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41. Napoleon's Theorem
NAPOLEON S theorem. If equilateral triangles are erected externally on the sides of any triangle then their centers form an equilateral
http://axp.mat.uniroma2.it/~tauraso/Java/Napoleon/napo.html
 NAPOLEON'S THEOREM . If equilateral triangles are erected externally on the sides of any triangle then their centers form an equilateral triangle called Napoleon triangle . The center of the Napoleon triangle coincides with the centroid of the original triangle Move the vertices of the triangle and see how the picture changes.

43. DC MetaData For: Napoleon's Theorem With Weights In N-Space
Abstract The famous theorem of Napoleon was recently extended to higher dimensions. Keywords Napoleon s theorem, Torricelli s configuration.
 Napoleon's Theorem with Weights in n-Space by Preprint series: 98-20, Preprints MSC 51N10 Affine analytic geometry 51N20 Euclidean analytic geometry Abstract The famous theorem of Napoleon was recently extended to higher dimensions. With the help of weighted vertices of an n-simplex T in E n , n >= 2, we present a weighted version of this generalized theorem, leading to a natural configuration of (n-1)-speres corresponding with T by an almost arbitrarily chosen point. Besides the Euclidean point of view, also affine aspects of the theorem become clear, and in addition a critical discussion on the role of the Fermat-Tooicelli point in this framework is given. Keywords: Napoleon's Theorem, Torricelli's configuration Upload: Update: The author(s) agree, that this abstract may be stored as full text and distributed as such by abstracting services.

44. Napoleon's Theorem
Napoleon s theorem. Amer., pp. 6065, 1967. Pappas, T. ``Napoleon s theorem. The Joy of Mathematics. San Carlos, CA Wide World Publ./Tetra, p. 57, 1989.
http://icl.pku.edu.cn/yujs/MathWorld/math/n/n020.htm
##### Napoleon's Theorem
If Equilateral Triangles are erected externally on the sides of any Triangle , then the centers form an Equilateral Triangle (the outer Napoleon Triangle ). Furthermore, the inner Napoleon Triangle is also Equilateral and the difference between the areas of the outer and inner Napoleon triangles equals the Area of the original Triangle See also Napoleon Points Napoleon Triangles
References Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 60-65, 1967. Pappas, T. ``Napoleon's Theorem.'' The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 57, 1989. Didaktik der Mathematik Wentzel, J. E. ``Converses of Napoleon's Theorem.'' Amer. Math. Monthly
Eric W. Weisstein

45. CRC Concise Encyclopedia Of Mathematics On CD-ROM: N
Inequality; Napierian Logarithm; Napkin Ring; Napoleon Points; Napoleon s Problem; Napoleon s theorem; Napoleon Triangles; Nappe; Narcissistic
http://icl.pku.edu.cn/yujs/MathWorld/math/n/n.htm
 Eric W. Weisstein

46. Napthm
Napoleon s theorem is the name popularly given to a theorem which states that if equilateral triangles are constructed on the three legs of any triangle, the
http://www.pballew.net/napthm.html
##### Napoleon's Thm and the Napoleon Points
Napoleon's Theorem is the name popularly given to a theorem which states that if equilateral triangles are constructed on the three legs of any triangle, the centers of the three new triangles will also form an equilateral triangle. In the figure the original triangle is labeled A, B, C, and the centers of the three equilateral triangles are A', B', C'. If the segments from A to A', B to B', and C to C' are drawn they always intersect in a single point, called the First Napoleon Point. If the three equilateral triangles are drawn interior to the original triangle, the centers will still form an equilateral triangle, but the segments connecting the centers with the opposite vertices of the original triangle meet in a (usually) different point, called the 2nd Napoleon Point.
Although it is known that Napoleon had a keen interest in geometry, math historians seem unable to find evidence he really discovered the theorem. Here is a letter on the subject from Antreas P. Hatzipolakis, a real living Greek mathematician, to the Geometry Forum.
The early history of Napoleon's theorem and the Fermat points F, F' (which are also called isogonic centers of ABC) is summarized in Mackey [21], who traces the fact that LMN and L'M'N' are equilateral to 1825 to one Dr. W. Rutherford [27] and remarks that the result is probably older.

