1. Napoleon's Theorem napoleon's theorem. napoleon's theorem states that if we construct equilateral triangles on the sides of any triangle (all outward or all inward) http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

2. Napoleon's Theorem napoleon's theorem Besides conquering most of Europe, Napoleon reportedly came up with this theorem http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

3. Math Forum Napoleon's Theorem A Template for napoleon's theorem Explorations http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

4. Geometry Step By Step From The Land Of The Incas (Inca), Dynamic Presents problems involving circles and triangles, with proofs, SAT practice quizzes and famous quotes. Also, has examples of geometry in Peruvian http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

5. Napoleon's Theorem. Geometry From The Land Of The Incas - Antonio napoleon's theorem. Any triangle with the centers of equilateral triangles on the sides. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

6. Famous Napoleon Theorem Famous Napoleon Theorem http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

7. Napoleon's Theorem napoleon's theorem http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

8. Cut The Knot! Attempt to spread a novel Cut The Knot! meme via the Web site of the Mathematical Association of America, napoleon's theorem, Douglass' Theorem http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

9. Generalization Of Napoleon's Theorem Generalization of napoleon's theorem http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

10. Napoleon's Theorem napoleon's theorem, a couple of simple proofs http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

11. Napoleon S Theorem Peer Teaching Napoleon s Theorem Peer Teaching. Lesson Plan This particular lesson was anexploration of Napoleon’s Theorem using the dynamic geometry software tool, http://www.personal.psu.edu/users/l/e/leg147/CI495CPortfolio/Napoleon's Theorem

12. Napoleon's Triangle Napoleon s theorem states that given any triangle ABC you can construct anequilateral triangle on each side of the triangle ABC such that the figure formed http://jwilson.coe.uga.edu/EMAT6680/Huffman/Napoleon's Triangle/Napoleon's_Trian

Napoleon's Triangle Napoleon's Theorem states that given any triangle ABC you can construct an equilateral triangle on each side of the triangle ABC such that the figure formed by connecting the centroids of the three equilateral triangles is an equilateral triangle. This equilateral triangle is called Napoleon's triangle, named after Napoleon Bonapart. It is thought that either Napoleon himself discovered this triangle since it is known that he had a great interest in mathematics, or that a close friend discovered this triangle and named it for him. Now we will look at Napoleon's triangle and prove that it is indeed an equilateral triangle. Let's begin with triangle ABC. Now we will construct three equilateral triangles, one on each side of triangle ABC. Next we will find the centriods, the center, of each of the three equilateral triangles. Finally we will connect these three points to form our triangle. The yellow triangle is known as Napoleon's Triangle. From looking at the above figure we can see that it appears that Napoleon's Triangle is indeed equilateral, now we will prove this to be a fact. o since triangle BEC is an equilateral triangle and BQ is an angle bisector. Now we can apply the law of cosines to get:

13. Napoleon's Theorem Napoleon s theorem, introduction. Napoleon s theorem. On each side of agiven (arbitrary) triangle describe an equilateral triangle exterior to the http://www.cut-the-knot.org/proofs/napoleon_intro.shtml

Username: Password: Sites for teachers Sites for parents Awards Interactive Activities ... Sites for parents Napoleon's Theorem On each side of a given (arbitrary) triangle describe an equilateral triangle exterior to the given one, and join the centers of the three thus obtained equilateral triangles. Show that the resulting triangle is also equilateral. It's indeed quite surprising that the shape of the resulting triangle does not depend on the shape of the original one. However it appears to depend on the shape of the constructed triangles: it's equilateral whenever the latter are equilateral. Herein lies an opportunity for a generalization On sides of an arbitrary triangle, exterior to it, construct (directly) similar triangles subject to two conditions: The apex angles of the three triangles are all different. The triangle of apices has the same orientation as the three triangles. Connect centroids of the three triangles. Thus obtained triangle is similar to the constructed three. Actually it's not even necessary to connect the centers. Any three corresponding (in the sense of similarity) points, when connected, define a triangle similar to the constructed ones [ Wells , pp. 178-181]. Perhaps less surprisingly by now, the triangles can be constructed on the same side as the original triangle.

