21. Napoleon's Theorem Mathematical technology for industry and education. 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 Also see Napoleon's Theorem in our Geometry Formula Atlas

22. Napoleon’s Theorem Mathematical technology for industry and education. http://www.saltire.com/Geometry_Atlas/GeometryAtlas.php?path=Theorems_&_Interest

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

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

24. Geometry Problems - Antonio Gutierrez - Morley, Clifford, Bevan, Langley, Napole Napoleon s theorem. 17. Nagel Point theorem. See also Nagel Point FlowchartProof, Nagel Point Puzzle. 16. Napoleon s theorem. Proof http://agutie.homestead.com/files/Geoproblem_B.htm

Geometry Theorems and Problems Archimedes' Book of Lemmas Butterfly Theorem . See also: Butterfly Theorem Puzzle Carnot's Theorem Ceva's Theorem Clifford's Circles Chain Theorems ... Langley Problem: 20° Isosceles Triangle A dventitious angles. Menelaus' Theorem Miquel's Pentagram. Proof Miquel's Pentagram. Dynamic Geometry Requires Java applet 1.3 or higher. Requires Java applet 1.3 or higher. Morley's Theorem Napoleon's Theorem Nagel Point Theorem . See also: Nagel Point Flowchart Proof Nagel Point Puzzle Newton's Theorem: Newton's Line Parallelogram with Squares theorem ... Pentagons and Pentagrams . Menelaus and Collinearity Poncelet's Theorem Sangaku Problem Sangaku Problem 2 Sangaku Problem 3 ... Triangle with the bisectors of the exterior angles. Collinearity Triangle with Squares 0 Two squares Triangle with Squares 1 Two squares Triangle with Squares 2 Two squares Triangle with Squares 3 Three squares Triangle with Squares 4 Finsler-Hadwiger Theorem Triangle with Squares 5 Two squares, median and altitude

25. Napoleon I Of France - Wikipedia, The Free Encyclopedia Napoleon s theorem Enlarge. Napoleon s theorem. In Britain he is remembered asa despot. During his lifetime, he was often caricatured as a tyrannical http://en.wikipedia.org/wiki/Napoleon_Bonaparte

Napoleon I of France From Wikipedia, the free encyclopedia. (Redirected from Napoleon Bonaparte "Napoleon" redirects here. For other uses, see Napoleon (disambiguation) Napoleon I of France, by Jacques-Louis David Napol©on Bonaparte 15 August 5 May ) was a general of the French Revolution , and the ruler of France as First Consul Premier Consul ) of the French Republic from 11 November to 18 May , then as Emperor of the French ( Empereur des Fran§ais ) and King of Italy under the name Napoleon I from 18 May to 6 April , and again briefly from 20 March to 22 June Napoleon is considered to have been a military genius, and is known for commanding many successful campaigns, together with some spectacular failures. Over the course of little more than a decade, he acquired control of most or all of the western and central mainland of Europe by conquest or alliance until his defeat at the Battle of the Nations near Leipzig in October , which led to his abdication several months later. He staged a comeback known as the Hundred Days les Cent Jours ), but was defeated decisively at the

26. Napoleon's Theorem Napoleon s theorem, by complex numbers. Napoleon s theorem A proof bytesselation A proof with complex numbers; A second proof with complex numbers http://www.cut-the-knot.com/proofs/napoleon_complex2.shtml

Username: Password: Sites for teachers Sites for parents Awards Interactive Activities ... Sites for parents 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 Search: All Products Apparel Baby Beauty Books DVD Electronics Gourmet Food Personal Care Housewares Magazines Musical Instruments Music Computers Software VHS Computer Games Cell Phones Keywords: G o o g ... e Web CTK Latest on CTK Exchange probability Posted by tennyson 6 messages 03:13 PM, Aug-29-05

27. Napoleon's Theorem Napoleon s theorem. © Copyright 2003, Jim Loy. Apparently Napoleon Bonaparte cameup with this theorem. On any triangle, draw equilateral triangles on each 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

28. PlanetMath: Napoleon's Theorem Napoleon s theorem is owned by drini. owner history (2) This is version2 of Napoleon s theorem, born on 200307-31, modified 2003-08-02. http://planetmath.org/encyclopedia/NapoleonsTheorem.html