47. MATH WORDS, AND SOME OTHER WORDS OF INTEREST
If you have suggestions or comments Email to Pat Ballew A B C D E F G H I J K L M N O P Q R S T U V W X Y Z. N Napoleon s Point; Napoleon s theorem;
http://www.pballew.net/etyind2.html
 MATH WORDS, AND SOME OTHER WORDS, OF INTEREST - cont. If you have suggestions or comments E-mail to: Pat Ballew A B C ... M N O P Q R ... Z N Napoleon's Point Napoleon's Theorem Nappe (of a conic surface) Natural Logarithm base, e ... Nine Chapters on the Mathematical Arts, Chinese Manuscript, Wikipedia link Nine Men's Morris, the game Nine Point Circle Nocebo Node ... Null Set O Obelus, "Ã·" Oblate Spheroid Oblique (triangles or lines) Oblong ... Oxygen P Pappus Extension of Pythagorean Theorem parable parabola Paradox ... Pythagorean relation for Spherical Triangles Q Q, Symbol for Rational numbers QED ,symbol at end of proofs Quadrangle ... Quotient R r for coefficient of correlation Radians Radical ... Russian Peasant Multiplication S Sabot sabotage Saint Petersburg Game Salient Angle ... Syracuse Problem T t - Test Tally Tangent Tangram ... Two U Umbra Umbrella Uranium Uranus V Variable / Variance Varignon Quadrilateral and Varignon's Thm Vector Velocity ... Vortex W Wednesday Whetstone of Wit Whole Numbers Wilson's Theorem ... World's Largest Number X X-Bar for Sample Average Xenon Y Yard Year Year of Miracles - Annus Mirabilis Z Zenzizenzizenzic Zero Zero, The Symbol for

48. Index
A Generalization of Napoleon s theorem. Napoleon s theorem Explorations. Napoleon s theorem (Jessica D. Dwy). Interactive Geometry Problem.
http://poncelet.math.nthu.edu.tw/chuan/99s/
 Geometric Constructions Â±ÃÂ®vÂ¡GÂ¥Ã¾Â¥Ã´Â­Â« e-mail: jcchuan@math.nthu.edu.tw Â¹qÂ¸ÃÂ¡G3029 ÂºÃ´Â­Â¶Â¡Gponcelet.math.nthu.edu.tw/chuan/99s Â¿Ã¬Â¤Â½Â«ÃÂ¡GÂºÃ®Â¦XÂ¤TÃ] Â¤WÂ½ÃÂ¦aÃIÂ¡GÂºÃ®Â¦XÂ¤TÃ]203Â«ÃÂ¤ÃÂ¤TÂ¼ÃÂ¼ÃÂ¾ÃÂ¹ÃÂ®ÃÃ]Â¤Âº Introduction to Maple V JavaSketchpad files Photos Â¤Â§Â®a ... Â¾ÃÂ¥ÃÂªÂºÂ®a

49. Napoleon Theorem
Napoleon s theorem. (?). 2, 114, Generalize Napoleon s theorem. ().
http://poncelet.math.nthu.edu.tw/usr3/summer99/18/work14.html
 Napoleon's Theorem Illustrate Napoleon's Theorem : the centers L, M, N of the three equilateral triangles DBXC, DCYA, DAZB built outwards on the sides BC, CA, AB of an arbitrary triangle DABC are the vertices of an equilateral triangle. The same is true of the centers of the three inward equilateral triangles. Generalize Napoleon's Theorem. (a) Explore interesting porperties associated with the Torricelli's configuration. (Â¤TÂ¹Ã¯ÃÂ³Ã¤ÂªÂºÂ¤Â¤ÃIÂ³sÂ¦Â¨Â¥Â¿Â¤TÂ¨Â¤Â§Ã) (b) Explore interesting porperties associated with the Torricelli's configuration. ( Â¤TÂ­ÃÂ¤pÂ¤TÂ¨Â¤Â§ÃÂªÂºÂ­Â«Â¤ÃÂ³sÂ¦Â¨Â¥Â¿Â¤TÂ¨Â¤Â§Ã) (c) Explore interesting porperties associated with the Torricelli's configuration. (d) Explore interesting porperties associated with the Torricelli's configuration. (Â¤TÂ±Ã¸ÂªÂ½Â½uÂ¦@ÃI) Explore interesting porperties associated with the Torricelli's configuration. (Â¥HÂ¤TÂ¨Â¤Â§ÃÂªÂºÂ¤TÃ¤Â§@Â¤TÂ­ÃÂ¬ÃÂ¦Ã¼Â¤TÂ¨Â¤Â§Ã) (e) Explore interesting porperties associated with the Torricelli's configuration. (Â¤TÂ­ÃÂºÃ±Â¦Ã¢Â¤TÂ¨Â¤Â§ÃÂªÂºÂ¥~Â¤ÃÂ³sÂ¦Â¨Â¬ÃÂ¦Ã¼Â¤TÂ¨Â¤Â§Ã) (f) Explore interesting porperties associated with the Torricelli's configuration.

