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21. Mudd Math Fun Facts: All Fun Facts
Method; Multiplication by 11; Music Math Harmony; Napoleon s theorem; Nine Points! Odd Numbers in Pascal s Triangle; One Equals Zero!
http://www.math.hmc.edu/funfacts/allfacts.shtml
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22. Affine
Note that when the number of sides is equal to three this is Napoleon s theorem. Sorry, this page requires a Javacompatible web browser. affine.
http://www.angelfire.com/mn3/anisohedral/affine.html
var cm_role = "live" var cm_host = "angelfire.lycos.com" var cm_taxid = "/memberembedded"
This page uses JavaSketchpad , a World-Wide-Web component of The Geometer's Sketchpad Move the point M or either of the sliders at the top to vary the parameters. Note that the inside figure is a "flattened" or affine version the original n-gon. We construct triangles that are similar to the triangle at the center of the regular n-gon on the left. The outer vertices of the triangles define another regular n-gon. Note that when the number of sides is equal to three this is Napoleon's theorem.

23. Math Applets
Affine polygons This applet extends Napoleon s theorem from a triangle to a generalized ngon. I have not proved that the theorem implied is true.
http://www.angelfire.com/mn3/anisohedral/applets.html
var cm_role = "live" var cm_host = "angelfire.lycos.com" var cm_taxid = "/memberembedded"
##### Math Applets
Circle applet: This applet will let you change the radius of circles arranged in a lattice shape. The overlapping circles give ideas for tilings. Affine polygons: This applet extends Napoleon's theorem from a triangle to a generalized n-gon. I have not proved that the theorem implied is true. (I have also not implied that the true theorem is proved.) Midpoint Porism: These applets demonstrate a new (to me) porism involving the midpoints of inscribed polygons. I have proved that the porism holds for triangles, and quadrilaterals, but not for higher order polygons. I'd like to hear by email if you know about these in the literature or can prove something new. Enjoy. Stars: This applet draws stars with different numbers of vertices and spacing. Tiling Transformer This applet allows you to modify tilings to create other tilings. Back to anisohedral tilings page. This page designed by John Berglund.
You can reach me by email.

24. PlanetMath: Napoleon's Theorem
Napoleon s theorem, (theorem). For more on the story, see MathPages. Napoleon s theorem is owned by drini. owner history (1) . (view preamble).
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 EncyclopÂ¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Corrections talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information Docs Classification News Legalese ... TODO List 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 ... equilateral triangles This is version 2 of Napoleon's theorem , born on 2003-07-31, modified 2003-08-02.

25. PlanetMath:
Napier s constant (=natural log base) owned by akrowne. Napoleon s theorem owned by drini. \$n\$ary relation (in relation) owned by yark.
http://planetmath.org/encyclopedia/N/
 (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 EncyclopÂ¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Corrections talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information Docs Classification News Legalese ... TODO List PlanetMath Encyclopedia (browse by subject) - 1 entry - 2 entries - 1 entry - 1 entry - 2 entries - 1 entry - 1 entry - 1 entry A - 336 entries B - 237 entries C - 804 entries D - 316 entries E - 298 entries F - 318 entries G - 216 entries H - 262 entries I - 278 entries J - 58 entries K - 98 entries L - 357 entries M - 302 entries N - 243 entries natural number ) owned by djao owned by Henry owned by Henry nabla owned by drini (in nabla ) owned by drini Nagao's theorem owned by bwebste Nagell-Lutz theorem owned by alozano Nagell-Lutz theorem (in Nagell-Lutz theorem ) owned by alozano Nakayama's lemma owned by n3o proof of Nakayama's lemma owned by mclase proof of Nakayama's lemma owned by nerdy2 Napier's constant natural log base ) owned by akrowne Napoleon's theorem owned by drini -ary relation (in relation ) owned by yark Nash equilibrium owned by Henry example of Nash equilibrium owned by Henry natural deduction owned by Henry natural density asymptotic density ) owned by mathcam natural embedding dual space ) owned by Daume natural equivalence owned by mathcam natural equivalence (in natural transformation ) owned by mps natural isomorphism natural equivalence

 26. NapoleonsTheorem Napoleon s theorem (English). Search for Napoleon s theorem in NRICH PLUS maths.org Google. Definition level 1. If we takehttp://thesaurus.maths.org/mmkb/entry.html?action=entryByConcept&id=2099

 27. NapoleonsTheorem Size). Napoleon s theorem (English). Search for Napoleon s theorem in NRICH PLUS maths.org Google. Definition level 1. Ifhttp://thesaurus.maths.org/mmkb/entry.html?action=entryByConcept&id=2099&langcod

