1. Napoleon's Theorem - Wikivisual In mathematics, Napoleon's theorem is a theorem that states that if equilateral triangles are constructed on the sides of any triangle, either all outward, or all inward, the http://en.wikivisual.com/index.php/Napoleon's_theorem

Francais English Napoleon's theorem From Wikipedia, the free encyclopedia Jump to: navigation search Image:Napoleon's theorem.svg In mathematics Napoleon's theorem is a theorem that states that if equilateral triangles are constructed on the sides of any triangle , either all outward, or all inward, the centroids of those equilateral triangles themselves form an equilateral triangle. The absolute difference in area of the inner and outer Napoleon triangles equals the area of the given triangle. Although he is traditionally given the credit, there is no specific evidence that Napoleon Bonaparte discovered or proved the theorem edit External links MathPages.com Napoleon's Theorem and Generalizations This article incorporates material from Napoleon's theorem on PlanetMath , which is licensed under the GFDL de:Napoleon-Dreieck fr:Théorème de Napoléon fi:Napoleonin lause ... zh:拿破侖定理 This geometry-related article is a stub . You can help Wikipedia by expanding it Retrieved from " http://en.wikivisual.com/index.php/Napoleon%27s_theorem Categories PlanetMath sourced articles Euclidean plane geometry ... Geometry stubs Views Article Discussion Edit this page History Personal tools Navigation Main Page Random article Search Toolbox Francais English Printable version Permanent link This page was last modified 23:16, 1 December 2006.

2. Napoleon's Theorem - Wikipedia, The Free Encyclopedia In mathematics, Napoleon 's theorem states that if equilateral triangles are constructed on the sides of any triangle, either all outward, or all inward, the centroids of those http://en.wikipedia.org/wiki/Napoleon's_theorem

Napoleon's theorem From Wikipedia, the free encyclopedia Jump to: navigation search In mathematics Napoleon 's theorem states that if equilateral triangles are constructed on the sides of any triangle , either all outward, or all inward, the centroids of those equilateral triangles themselves form an equilateral triangle. 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. The theorem is often attributed to 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. 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 homothety of ratio √ with the same center and that and LN also becomes CZ after a counterclockwise rotation of 30° around B and an homotecy of ratio √ with the same center. the respective spiral similarities A(√ ,-30°) and B(√ ,30°). That implies MN = LN and the angle between them must be 60°.

3. Napoleon's Theorem - Includipedia, The Inclusionist Encyclopaedia In mathematics, Napoleon's theorem states that if equilateral triangles are constructed on the sides of any triangle, either all outward, or all inward, the centroids of those http://www.includipedia.com/wiki/Napoleon's_theorem

Napoleon's theorem From Includipedia, the inclusionist encyclopedia Jump to: navigation search Image:Napoleon's theorem.svg In mathematics Napoleon's theorem states that if equilateral triangles are constructed on the sides of any triangle , either all outward, or all inward, the centroids of those equilateral triangles themselves form an equilateral triangle. The absolute difference in area of the inner and outer Napoleon triangles equals the area of the given triangle. Although he is traditionally given the credit, there is no specific evidence that Napoleon Bonaparte discovered or proved the theorem External links Napoleon's Theorem at MathPages Napoleon's Theorem and Generalizations Napoleon's theorem A proof using the Fermat point without transformations This article incorporates material from Napoleon's theorem on PlanetMath , which is licensed under the GFDL Image:POV-Ray-Dodecahedron.svg This geometry-related article is a stub . You can help Wikipedia by expanding it de:Napoleon-Dreieck fr:Théorème de Napoléon he:משולשי נפוליאון ... zh:拿破侖定理 Retrieved from " http://www.includipedia.com/wiki/Napoleon%27s_theorem

4. Napoleon's Theorem @ Top40-Charts.info Napoleon's theorem information article In mathematics, Napoleon 's theorem states that if equilateral triangles are constructed on the sides of any triangle, either all outward, or all http://www.top40-charts.info/?title=Napoleon's_theorem

5. File:Napoleon's Theorem.svg - Wikipedia, The Free Encyclopedia napoleon's_theorem.svg ‎ (SVG file, nominally 315 333 pixels, file size 8 KB) http://en.wikipedia.org/wiki/File:Napoleon's_theorem.svg

