1. Napoleon's Theorem napoleon's theorem. Besides conquering most of Europe, Napoleon reportedly came up with this theorem http://www.saltire.com/applets/advanced_geometry/napoleon_executable/napoleon.ht

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

2. Beiträge Zur Algebra Und Geometrie / Contributions To Algebra And Geometry, Vol hexagons can be associated to any triangle, thus extending napoleon's theorem. The aim of this paper is to prove that http://www.emis.de/journals/BAG/vol.43/no.2/11.html

Napoleon's Theorem and Generalizations Through Linear Maps Hellmuth Stachel Institute of Geometry, Vienna University of Technology, Wiedner Hauptstr. 8-10/113, A-1040 Wien, Austria, e-mail: stachel@geometrie.tuwien.ac.at Abstract: Recently J. Fukuta and Z. Cerin showed how regular hexagons can be associated to any triangle, thus extending Napoleon's theorem. The aim of this paper is to prove that these results are closely related to linear maps. This reflects better the affine character of some constructions and gives also rise to a few new theorems. Keywords: Napoleon's theorem, triangle, regular hexagon, linear map Classification (MSC2000): Full text of the article: Compressed PostScript file (95 kilobytes) PDF file (252 kilobytes) Previous Article Next Article Contents of this Number ELibM for the EMIS Electronic Edition

3. Napoleon's Theorem napoleon's theorem. On each side of a given (arbitrary) triangledescribe an equilateral triangle exterior to the given one, and http://www.cut-the-knot.com/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 in such a way as to have three apex angles all different. 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 ([D.Well's]). Perhaps less surprisingly by now, the triangles can be constructed on the same side as the original triangle. The original problem is traditionally ascribed to Napoleon Bonaparte who was known to be an amateur mathematician.

4. Generalization Of Napoleon's Theorem This sounds even more surprising than the napoleon's theorem itself. Here I would like to consider a further 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 the 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 lines connecting their corresponding vertices (white), construct triangles (red) similar to each other and similarly oriented. 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.

5. Generalization Of Napoleon's Theorem A Generalization of napoleon's theorem. A theorem shape. This soundseven more surprising than the napoleon's theorem itself. Here http://www.cut-the-knot.com/Generalization/napoleon.shtml

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 the 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 lines connecting their corresponding vertices (white), construct triangles (red) similar to each other and similarly oriented. 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.

6. Napoleon's Theorem napoleon's theorem. On each side of a given (arbitrary) triangle describe an equilateral triangle exterior to the given 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 in such a way as to have three apex angles all different. 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 ([D.Well's]). Perhaps less surprisingly by now, the triangles can be constructed on the same side as the original triangle. The original problem is traditionally ascribed to Napoleon Bonaparte who was known to be an amateur mathematician.

7. Math Forum: Napoleon's Theorem A Template for napoleon's theorem Explorations. Steve Weimar. The followingsketch 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. Suggestion Box Home The Math Library Help Desk ... Search http://mathforum.org/ Steve Weimar The Geometry Forum November, 1995

8. Math Forum: Napoleon's Theorem A thread from the Geometry Forum newsgroup archive. napoleon's theorem http://forum.swarthmore.edu/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. Suggestion Box Home The Math Library Help Desk ... Search http://mathforum.org/ Steve Weimar The Geometry Forum November, 1995

9. Head Annie's Sketchpad Materials Napoleon's Theorem napoleon's theorem. This is an investigation of a theorem attributed toNapoleon Bonaparte (yes, the same person we associate with Waterloo). http://mathforum.org/~annie/gsp.handouts/napoleon/

The Math Forum Annie's Sketchpad Activites Printable Version (no Java) Napoleon's Theorem This is an investigation of a theorem attributed to Napoleon Bonaparte (yes, the same person we associate with Waterloo). Construct a triangle, any triangle. You can do this using the segment tool. Then, using the selection arrow, drag each vertex of the triangle to make sure that everything is connected the way you want it to be. Now select the three segments and change their weight to thick and their color to blue (both options are under the Display menu). Construct equilateral triangles on the sides of the triangles. You can do this by constructing a circle centered at one vertex that goes through the other vertex, then do another circle the other way. You can hide the circles by selecting them with the selection arrow and choosing Hide under the Display menu. (You might also have a script tool that constructs the triangle.) Sorry, this page requires a Java-compatible web browser. Find the centers of these equilateral triangles. You can find the center of a triangle by connecting two of the vertices to the midpoint of the opposite side. Midpoint is an option under the Construct menu. Connect the centers with segments. Color these three new segments red. What is true of them? (You can hide the midpoints and segments you drew to find the centers by using Hide under the Display menu.) Sorry, this page requires a Java-compatible web browser.