14. Generalization Of Napoleon's Theorem Generalization of Napoleon s theorem. This sounds even more surprising thanNapoleon s theorem itself. Here I would like to consider a further http://www.cut-the-knot.org/Generalization/napoleon.shtml

Username: Password: Sites for teachers Sites for parents Awards Interactive Activities ... Sites for parents A Generalization of Napoleon's Theorem A theorem ascribed to Napoleon Bonaparte reads as follows: On each side of a given (arbitrary) triangle describe an equilateral triangle exterior to the given one, and join the centers of the three thus obtained equilateral triangles. Show that the resulting triangle is also equilateral. It was already pointed out that the theorem allows several generalizations. In particular, equilateral triangles can be replaced with similar triangles of arbitrary shape. This sounds even more surprising than Napoleon's theorem itself. Here I would like to consider a further generalization that makes the other two quite obvious. Start with two similar triangles (black). On each of the (white) lines connecting their corresponding vertices, construct triangles (red) similar to each other and similarly oriented such that their (white) bases correspond to each other. Then three free vertices of these triangles form a triangle similar to the original two. (See a Java simulation In a special case where two vertices of the given similar triangles coincide, only one (white) line is needed to connect vertices of the two triangles. The other two pairs are connect by sides of the triangles. Three similar isosceles triangles are constructed on the vertex connecting lines.

15. Napoleon's Theorem Napoleon s theorem states that if we construct equilateral triangles on the sides of It s not too difficult to give a proof of Napoleon s theorem using 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 his 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, Markham refers to Fourier as

16. Napoleonic Vectors Napoleon s theorem states that the centers of three equilateral triangles In the note on Napoleon s theorem we saw that this proposition can be http://www.mathpages.com/home/kmath408/kmath408.htm

Napoleonic Vectors Napoleon's Theorem states that the centers of three equilateral triangles constructed on the edges of any given triangle form an equilateral triangle.  In the note on Napoleon's Theorem we saw that this proposition can be expressed in terms of the three complex numbers  v , v , v   representing the vertices of the given triangle in the complex plane.  In general, three complex numbers z , z , z are the vertices of an equilateral triangle if and only if From this, given any two vertices of an equilateral triangle, we can solve for the third, choosing the appropriate root, depending on whether we want a clockwise loop or a counter-clockwise loop.  The centers of the counter-clockwise equilateral triangles are then given by the averages of their vertices, so the centers are given by The differences between these centers are Essentially Napoleon's Theorem asserts that the sum of the squares of these three quantities vanishes for any values of v , v , v , and this is easily verified algebraically.  Notice that the coefficients of the vertices are simply the cube roots of 1.  Denoting these roots by

17. Napoleon's Theorem -- From MathWorld Napoleon s theorem has a very beautiful generalization in the case of Napoleon s theorem is related to van Aubel s theorem and is a special case of the http://mathworld.wolfram.com/NapoleonsTheorem.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index DESTINATIONS About MathWorld About the Author Headline News ... Random Entry CONTACT Contribute an Entry Send a Message to the Team MATHWORLD - IN PRINT Order book from Amazon Geometry Plane Geometry Triangles ... Triangle Properties 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

18. Petr-Neumann-Douglas Theorem -- From MathWorld Pech, P. The Harmonic Analysis of Polygons and Napoleon s theorem. J.Geometry Graphics 5, 1322, 2001. Petr, K. Ein Satz über Vielecke. Arch. Math. http://mathworld.wolfram.com/Petr-Neumann-DouglasTheorem.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index DESTINATIONS About MathWorld About the Author Headline News ... Random Entry CONTACT Contribute an Entry Send a Message to the Team MATHWORLD - IN PRINT Order book from Amazon Geometry Plane Geometry Triangles ... van Lamoen Petr-Neumann-Douglas Theorem If isosceles triangles with apex angles are erected on the sides of an arbitrary -gon , and if this process is repeated with the -gon formed by the free apices of the triangles, but with a different value of , and so on until all values have been used in arbitrary order, then a regular -gon is formed whose centroid coincides with the centroid of Napoleon's theorem and van Aubel's theorem are special cases of the Petr-Neumann-Douglas theorem. SEE ALSO: Douglas-Neumann Theorem Isosceles Triangle Napoleon's Theorem van Aubel's Theorem ... [Pages Linking Here] Portions of this entry contributed by Floor van Lamoen REFERENCES: Baker, H. F. "A Remark on Polygons." J. London Math. Soc. Chang, G. "A Proof of Douglas and Neumann by Circulant Matrices."

19. Math Forum: Napoleon's Theorem A Template for Napoleon s theorem Explorations The following sketch isone nongeometer s first exploration of Napoleon s theorem. 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/ Steve Weimar The Geometry Forum November, 1995

20. Napoleon's Theorem Re Napoleon s theorem by Doris Schattschneider on 11/03/94. Tufte by Bob Haydenon 11/03/94. Re Napoleon s theorem by John Conway on 11/06/94. http://mathforum.org/~sarah/HTMLthreads/articletocs/napoleons.theorem.tufte.html

A thread from the Geometry Forum newsgroup archive Napoleon's Theorem Back to geometry.pre-college Search topics All Newsgroups Napoleon's Theorem by ... WebMaster

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