File:Napoleon's theorem.svg From Wikipedia, the free encyclopedia Jump to: navigation search File File history ... Global file usage (SVG file, nominally 315 × 333 pixels, file size: 8 KB) This is a file from the Wikimedia Commons . Information from its description page there is shown below. Commons is a freely licensed media file repository. You can help edit Summary This article incorporates material from theorem on PlanetMath , which is licensed under the GFDL edit Licensing Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License , Version 1.2 or any later version published by the Free Software Foundation ; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License www.gnu.org/copyleft/fdl.html GFDL GNU Free Documentation License true true This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license. You are free: to share – to copy, distribute and transmit the work

6. Napoleon's Theorem By The SwissPedia, The Free Encyclopedia By Just Click Search Napoleon's theorem. The biggest multilingual freecontent encyclopedia on the Internet. Over 7 million articles in over 200 languages, and still growing. http://www.swisscorner.com/wiki.php?title=Napoleon's_theorem

7. Napoleon's Theorem In mathematics, Napoleon's theorem states that if equilateral triangles are constructed on the sides of any triangle, either all outward, or all inward, the http://pediaview.com/openpedia/Napoleon's_theorem

Napoleon's theorem In mathematics Napoleon 's theorem states that if equilateral triangles are constructed on the sides of any triangle , either all outward, or all inward, the centroids of those equilateral triangles themselves form an equilateral triangle. 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. The theorem is often attributed to 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. 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 homothety of ratio √ with the same center and that and LN also becomes CZ after a counterclockwise rotation of 30° around B and an homotecy of ratio √ with the same center. the respective spiral similarities A(√ ,-30°) and B(√ ,30°). That implies MN = LN and the angle between them must be 60°. See also Napoleon's problem Notes See Napoleon's Theorem via Two Rotations on the Napoleon's Theorem and Generalizations webpage (reference below) External links Napoleon's Theorem at MathPages Napoleon's Theorem and Generalizations To see the construction Napoleon's Theorem by Jay Warendorff

8. Napoleon\'s Theorem - Wikipedia, The Free Encyclopedia Wikipedia does not have an article with this exact name. Please search for Napoleon\'s theorem in Wikipedia to check for alternative titles or spellings. http://danpritchard.com/wiki/Napoleon's_theorem

Napoleon's theorem edit extracted from Wikipedia, the Free Encyclopedia (using Wikipedia Reflection Script From Wikipedia, the free encyclopedia Jump to: navigation search Look for on one of Wikipedia's sister projects Wiktionary (free dictionary) Wikibooks (free textbooks) Wikiquote (quotations) Wikisource (free library) Wikiversity (free learning resources) Commons (images and media) Wikinews (free news source) Wikipedia does not have an article with this exact name. Please search for in Wikipedia to check for alternative titles or spellings. Start the article , using the Article Wizard if you wish, or add a request for it Search for " in existing articles. Look for pages within Wikipedia that link to this title Other reasons this message may be displayed: If a page was recently created here, it may not yet be visible because of a delay in updating the database; wait a few minutes and try the purge function. Titles on Wikipedia are case sensitive except for the first character; please check alternative capitalizations and consider adding a redirect here to the correct title.

9. Napoleon\'s_theorem - Encyclopedia Of Plants Plantspedia does not have an article with this exact name, however it may be added soon. http://plantspedia.org/info/Napoleon's_theorem

10. Napoleon's Theorem Napoleon s theorem states that if we construct equilateral triangles on the sides of any triangle (all outward or all inward), the centers of those http://www.mathpages.com/home/kmath270/kmath270.htm

Napoleon's Theorem Napoleon's theorem states that if we construct equilateral triangles on the sides of any triangle (all outward or all inward), the centers of those equilateral triangles themselves form an equilateral triangle, as illustrated below. This is said to be one of the most-often rediscovered results in mathematics.� The earliest definite appearance of this theorem is an 1825 article by Dr. W. Rutherford in "The Ladies Diary".� Although Rutherford was probably not the first discoverer, there seems to be no direct evidence supporting any connection with Napoleon Bonaparte, although we know that he did well in mathematics as a school boy.� According to Markham 's biography, To his teachers Napoleon certainly appeared a model and promising pupil, especially in mathematics... The school inspector reported that Napoleon's aptitude for mathematics would make him suitable for the navy, but eventually it was decided that he should try for the artillery, where advancement by merit and mathematical skill was much more open... Even after becoming First Consul he was proud of him membership in the Institute de France (the leading scientific society of France ), and was close friends with several mathematicians and scientists, including Fourier, Monge, Laplace, Chaptal and Berthollet.� (Oddly enough

