1. Napoleon's Theorem napoleon's theorem. napoleon's theorem states that if we construct equilateral triangles on the sides of any to give a proof of napoleon's theorem using coordinate geometry and a 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

2. Napoleon's Theorem Mathematical technology for industry and education. napoleon's theorem. Besides conquering most of Europe, Napoleon reportedly came up with this theorem See how to explore napoleon's theorem http://www.saltire.com/applets/advanced_geometry/napoleon_executable/napoleon.ht

Saltire Home Geometry Applet Gallery Basic Triangles ... About Saltire Napoleon's Theorem Besides conquering most of Europe, Napoleon reportedly came up with this theorem: If you take any triangle ABC and draw equilateral triangles on each side, then join up the incenters of these triangles, the resulting triangle GHI is equilateral. See how to explore Napoleon's theorem using the Casio Classpad 300

3. Napoleon's Theorem napoleon's theorem. ©. Copyright 2003, Jim Loy. Apparently Napoleon Bonaparte came up with this theorem On any triangle, draw equilateral triangles on each side. The theorem also works for 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 from 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

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

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index ABOUT THIS SITE About MathWorld About the Author DESTINATIONS What's New MathWorld Headline News Random Entry ... Live 3D Graphics CONTACT Email Comments Contribute! Sign the Guestbook 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 X (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 X 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

5. 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 triangles (outer or inner Napoleon triangles). napoleon's theorem states that the centers of the regular or starshaped. napoleon's theorem (both for outer and inner constructions http://www.maa.org/editorial/knot/Napolegon.html

Search MAA Online MAA Home Cut The Knot! An interactive column using Java applets by Alex Bogomolny Napoleon's Theorem March 1999 A remarkable theorem has been attributed to Napoleon Bonaparte, although his relation to the theorem is questioned in all sources available to me. This can be said, though: mathematics flourished in post-revolutionary France and mathematicians were held in great esteem in the new Empire. Laplace was a Minister of the Interior under Napoleon, albeit only for six short weeks. On the sides of a triangle construct equilateral triangles (outer or inner Napoleon triangles). Napoleon's theorem states that the centers of the three outer Napoleon triangles form another equilateral triangle. The statement also holds for the three inner triangles. The theorem admits a series of generalizations. The add-on triangles may have an arbitrary shape provided they are similar and properly oriented. Then any triple of the corresponding (in the sense of the similarity) points form a triangle of the same shape . Another generalization was kindly brought to my attention by Steve Gray. This time, the construction starts with an arbitrary n-gon (thought to be oriented) and proceeds in (n - 2) steps. The end result at every step is another n-gon, the last of which is either regular or star-shaped. Napoleon's theorem (both for outer and inner constructions) follows when n = 3. I shall follow the articles by B.H.Neumann (1942) and J.Douglass (1940).

6. Napoleon's Theorem napoleon's theorem. This is a theorem attributed by legend to Napoleon Bonaparte. It is rather doubtful that the Emperor actually discovered this theorem, but it is true that he was interested in mathematics. Statement of napoleon's theorem. For any triangle ABC, build equilateral triangles on the sides http://www.math.washington.edu/~king/coursedir/m444a02/class/11-25-napoleon.html

Napoleon's Theorem This is a theorem attributed by legend to Napoleon Bonaparte.  It is rather doubtful that the Emperor actually discovered this theorem, but it is true that he was interested in mathematics.  He established such institutions as the Ecole Polytechnique with a view to training military engineers, but these institutions benefited mathematics greatly. French mathematicians made many important discoveries at the turn of the Eighteenth to the Nineteenth Century. Statement of Napoleon's Theorem For any triangle ABC, build equilateral triangles on the sides.  (More precisely, for a side such as AB, construct an equilateral triangle ABC', with C and C' on opposite sides of line AB; do the same for the other two sides.). Then if the centers of the equilateral triangles are X, Y, Z, the triangle XYZ is equilateral.

7. Napoleon's Theorem napoleon's theorem. On each side of a given (arbitrary) triangle describe an equilateral triangle a Mathematician, John Wiley Sons, 1997. napoleon's theorem. A proof by tesselation http://www.cut-the-knot.com/proofs/napoleon_intro.html

CTK Exchange Front Page Movie shortcuts Personal info ... Recommend this site 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.

8. Generalization Of Napoleon's Theorem This sounds even more surprising than napoleon's theorem itself. Here I would like to consider the most general reformulation of the napoleon's theorem. The three similar triangles http://www.cut-the-knot.com/Generalization/napoleon.html

CTK Exchange Front Page Movie shortcuts Personal info ... Recommend this site 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.