(more info) Math for the people, by the people. Encyclopedia Requests Forums Docs ... Random Login create new user name: pass: forget your password? Main Menu sections Encyclop¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Classification talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information News Docs Wiki ChangeLog ... About Napoleon's theorem (Theorem) Theorem If equilateral triangles are erected externally on the three sides of any given triangle , then their centres are the vertices of an equilateral triangle. If we embed the statement in the complex plane , the proof is a mere calculation. In the notation of the figure, we can assume that , and is in the upper half plane . The hypotheses are where , and the conclusion we want is where From ( ) and the relation , we get and so proving ( Remarks: MathPages "Napoleon's theorem" is owned by drini owner history view preamble View style: HTML with images page images TeX source Cross-references: upper half plane complex plane centres triangle ... theorem This is version 2 of Napoleon's theorem , born on 2003-07-31, modified 2003-08-02.

29. Napoleon's Theorem: Information From Answers.com Napoleon s theorem Information From Answers.com. Mentioned In. Napoleon stheorem is mentioned in the following topics http://www.answers.com/topic/napoleon-s-theorem

showHide_TellMeAbout2('false'); Business Entertainment Games Health ... More... On this page: Best of Web Mentioned In Or search: - The Web - Images - News - Blogs - Shopping Napoleon's Theorem Best of the Web Some good "Napoleon's Theorem" pages on the web: Math mathworld.wolfram.com Mentioned In Napoleon's Theorem is mentioned in the following topics: triangle Your Ad Here Jump to: Best of Web Mentioned In Or search: - The Web - Images - News - Blogs - Shopping Send this page Print this page Link to this page Tell me about: Home About Tell a Friend Buzz ... Site Map

30. NapoleonsTheorem Napoleon s theorem (English). Search for Napoleon s theorem in NRICH PLUS maths.org Google. Definition (keystage 2) http://thesaurus.maths.org/mmkb/entry.html?action=entryByConcept&id=2099

31. NapoleonsTheorem Napoleon s theorem (Angielski). szukaj Napoleon s theorem w NRICH PLUS maths.org Google. Definicja wiek 11 lat poziom 1 http://thesaurus.maths.org/mmkb/entry.html?action=entryByConcept&id=2099&expand=

32. Casio ClassPad 300 Explorations -- Napoleon’s Theorem Napoleon’s theorem offers a tour de force for constraint geometry. The theoremstates that for any arbitrary triangle, if you construct an equilateral http://www.classpad.org/explorations/napoleon/napoleon.html

Home ClassPad News Overview Buy a ClassPad Now ... Saltire Family of Websites Napoleon’s Theorem with the Casio ClassPad Napoleon’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 FC (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 let’s join the centers of these circles. Shading the resulting triangle makes it stand out from the cat’s cradle of lines and circles. If you’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

33. Spreadsheets In Education Title, Napoleon s theorem and Beyond. Abstract, The use of Microsoft Excel toexplore old, wellknown geometrical theorems is set in the context of http://www.sie.bond.edu.au/articles.asp?id=7

34. Napoleon's Theorem Napoleon s theorem. Napoleon=proc() local A,B,C. ItIsEquilateral( CET(A,B) ,CET(B,C) , CET(C,A) ). end. Previous Definitions theorems Next. http://www.math.rutgers.edu/~zeilberg/PG/Napoleon.html

Napoleon's Theorem ItIsEquilateral CET (A,B) , CET (B,C) , CET (C,A) ): end: Previous Definitions Theorems Next

35. Introduction To Shalosh B. Ekhad XIV's Geometry Textbook Hence in order to understand the statement of Napoleon s theorem you only needto look up the definitions Ce, Center, CET, Circumcenter, http://www.math.rutgers.edu/~zeilberg/PG/Introduction.html

Introduction to Shalosh B. Ekhad XIV's Geometry Textbook (ca. 2050) Cover Foreword Definitions Theorems September 21, 2050 Dear Children, Do you know that until fifty years ago most of mathematics was done by humans? Even more strangely, they used human language to state and prove mathematical theorems. Even when they started to use computers to prove theorems, they always translated the proof into the imprecise human language, because, ironically, computer proofs were considered of questionable rigor! Only thirty years ago, when more and more mathematics was getting done by computer, people realized how silly it is to go back-and-forth from the precise programming-language to the imprecise humanese. At the historical ICM 2022, the IMS (International Math Standards) were introduced, and Maple was chosen the official language for mathematical communication. They also realized that once a theorem is stated precisely, in Maple, the proof process can be started right away, by running the program-statement of the theorem. All the theorems that were known to our grandparents, and most of what they called conjectures, can now be proved in a few nano-seconds on any PC. As you probably know, computers have since discovered much deeper theorems for which we only have semi-rigorous proofs, because a complete proof would take too long.