11. Napoleon's Theorem Napoleon's Theorem, introduction http://www.cut-the-knot.org/proofs/napoleon_intro.shtml

12. Napoleon's Theorem -- From Wolfram MathWorld If equilateral triangles DeltaABE_(AB), DeltaBCE_(BC), and DeltaACE_(AC) are erected externally on the sides of any triangle DeltaABC, then their centers N_(AB), N_(BC), and N_(AC http://mathworld.wolfram.com/NapoleonsTheorem.html

Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Interactive Demonstrations Napoleon's Theorem If equilateral triangles , and are erected externally on the sides of any triangle , then their centers , and , respectively, form an equilateral triangle (the outer Napoleon triangle . An additional property of the externally erected triangles also attributed to Napoleon is that their circumcircles concur in the first Fermat point (Coxeter 1969, p. 23; Eddy and Fritsch 1994). Furthermore, the lines , and connecting the vertices of with the opposite vectors of the erected triangles also concur at This theorem is generally attributed to Napoleon Bonaparte (1769-1821), although it has also been traced back to 1825 (Schmidt 1990, Wentzel 1992, Eddy and Fritsch 1994). Analogous theorems hold when equilateral triangles , and are erected internally on the sides of a triangle . Namely, the inner Napoleon triangle is equilateral , the circumcircles of the erected triangles intersect in the second Fermat point , and the lines connecting the vertices , and concur at Amazingly, the difference between the areas of the outer and inner Napoleon triangles equals the

13. Napoleon's Theorem - Wolfram Demonstrations Project Construct external equilateral triangles on the sides of a triangle. The centers of the equilateral triangles also form an equilateral triangle. http://demonstrations.wolfram.com/NapoleonsTheorem/

14. Napoleon's Theorem And Douglas' Theorem Attempt to spread a novel Cut The Knot! meme via the Web site of the Mathematical Association of America, Napoleon's Theorem, Douglas' Theorem http://www.cut-the-knot.org/ctk/Napolegon.shtml

15. Napoleon’s Theorem And Beyond Napoleon’s Theorem and Beyond. John Baker, Natural Maths. Abstract. The use of Excel to explore old, wellknown geometrical theorems is set in the context of Napoleon’s Theorem. http://epublications.bond.edu.au/ejsie/vol1/iss2/4/

Skip to main content My Account Help About ... Iss. 2 Napoleon’s Theorem and Beyond John Baker Natural Maths Abstract The use of Excel to explore old, well-known geometrical theorems is set in the context of Napoleon’s Theorem. A number of geometrical ideas are drawn into the investigation, and spreadsheet examples are given to show how Excel can effectively model such concepts. The article concludes with a generalisation with respect to lines drawn from the vertices of a triangle. The article is not written primarily from the geometrical point of view, but emphasis is placed on the role that spreadsheets can play in developing an environment in which ideas can be explored and discovered by students. The benefits of using Excel in the context of a geometric investigation are outlined in terms of student control and ease of learning of the environment together with the spreadsheet’s ability to cope with exceptions without halting. Recommended Citation Baker, John (2004) "Napoleon’s Theorem and Beyond," Spreadsheets in Education (eJSiE) : Vol. 1: Iss. 2, Article 4.

16. Napoleon's Theorem the inner Napoleon triangles The theorem also works for the inner Napoleon triangle (right, in red). Here the equilateral triangles are drawn toward the http://www.jimloy.com/geometry/napoleon.htm

Return to my Mathematics pages Go to my home page Napoleon's Theorem Apparently Napoleon Bonaparte came up with this theorem: On any triangle, draw equilateral triangles on each side. Then the lines connecting the centers of these equilateral triangles form another equilateral triangle (the one in red here). This equilateral triangle is called the outer Napoleon triangle. The theorem also works for the inner Napoleon triangle (right, in red). Here the equilateral triangles are drawn toward the inside of the original triangle. And the area of the outer Napoleon triangle, minus the area of the inner Napoleon triangle, is equal to the area of the original triangle. These diagrams were drawn with the program Cinderella Return to my Mathematics pages Go to my home page