50. Geometria
theorem. GeoScript-File GeoStyle-File.
http://www.joensuu.fi/mathematics/DidMat/Ehmke/seminar-joensuu/napoleons_theorem
 Home First Examples Resp. Analysis Exercises ... JavaScript Napoleon's Theorem GeoScript -File GeoStyle -File

51. Geometria
theorem. Home 1 Intro 2 Resp.-Anal. 3 JavaScript 4 JS Interf. 5 Design 6 Special 7 Java-Interf. Reference.
http://www.joensuu.fi/mathematics/DidMat/Ehmke/JOENSUU2002/napoleons_theorem.htm
 Napoleon's Theorem Home 1 Intro 2 Resp.-Anal. 3 JavaScript ... 4 JS Interf. ] [5 Design] [6 Special] [7 Java-Interf.] [ Reference GeoScript -File GeoStyle ... -File

 52. Www.seanet.com/~ksbrown/kmath270.htm theorem (Dec 98)Third Tier Napoleon s theorem (Dec 98) Content Level 5 Challenge Level Triangle ABC has equilateral triangles drawn on its edges. Pointshttp://www.seanet.com/~ksbrown/kmath270.htm

53. Casio ClassPad 300 Explorations -- NapoleonÂs Theorem
NapoleonÂs theorem with the Casio ClassPad. NapoleonÂs theorem offers a tour de force for constraint geometry. The theorem states
 Home ClassPad News Overview Buy a ClassPad Now ... Saltire Family of Websites NapoleonÂs Theorem with the Casio ClassPad NapoleonÂs theorem offers a tour de force for constraint geometry. The theorem states that for any arbitrary triangle, if you construct an equilateral triangle on each edge, and join the centers of the incircles of these triangles, then the resulting triangle is equilateral. The theorem is named for, and supposedly discovered by, Napoleon Bonaparte, himself no stranger to tours de force. Take a triangle and draw a triangle on each of its sides We can make the subtended triangles equilateral simply by specifying congruence constraints. Start by making AC congruent to AF and AC congruent to FCÂ (Congruence is the second option on the Measurement Selection drop down button when two segments are selected.) We can follow the same procedure for the other two triangles: We have created the required equilateral triangles. Now for the incircles. We sketch the circles then set tangency constraints - three for each. Now letÂs join the centers of these circles. Shading the resulting triangle makes it stand out from the catÂs cradle of lines and circles. If youÂre not convinced that it is indeed equilateral - and why should you be, Napoleon was more famous for geopolitics than geometry - inspect its side lengths

54. Originator Of Napoleon's Theorem?
forum.swarthmore.edu/ces95/napoleon.html @fph s ps, pdf, A proof of Napoleon s theorem (given a triangle, erect equilateral triangles on its sides then their centroids form another equilateral triangle) which
http://forum.swarthmore.edu/epigone/geometry-college/ninmaxsmel
a topic from geometry-college
##### Originator of Napoleon's Theorem?
post a message on this topic
post a message on a new topic

3 Jan 1999 Originator of Napoleon's Theorem? , by John Conway
3 Jan 1999 Re: Originator of Napoleon's Theorem? , by Antreas P. Hatzipolakis
3 Jan 1999 Re: Originator of Napoleon's Theorem? , by John Conway
The Math Forum

55. From Israel@math.ubc.ca (Robert Israel) Subject Re NAPOLEON On
The most wellknown (at least in France) is the Napoleon s theorem, which, if I remember well says that pour tout triangle, l aire de son cercle
http://www.math.niu.edu/~rusin/known-math/00_incoming/napoleon
 From: israel@math.ubc.ca (Robert Israel) Subject: Re: NAPOLEON on PBS TV; questions Date: 10 Nov 2000 00:17:28 GMT Newsgroups: sci.edu,soc.history,sci.math Summary: [missing] In article , guillaume agostini

56. ICTCM-8 Abstract
In elementary Euclidean geometry this result is known as Napoleon s theorem. Consider the following generalization of this construction.
http://archives.math.utk.edu/ICTCM/abs/8-C96.html
##### Electronic Proceedings of the Eighth Annual Conference on Technology in Collegiate Mathematics
CONTRIBUTED PAPER: 8-C96
##### Napoleon-Like Properties of Spherical Triangles
Mark R. Treuden
Department of Mathematics and Computing
University of Wisconsin-Stevens Point
Stevens Point, Wisconsin 54481-3897
Phone: (715) 346-3734
E-mail: m2treude@uwspmail.uwsp.edu
##### ABSTRACT
If equilateral triangles are constructed outwards or inwards on the sides of any given triangle, the centroids of these triangles are the vertices of an equilateral triangle. In elementary Euclidean geometry this result is known as Napoleon's Theorem. In this paper we adapt the foregoing construction to certain classes of spherical triangles and use a CAS to determine various values of s,t with the properties given above.