10. Re: Originator Of Napoleon's Theorem? By Antreas P. Hatzipolakis Subject Re Originator of napoleon's theorem? Author Antreas P. Hatzipolakis xpolakis@hol.gr Date Mon, 4 Jan 1999 http://mathforum.com/epigone/geometry-college/ninmaxsmel/v01540B03B2B5B50A770C@[

Re: Originator of Napoleon's Theorem? by Antreas P. Hatzipolakis reply to this message post a message on a new topic Back to messages on this topic Back to geometry-college Subject: Re: Originator of Napoleon's Theorem? Author: xpolakis@hol.gr Date: The Math Forum

11. Napoleon's Theorem napoleon's theorem. Regarding the idea that Napoleon might actually have discoveredwhat we now call napoleon's theorem, Coxeter and Greitzer have said that. 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

12. Originator Of Napoleon's Theorem? a topic from geometrycollege 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 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

13. Napoleonic Vectors napoleon's theorem states that the centers of three equilateral triangles constructedon 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

14. Napoleon's Theorem -- From MathWorld napoleon's theorem, Assoc. Amer., pp. 6065, 1967. Pappas, T. napoleon's theorem. The Joy of Mathematics. San Carlos, CA Wide World Publ./Tetra, p. 57, 1989. http://mathworld.wolfram.com/NapoleonsTheorem.html

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 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

15. Von Aubel's Theorem -- From MathWorld Von Aubel's theorem is related to napoleon's theorem and is a specialcase of the PetrDouglas-Neumann theorem. napoleon's theorem http://mathworld.wolfram.com/vonAubelsTheorem.html

Geometry Plane Geometry Quadrilaterals von 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 Von 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 ... Square References Kitchen, E. "Dörrie Tiles and Related Miniatures." Math. Mag. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 11, 1991. Author: Eric W. Weisstein Wolfram Research, Inc.

16. Napoleon's Theorem By Plane Tesselation We already have several proofs of napoleon's theorem and its generalizations. http://www.cut-the-knot.com/Generalization/NapTess.html

CTK Exchange Front Page Movie shortcuts Personal info ... Recommend this site Napoleon's Theorem by Plane Tesselation We already have several proofs of Napoleon's Theorem and its generalizations . Here is an illustration of another one that is somewhat akin to the second proof by Scott Brodie but (to me is) more revealing. The given triangle is the one at whose vertices I placed red dots. (These are moveable so that you must be able to modify the triangle.) It appears that the triangle and the attached Napoleon's triangles form a starting configuration for a simple tesselation of the plane. Napoleon's triangles have different backgrounds. To clarify the argument, pick two colors and erase all other triangles. Then it becomes quite obvious that the centers of triangles of a single color form a hexagonal lattice (black lines). Centers of all other triangles lie at the centers of the lattice triangles. In the diagram, all such points are connected by greenish lines. The greenish lines form a finer lattice (a lattice with more points.) Centers of all Napoleon's triangles lie at nodes of the latter lattice. This proves the theorem. Various plane tesselations are discussed in a recent book (Cambridge University Press, 1997) by Greg Frederickson from Purdue University. Although Napoleon's theorem is not mentioned in the book, you can get a sense how this approach applies to other problems. For another example you are referred to the proof of

17. Advanced A derived quadrilateral. napoleon's theorem. Circle through three points. 4 specialpoints of a triangle. The shape of birds' eggs. Saltire Software Feedback. http://www.saltire.com/applets/advanced_geometry/advanced_geometry.htm

Up Basic Triangles [ Advanced ] Mechanisms Advanced geometric configurations Some interesting, somewhat more difficult geometric configurations are presented here. A cubic spline construction Intersections of exterior common tangents to 3 circles Intersections of interior common tangents of 3 circles An isosceles triangle theorem ... The shape of birds' eggs Send mail to webmaster@saltire.com with questions or comments about this web site. Last modified: April 23, 2001

18. Cut The Knot! meme via the Web site of the Mathematical Association of America, Napoleon'sTheorem, Douglass' Theorem. Cut The Knot! napoleon's theorem. March 1999. http://www.maa.org/editorial/knot/Napolegon.html

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).

19. Napoleon's Propeller (2), Of course, it could be used to derive napoleon's theorem. napoleon's theoremis equivalent to the Asymmetric Propeller's theorem! How small is the world! http://www.maa.org/editorial/knot/NapoleonPropeller.html

Cut The Knot! An interactive column using Java applets by Alex Bogomolny Napoleon's Propeller July 2002 As the two most recent columns have been devoted to synthetic proofs of a curious result , I've been looking for an example or two of an illuminating analytic proof. I found quite a few. Two such appear below. In the process I made a small, but surprising, discovery that is reflected in the title of the present column. The three altitudes of a triangle meet at a point known as the orthocenter of the triangle. There are many proofs of that result. Here's one that uses complex numbers. Given ABC, we may assume its vertices lie on a circle centered at the origin of a Cartesian coordinate system. Let's think of points in the plane as complex numbers. Define H = A + B + C, a simple symmetric function of all the vertices. In fact, H is the common point of the three altitudes of the triangle. Indeed, for AH and BC to be orthogonal, the ratio (H - A)/(B - C) must be purely imaginary. But (H - A)/(B - C) + B C - BC = (B C - BC where denotes the conjugate operator. If X denotes the latter expression

20. Mudd Math Fun Facts: Napoleon's Theorem 19992003 Francis Edward Su. From the Fun Fact files, here is aFun 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 887385 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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