17. Math Forum: Napoleon's Theorem A Template for Napoleon's Theorem Explorations. Steve Weimar. Sketchpad Resources Main CIGS Page. The Investigation Draw a triangle. On the edges of the triangle, construct http://mathforum.org/ces95/napoleon.html

A Template for Napoleon's Theorem Explorations Steve Weimar Sketchpad Resources Main CIGS Page The Investigation Draw a triangle. On the edges of the triangle, construct equilateral triangles. Find the centroids of the equilateral triangles and connect them to form a new triangle. What is true about the new triangle? What happens when you reflect each centroid over the closest edge of your original triangle? What is the difference in areas between the outer and inner Napoleanic triangles? Now construct segments from the outside corners of the equilaterals triangles to the opposite vertices of the original triangle. What is true about these segments? What is true about these segments relative to the original triangle? Our Sketch and Comments The following sketch is one "non-geometer's" first exploration of Napoleon's theorem. Here's a link to this sketch by Sarah Seastone, for which you need The Geometer's Sketchpad. Home The Math Library Quick Reference Search ... Help http://mathforum.org/ The Math Forum is a research and educational enterprise of the Goodwin College of Professional Studies Steve Weimar The Geometry Forum November, 1995

18. Napoleon's Theorem -- Math Fun Facts Take any generic triangle, and construct equilateral triangles on each side whose side lengths are the same as the length of each side of the original triangle. http://www.math.hmc.edu/funfacts/ffiles/10009.2.shtml

hosted by the Harvey Mudd College Math Department Francis Su The Math Fun Facts App is now in the App Store Subscribe to our RSS feed or Any Easy Medium Advanced The Math Fun Facts App! List All List Recent List Popular ... Francis Edward Su From the Fun Fact files, here is a Fun Fact at the Easy level: Napoleon's Theorem Figure 1 Take any generic triangle, and construct equilateral triangles on each side whose side lengths are the same as the length of each side of the original triangle. Surprise: the centers of the equilateral triangles form an equilateral triangle Presentation Suggestions: Show the truth of the statement using some extreme cases for the initial triangle (a particularly instructive example is a triangle with one sidelength very close to zero). The Math Behind the Fact: This theorem is credited to Napoleon, who was fond of mathematics, though many doubt that he knew enough math to discover it! How to Cite this Page: Su, Francis E., et al. "Napoleon's Theorem." Math Fun Facts Share Subjects: geometry Level: Easy Fun Fact suggested by: Michael Moody Suggestions? Use

19. Napoleon's Theorem. Geometry From The Land Of The Incas - Antonio Gutierrez Napoleon s Theorem. Any triangle with the centers of equilateral triangles on the sides. http://agutie.homestead.com/files/Napoleon0.htm

Napoleon's Theorem. Geometry from the Land of the Incas. No Frame

20. Napoleon's Theorem And The Fermat Point Napoleon's Theorem and the Fermat Point. This page has a proof of Napoleon's theorem and also proofs of the main properties of the special ines and circles in this figure that http://www.math.washington.edu/~king/coursedir/m444a03/notes/12-05-Napoleon-Ferm

Napoleon's Theorem and the Fermat Point This page has a proof of Napoleon's theorem and also proofs of the main properties of the special ines and circles in this figure that all pass through the Fermat point. The proofs use several important tools that should be reviewed, if needed. See the References section at the end for places to look. The Napoleon figure is a triangle ABC with an equilateral triangle built on each side: BCA', CAB', ABC'.� The centers of the equilateral triangles are X, Y, Z, respectively. Napoleon's Theorem For any triangle ABC, the triangle XYZ is an equilateral triangle. Proof: � The rotation Y maps A to C.� The rotation X maps C to B.� So if we define S = X Y , then S(A) = X (Y (A)) = X (C) = B.� But by the theory of composition of rotations (see Brown 2.4), S is a rotation by angle 240 degrees and the center D of S is constructed as the vertex of a triangle YXD, where angle X = 120/2 = 60 degrees and angle Y also = 60 degrees.� Thus YXD is an equilateral triangle. But also Z (A) = B, since Z

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