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

29. Geometry Problems - Antonio Gutierrez - Morley, Clifford, Bevan, Langley, Napole
8. Morley s theorem. 9. Napoleon s theorem. 10. Parallelogram with Squares theorem. 11. 9. Napoleon s theorem. Proof. 10. Parallelogram with Squares theorem.
 Geometry Problems Butterfly Theorem Carnot's Theorem Clifford's Circles Chain Theorems Eyeball Theorem ... Langley Problem: 20Â° Isosceles Triangle A dventitious angles. Morley's Theorem Napoleon's Theorem Parallelogram with Squares theorem Poncelet's Theorem ... Triangle with Squares 4 Finsler-Hadwiger Theorem Triangle with Squares 5 Median and Altitude Triangle with Squares 6 Four squares Triangle Centers Van Aubel's Theorem Quadrilateral with Squares Varignon and Wittenbauer parallelograms Adjacent Angles and Five Bisectors ... Top 1. Butterfly Theorem. Proof Home Top 2. Carnot's Theorem. Proof In any triangle ABC the algebraic sum of the distances from the circumcenter O to the sides , is R+r , the sum of circumradius and the inradius 3. Clifford's Circle Chain Theorems. See more. Home Top 4. Eyeball Theorem. Proof Given two circles A and B, draw the tangents from the center of each circle to the sides of the other. Then the line segments MN and PQ are of equal length.

30. A Generalization Of Napoleon's Theorem
Napoleon s theorem. On each side of a triangle, erect an equilateral triangle, lying exterior to the original triangle. A Generalization of Napoleon s theorem.
http://mathsforeurope.digibel.be/Napoleon2.html
 Napoleon Bonaparte Emperor of the French Katrien Vandermeulen Annelies Deceuninck Karen Van Hoey Napoleon, a mathematician ? Napoleon is best known as a military genius and Emperor of France but he was also an outstanding mathematics student. He was born on the island of Corsica and died in exile on the island of Saint-HÃ©lÃ¨ne after being defeated in Waterloo. He attended school at Brienne in France where he was the top maths student. He took algebra, trigonometry and conics but his favorite was geometry. After graduation from Brienne, he was interviewed by Pierre Simon Laplace (1749-1827) for a position in the Paris Military School and was admitted by virtue of his mathematics ability. He completed the curriculum, which took others two or three years, in a single year and subsequently he was appointed to the maths section of the French National Institute. During the Egyptian military campaign of 1798-1799, Napoleon was accompanied by a group of educators, civil engineers, chemists, mineralogists and mathematicians, including Gaspard Monge (1746-1818) and Joseph Fourier (1768-1830). On his return from Egypt, Napoleon led a successful coup d'Ã©tat and became head of France. As emperor, he instituted a number of juridical, economical and educational reforms and placed men such as

31. Napoleon's Theorem
Napoleon s theorem. by Kala Fischbein and Tammy Brooks. Given any triangle, we can construct equilateral triangles on the sides of each leg.
http://jwilson.coe.uga.edu/emt725/Class/Brooks/Napoleon/napoleon.html
##### by Kala Fischbein and Tammy Brooks
Given any triangle, we can construct equilateral triangles on the sides of each leg. In these equilateral triangles, we can then find the centers: centroid, orthocenter, circumcenter, and incenter. Each of these centers is in the same location because the triangles are equilateral. After the centers have been located, we connect them thus forming Napoleon's Triangle.
Construction of Napoleon's Triangle.
##### Napoleon's Triangle is the grey triangle. Notice that it is also an equilateral triangle. Napoleon's Triangle appears to be congruent to the original equilateral triangle ABC by the SSS postulate. Now, let's see what happens when our original triangle is a right triangle.
The green triangle, which is Napoleon's Triangle, is still an equilateral triangle.
Let us explore when the original triangle is an isosceles triangle. Notice that the yellow triangle represents Napoleon's Triangle which remains an equilateral triangle.
After exploring all of the special types of triangles, what happens when we have a scalene or general triangle? Again, notice that Napoleon's Triangle, the red triangle, is still equilateral no matter which type of triangle is used for the original triangle.