57. Math Resources From Grau-Hall Scientific
Mutually prime integers, My logo, 9Point Circle, Nine point cirle, Nagel point, Napier Bones, Napoleon s theorem, Napoleon s theorem, a generalization
http://www.grauhall.com/math.htm
grauhall grauhall.com P. O. Box 279592 Sacramento, CA 95827
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##### Teacher's Resources: Math - last updated on 06/20/03.
e p i Click once on any underlined text to go to that site. (Last updated 05/25/03): The Abel Prise
Arapahulian Rope Computer
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- the online learning center Geometry Formulas and Facts Geometry Center Graphics Archives GeometryJunkyard - from David Eppstein Gliders in Life-Like Cellular Automata HAKMEM - a variety of results developed by the Artificial Intelligence Laboratory at M.I.T.under the aegis of A. R. P. A. Hilbert Functions and the Chinese Remainder Theorem Integer Sequences - an on-line encyclopedia KdV Institute - math links Lawrence Hall of Science - math and science education links Los Alamos National Laboratory - math e-print archive MacTutor History of Math Virtual Math Lab - from KCOS TV Math Spans All Dimensions , from the Joint Policy Board for Mathematics MathJournals - from Math on the Web Mathematical Associationof America The Mathematical Atlas - a gateway to modern mathematics The Math Forum American Mathematical Society math pre-print server Elsevier math pre-print server Mathematics Websites Websites for Mixed Integer Programming - from Harvey J. Greenberg

58. Version Of 10/9/98 (includes Poncelet) (DELETE THIS LINE!) PLANE
Don t worry, none of the definitions are circular. Example Look up Napoleon s theorem. It involves two definitions ItIsEqui and CET. Look up ItIsEqui.
http://www.math.temple.edu/~zeilberg/mamarim/mamarimTeX/Gtext
 " (without the quotes); once inside Maple, #type: "read text; <2 then ERROR(`Need at least two Pts`): fi: for i from 3 to nargs do if AREA(args[1],args[2],args[i])

59. Napoleon.html
Here is a very short code that proves Napoleon s theorem that says that if you erect equilateral triangles on the three sides of an arbitrary triangle, then
http://www.math.temple.edu/~zeilberg/mamarim/mamarimhtml/Napoleon.html
##### A VERY SHORT PROOF OF A SHORT EMPEROR'S THEOREM
BY SHALOSH B. EKHAD (Exclusive to the Personal Journal of Shalosh B.Ekahd and Doron Zeilberger.) Written: Jan. 4, 1998 Thanks to Rene Descartes all plane geometry is routine in principle, and thanks to computer algebra, it is also routine in practice. However, it is still a challenge to write Maple code that takes as few characters as possible. Here is a very short code that proves Napoleon's Theorem that says that if you erect equilateral triangles on the three sides of an arbitrary triangle, then their cirmcumcenters forms yet another equilateral triangle. Maple Source Code (Plain Text File) The Personal Journal of Shalosh B. Ekhad and Doron Zeilberger Doron Zeilberger's Home Page Shalosh B. Ekhad's Home Page

60. Gleichseitiges Dreieck
theorem (A proof by tesselation, A proof with complex numbers, A second proof with complex numbers, Two proofs
http://www.mathematische-basteleien.de/dreieck.htm
 Gleichseitiges Dreieck Inhalt dieser Seite Was ist ein gleichseitiges Dreieck? Formeln zum Dreieck Ein Punkt im Dreieck Quadrat und Dreieck Dreieck(e) im Dreieck ... Zur Hauptseite "Mathematische Basteleien" Was ist ein gleichseitiges Dreieck? Wie der Name sagt, ist das gleichseitige Dreieck ein Dreieck mit gleich langen Seiten. Wenn auf dieser Seite vom Dreieck die Rede ist, so ist das gleichseitige Dreieck gemeint. Formeln zum Dreieck top Es gilt damit R = (2/3)*h = (1/3)*sqr(3)*a und r = (1/3)*h=(1/6)*sqr(3)*a. Beweis: ED ist Mittellinie und parallel zu AB. Es gilt nach dem zweiten Strahlensatz: MA:MD=AB:ED. Daraus folgt mit AB=a und ED=a/2 die Beziehung MA MD=2 1, qed. Ein Punkt im Dreieck top Beweis am Ende! Zeichnet man von einem Punkt im Dreieck aus die Lote auf die Seiten und die Verbindungslinien zu den Eckpunkten, so entstehen sechs Dreiecke. Beweis am Ende! 3-4-5-Punkt Quadrat und Dreieck top Es sei a die Seite des Dreiecks und b die Seite des Quadrates. Diese Formel leitet man mit Hilfe des Strahlensatzes (blau) und den Beziehungen h = (1/2)*sqr(3)*a und b = sqr(2)*x her. Dreieck(e) im Dreieck top Gedrehtes und gestauchtes Dreieck Beweisgang am Ende!

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