleon Theorem thus: ''If equilateral triangles are erected externally on the sides of any triangle, their centers form an equilateral triangle''. They observe that Napoleon Bonaparte was a bit of a mathematician with a great interest in geometry. However, they doubt whether Napoleon knew enough geometry to discover the theorem attributed to him.<ref name=Coexeter>{{cite book|first1=H. S. M. |last1=Coxeter |first2=S. L. |last2=Greitzer|title=Geometry Revisited|year=1967 | publisher=Mathematical Association of America|pages=61–64}}</ref>
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| The earliest recorded appearance of the result embodied in Napoleon's theorem is in an article in [[The Ladies' Diary]] appeared in 1825. The Ladies' Diary was an annual periodical which was in circulation in London from 1704 to 1841. The result appeared as part of a question posed by W. Rutherford, Woodburn.
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| :VII. Quest.(1439); by Mr. W. Rutherford, Woodburn." ''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 centers of gravity of those three equilateral triangles will constitute an equilateral triangle. Required a demonstration.''"
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| However, there is no reference to the existence of the so-called Napoleon points in this question. [[Christoph J. Scriba]], a German [[historian of mathematics]], has studied the problem of attributing the Napoleon points to [[Napoleon]] in a paper in [[Historia Mathematica]].<ref>{{cite journal |last= Scriba|title=Wie kommt 'Napoleons Satz' zu seinem namen?|journal=Historia Mathematica|year=1981|volume=8|pages=458–459 |doi=10.1016/0315-0860(81)90054-9 |first=Christoph J |issue=4}}</ref>
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| ==See also==
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| *[[Triangle center]]
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| *[[Napoleon's theorem]]
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| *[[Napoleon's problem]]
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| *[[Van Aubel's theorem]]
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| *[[Fermat point]]
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| ==References==
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| {{reflist|colwidth=33em}}
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| ==Further reading==
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| *{{cite journal|last=Stachel|first=Hellmuth|title=Napoleon's Theorem and Generalizations Through Linear Maps|journal=Contributions to Algebra and Geometry|year=2002|volume=43|issue=2|pages=433–444|url=http://www.emis.de/journals/BAG/vol.43/no.2/b43h2sta.pdf|accessdate=25 April 2012}}
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| *{{cite journal|last=Grünbaum|first=Branko|title=A relative of "Napoleon's theorem"|journal=Geombinatorics|year=2001|volume=10|pages=116–121|url=http://www.math.washington.edu/~grunbaum/A%20Relative%20of%20Napoleons%20Theorem.pdf|accessdate=25 April 2012}}
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| *{{cite web|last=Katrien Vandermeulen, et al.|title=Napoleon, a mathematician ?|url=http://mathsforeurope.digibel.be/Napoleon2.html|publisher=Maths for Europe|accessdate=25 April 2012}}
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| *{{cite web|last=Bogomolny|first=Alexander|title=Napoleon's Theorem|url=http://www.cut-the-knot.org/ctk/Napolegon.shtml|publisher=Cut The Knot! An interactive column using Java applets|accessdate=25 April 2012}}
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| *{{cite web|title=Napoleon's Thm and the Napoleon Points|url=http://www.pballew.net/napthm.html|accessdate=24 April 2012}}
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| *{{cite web|last=Weisstein|first=Eric W.|title=Napoleon Points|url=http://mathworld.wolfram.com/NapoleonPoints.html|publisher=From MathWorld—A Wolfram Web Resource|accessdate=24 April 2012|separator=}}
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| *{{cite web|last=Philip LaFleur|title=Napoleon’s Theorem|url=http://scimath.unl.edu/MIM/files/MATExamFiles/LaFleur%20Expository_FINAL.pdf|accessdate=24 April 2012}}
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| * {{cite web|last=Wetzel|first=John E.|title=Converses of Napoleon's Theorem|date=April 1992|url=http://apollonius.math.nthu.edu.tw/d1/disk5/js/geometry/napoleon/9.pdf|accessdate=24 April 2012}}
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| [[Category:Triangle centers]]
| |
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