9. Napoleon's Theorem napoleon's theorem. post a message on this topic. post a message on a new topic. 2 Nov 1994 napoleon's theorem, by Keith Grove. 2 Nov 1994. Re napoleon's theorem, by Jeff Greif. 3 Nov 1994. Re napoleon's theorem, by Annie Fetter http://mathforum.com/epigone/geometry-pre-college/72

a topic from geometry-pre-college Napoleon's Theorem post a message on this topic post a message on a new topic 2 Nov 1994 Napoleon's Theorem , by Keith Grove 2 Nov 1994 Re: Napoleon's Theorem , by Jeff Greif 3 Nov 1994 Re: Napoleon's Theorem , by Annie Fetter 3 Nov 1994 Re: Napoleon's Theorem , by Doris Schattschneider 29 Oct 2001 Napolean's Theorem , by Valerie Howie 3 Nov 1994 Tufte , by Bob Hayden 3 Nov 1994 Re: Napoleon's Theorem , by Philip Mallinson 3 Nov 1994 Re: Napoleon's Theorem , by James King 3 Nov 1994 Re: Napoleon's Theorem , by Dan Hirschhorn 6 Nov 1994 Re: Napoleon's Theorem , by John Conway 6 Nov 1994 Re: Napoleon's Theorem , by John Conway 7 Nov 1994 Re: Napoleon's Theorem , by Doris Schattschneider 8 Nov 1994 Re: Napoleon's Theorem , by John Conway 2 Nov 1994 Napoleon's Theorem , by Keith Grove The Math Forum

10. Napoleon's Theorem (Part III) By Julio Gonzalez Cabillon napoleon's theorem (Part III) by Julio Gonzalez Cabillon. reply to this message. post a message on a new topic. Back to mathhistory-list Subject napoleon's theorem (Part III) Author Julio Gonzalez Cabillon jgc@adinet.com.uy Date Sun, 27 Oct 1996 20 http://mathforum.com/epigone/math-history-list/lilbingkhing

Napoleon's Theorem (Part III) by Julio Gonzalez Cabillon reply to this message post a message on a new topic Back to math-history-list Subject: Napoleon's Theorem (Part III) Author: jgc@adinet.com.uy Date: The Math Forum

11. Napoleon's Theorem Technologies, Inc. Licensed only for noncommercial use. Sorry, this page requires a Java-compatible web browser. Napoleon s theorem. http://nemendur.khi.is/ingisigt/Napoleon's theorem.htm

This page uses JavaSketchpad , a World-Wide-Web component of The Geometer's Sketchpad Sorry, this page requires a Java-compatible web browser. Napoleon's theorem

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

CTK Exchange Front Page Movie shortcuts Personal info ... Recommend this site 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.

13. Napoleon's Theorem Napoleon s theorem. On each side of a triangle, erect an equilateral triangle, lying exterior to the original triangle. Then the http://www.cut-the-knot.org/proofs/napoleon_complex2.shtml

CTK Exchange Front Page Movie shortcuts Personal info ... Recommend this site Napoleon's Theorem On each side of a triangle, erect an equilateral triangle, lying exterior to the original triangle. Then the segments connecting the centroids of the three equilateral triangles themselves form an equilateral triangle. Let the original triangle be ABC with equilateral triangle ABC , CAB , and BCA built on its sides. Think of all the vertices involved as complex numbers. We shall apply a classical criterion to the three equilateral triangles. Let j be a suitable rotation through 120 o . Then the fact that triangles ABC , CAB , and BCA are equilateral may be expressed as A + jB + j C C + jA + j B B + jC + j A The center of ABC is given by P = (A + B + C and similarly for centers Q and R of triangles CAB and BCA Q = (C + A + B and R = (B + C + A We want to show that P + jQ + j R = 0. Indeed, 3(P + jQ + j R) = A + B + C + j(C + A + B ) + j (B + C + A = (B + jA + j C) + j(A + jB + j C ) + j (C + jA + j B Napoleon's Theorem A proof by tesselation A proof with complex numbers A second proof with complex numbers Two proofs Inscribed angles Douglas' Generalization A Generalization ... Alexander Bogomolny G o o g l e Web Search Latest on CTK Exchange Math Glossary on CTK website Posted by 1mathworld24 1 messages 03:34 PM, Mar-01-04

14. Saltire Software - Geometry Java Applet Gallery These cover theorems about common tangents to circles, Napoleon s theorem on the incenters of equilateral triangles and a number of textbook problems as well http://www.saltire.com/gallery.html