36. Cut The Knot! Napoleon s theorem states that the centers of the three outer Napoleon Napoleon s theorem (both for outer and inner constructions) follows when n = 3. 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).

37. Napoleon's Propeller Napoleon s theorem is equivalent to the Asymmetric Propeller s theorem! Now, both the original Asymmetric Propeller and Napoleon s theorem start with http://www.maa.org/editorial/knot/NapoleonPropeller.html

Search MAA Online MAA Home 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

38. NAPOLEON BONAPARTE The famous Napoleon theorem is stated by Coxeter and Greitzer as follows Ifequilateral triangles are erected externally on the sides of any triangle, http://faculty.evansville.edu/ck6/bstud/napoleon.html

Napoleon Bonaparte (1769-1821) Emperor of the French The famous Napoleon Theorem is stated by Coxeter and Greitzer as follows: If equilateral triangles are erected externally on the sides of any triangle, their centers form an equilateral triangle. They continue with a historical anecdote: It is known that Napoleon Bonaparte was a bit of a mathematician with a great interest in geometry. In fact, there is a story that, before he made himself ruler of the French, he engaged in a discussion with the great mathematicians Lagrange and Laplace until the latter told him, severely, "The last thing we want from you, general, is a lesson in geometry." Laplace became his chief military engineer. Coxeter and Greitzer then remark that Napoleon probably did not know enough geometry to discover Napoleon's Theorem, just as he probably did not know enough English to compose the palindrome often attributed to him: Able was I ere I saw Elba. The portrait is by Anne-Louis Girodet-Trioson (1767-1824). I thank the MAA for permission to quote from H. S. M. Coxeter and S. L. Greitzer

39. Principles Of Nature: Napoleon's Theorem Napoleon s theorem, also holds true for centripetallyerected triangles If we combine the Napoleon theorem with the new relative unit of area (the etu), http://www.principlesofnature.net/references/Napoleons_theorem_in_geometry.htm

Principles of Nature: towards a new visual language Appendix 2 Excerpt from (W Roberts, 2003) reformatted for web presentation. Napoleon's Theorem . Apparently Napoleon Bonaparte had a strong interest in geometry and this theory has been attributed to him. If equilateral triangles are erected externally on the sides of any triangle, their centers form an equilateral triangle. Similarly, if equilateral triangles are erected internally (or centripetally) around the sides of any triangle as in Figure A-2.2, their centres also form an equilateral triangle known as the inner Napoleon triangle We do not prove it here, but the difference in area between the outer and inner Napoleon triangles around any triangle is equal to the area of the original triangle in question If we combine the Napoleon Theorem with the new relative unit of area the etu ), we see a most interesting relation, p representing the difference in areas of the outer and inner napoleon triangles expressed in etu. An etu is the area of an equilateral triangle of unitary side length back to top Introduction ... is flawed as a method of proof

40. The Eutrigon Theorem For A New Class Of Euclidean Triangle *The closest theorem I have found to this is the Napoleon theorem See here forPDF 173kb of geometric form of Eutrigon theorem from the book (W. Roberts, http://www.principlesofnature.net/number_geometry_connections/new_angles_on_tria

Principles of Nature: towards a new visual language Site blog The Eutrigon Theorem - a new* twin to the theorem of Pythagoras Introduction Scale structure theory has implications not only to the units and measurement of areas but also to triangle classification as we have seen. Particularly pertinent to the theorem presented here on this page is the implementation of the equitriangular unit of area (etu) as we defined earlier. We will also later use our new knowledge of the area of a eutrigon (in terms of these relative units of area, etu ) in the algebraic interpretation of the geometric construction (figure resonant scale structure visually proves the theorem the Eutrigon Theorem The area of any eutrigon (shaded in figure ) is equal to the sum of the areas of the equilateral triangles on its legs a and b , minus the area of the equilateral triangle on its hypotenuse, c Figure Geometric form of Eutrigon Theorem The area of any eutrigon is equal to the sum of the areas of the equilateral triangles on its legs a and b , minus the area of the equilateral triangle on its hypotenuse, c (W. Roberts, 2003, p.122)

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