Saltire Home Geometry Applet Gallery Basic Triangles ... About Saltire Geometry Java Applet Gallery Saltire's Geometry Gallery is a collection of Java applets showing interesting geometry configurations. All applets are dynamic in that user input can cause the configuration to change. In some cases, points can be dragged. In other cases dimensions can be changed via edit controls. Some of the applets feature animation. Basic Geometry Applets in this gallery exhibit basic geometrical properties of angles in parallel lines and circular configurations. We also look at pythagorean triples and incircles. Triangle Calculators Four triangle calculators are presented, which allow for the triangle to be specified either by three sides, two sides and the included angle, two sides and the non-included angle or two angles and one side. In all cases the remaining sides and angles are computed. Advanced Geometry A number of advanced geometrical configurations are presented. These cover theorems about common tangents to circles, Napoleon's theorem on the incenters of equilateral triangles and a number of textbook problems as well as an illustration of Casteljeau's spline construction algorithm.

15. 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. 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 r r r , we have the identities Hence the sum of squares of the three preceding differences is To see why this sum vanishes, note the general algebraic identity

16. Math Forum: Napoleon's Theorem A Template for Napoleon s theorem Explorations. Steve Weimar. The following sketch is one 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 ... Contact Us http://mathforum.org/ Steve Weimar The Geometry Forum November, 1995

17. Napoleon's Theorem A thread from the Geometry Forum newsgroup archive. Napoleon s theorem. Back to Newsgroups. Napoleon s theorem by Keith Grove on 11/02/94. Re 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

18. Aubel's Theorem -- From MathWorld right angle. Van Aubel s theorem is related to Napoleon s theorem and is a special case of the PetrDouglas-Neumann theorem. Kiepert http://mathworld.wolfram.com/AubelsTheorem.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index ABOUT THIS SITE About MathWorld About the Author DESTINATIONS What's New MathWorld Headline News Random Entry ... Live 3D Graphics CONTACT Email Comments Contribute! Sign the Guestbook MATHWORLD - IN PRINT Order book from Amazon Geometry Plane Geometry Quadrilaterals Aubel's Theorem Given an arbitrary quadrilateral , place a square outwardly on each side, and connect the centers of opposite squares . Then the two lines are of equal length and cross at a right angle Van Aubel's theorem is related to Napoleon's theorem and is a special case of the Petr-Douglas-Neumann theorem Kiepert Hyperbola Napoleon's Theorem Petr-Douglas-Neumann Theorem ... search Kimberling, C. Geometry in Action: A Discovery Approach Using the Geometer's Sketchpad. Key Curriculum Press, p. 23, 2003. Kitchen, E. "Dörrie Tiles and Related Miniatures." Math. Mag. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 11, 1991. Eric W. Weisstein. "Aubel's Theorem." From

19. Jim Loy's Mathematics Page Morley s theorem; Some Triangle Formulas; Desargues theorem; Pappus theorem; Hero s Formula; Napoleon s theorem NEW; My theorem? (the http://www.jimloy.com/math/math.htm

Go to my home page Jim Loy's Mathematics Page Participate in The Most Pleasing Rectangle Web Poll which recently moved to jimloy.com. "He must be a 'practical' man who can see no poetry in mathematics." - W. F. White. Dedicated to the memory of Isaac Asimov. See the top of my Science pages for comments on Dr. Asimov. My Mathematics Pages were described briefly in the Math Forum Internet News No. 5.48 (27 November 2000) My Mathematics Pages were listed on ENC Online's Digital Dozen for Sep. 2003, as one of the most educational sites on the WWW. ENC is the Eisenhower National Clearinghouse, and is concerned with science and mathematics education. My theorem: There are no uninteresting numbers. Assume that there are. Then there is a lowest uninteresting number. That would make that number very interesting. Which is a contradiction. A number of readers have objected that "numbers" in the above theorem should be "natural numbers" (non-negative integers). My reply to one reader was this: Yes, but I wanted to keep it simple and quotable. And the proof that all numbers are interesting should not be boring. From natural numbers, it can be generalized to rationals, as fractions with interesting numerators and denominators are obviously interesting. And what could be more interesting than an irrational that cannot be formed from any finite combination of rationals? I see that David Wells' book

20. Mudd Math Fun Facts: Napoleon's Theorem 19992003 Francis Edward Su. From the Fun Fact files, here is a Fun Fact at the Easy level Napoleon s theorem. Figure 1 Figure 1. http://www.math.hmc.edu/funfacts/ffiles/10009.2.shtml

hosted by the Harvey Mudd College Math Department Francis Su Any Easy Medium Advanced Search Tips List All Fun Facts Fun Facts Home About Math Fun Facts ... Other Fun Facts Features 1553364 Fun Facts viewed since 20 July 1999. 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! * Subjects: geometry * Level: Easy * Fun Fact suggested by: Michael Moody current rating Click on a number to rate this Fun Fact...

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