32. Essay 3 Napoleon's Theorem
Napoleon s theorem goes as follows Given any arbitrary triangle ABC, construct equilateral triangles on the exterior sides of triangle ABC.
http://jwilson.coe.uga.edu/EMT668/EMAT6680.F99/Martin/essays/essay3.html
 Essay 3: Napoleon's Triangle by Anita Hoskins and Crystal Martin Napoleon's Theorem goes as follows: Given any arbitrary triangle ABC, construct equilateral triangles on the exterior sides of triangle ABC. The segments connecting the centroids of the equilateral triangles form an equilateral triangle. Let's explore this theorem. Construct an equilateral triangle and see if Napoleon's triangle is equilateral. We can see from this construction, that when given an equilateral triangle, the resulting Napoleon triangle is also equilateral. Construct an isosceles triangle. Again, we see that with an isosceles triangle, Napoleon's triangle is still equilateral. Now, let's construct a right triangle. Still, even with a right triangle, Napoelon's triangle is equilateral. Now, we will prove that for any given triangle ABC, Napoleon's triangle is equilateral. We will use the following diagram: A represents vertex A and it's corresponding angle. a denotes the length of BC, c denotes the length of AB, and b denotes the length of AC. G, I, and H are the centroids of the equilateral triangles. x is the length of segment AG and y is the length of segment AI.

33. Napoleon's Propeller
(2), Of course, it could be used to derive Napoleon s theorem. Napoleon s theorem is equivalent to the Asymmetric Propeller s theorem! How small is the world!
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

34. Napoleon's Theorem And The Fermat Point
Napoleon s theorem and the Fermat Point. This respectively. Napoleon s theorem. For any triangle ABC, the triangle XYZ is an equilateral triangle.
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'.Â  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: Â  The rotation Y maps A to C.Â  The rotation X maps C to B.Â  So if we define S = X Y , then S(A) = X (Y (A)) = X (C) = B.Â  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.Â  Thus YXD is an equilateral triangle. But also Z (A) = B, since Z

35. Investigation: Napoleon's Theorem
INTD 302 . Laboratory Activity Napoleon s theorem. French observed. Make a conjecture What is Napoleon s theorem? Present Your Findings I.
http://www.geneseo.edu/~wallace/intd302/Current.htm
 INTD 302 - Laboratory Activity: Napoleon's Theorem French emperor Napoleon Bonaparte fancied himself as something of an amateur geometer and liked to hang out with mathematicians. The theorem you'll investigate in this activity is attributed to him. Procedure: Your first task is to establish a New Tool that will allow you to construct an equilateral triangle and its centroid on any given line segment. To do this, follow these steps: i. Use the segment tool to construct a line segment If necessary, rename the endpoints so that they are labeled A and B ii. Use the pointer (i.e. the Selection Arrow Tool ) and choose points A and B in that order. Then from the Construct menu, choose the Circle by Center and Point option. iii. Repeat step 2 but this time select B first and point A second. iv. The two circles constructed in ii. and iii. will intersect at two points. Use the pointer to construct the point of intersection that is above Label this point C v. Construct the segments that connect points A, B, and C. is an equilateral triangle.

 36. Spreadsheets In Education Refereed Articles. Author(s) John Baker. Natural Maths. Title Napoleon s theorem and Beyond. Abstract The use of Microsoft Excelhttp://www.sie.bond.edu.au/articles.asp?id=7

37. BeitrÃ¤ge Zur Algebra Und Geometrie / Contributions To Algebra And Geometry, Vol
433444 (2002) Napoleon s theorem and Generalizations Through Linear Maps. Hellmuth Stachel. Keywords Napoleon s theorem, triangle, regular hexagon, linear map.
http://www.emis.de/journals/BAG/vol.43/no.2/11.html
##### 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:
Previous Article Next Article Contents of this Number ELibM for the EMIS Electronic Edition

38. Journal For Geometry And Graphics, Vol. 5, No. 1, Pp. 13-22, 2001
Journal for Geometry and Graphics, Vol. 5, No. 1, pp. 1322 (2001) The Harmonic Analysis of Polygons and Napoleon s theorem. Pavel Pech.
http://www.emis.de/journals/JGG/5.1/2.html
Journal for Geometry and Graphics, Vol. 5, No. 1, pp. 13-22 (2001)
##### Pavel Pech
Pedagogical Faculty, University of South Bohemia
Jeronymova 10, 371 15 Ceske Budejovice, Czech Republic email: habdelmoez@yahoo.com
email: pech@pf.jcu.cz Abstract: Plane closed polygons are harmonically analysed, i.e., they are expressed in the form of the sum of fundamental \$k-\$regular polygons. From this point of view Napoleon's theorem and its generalization, the so-called theorem of Petr, are studied. By means of Petr's theorem the fundamental polygons of an arbitrary polygon have been found geometrically. Keywords: finite Fourier series, polyon transformation Classification (MSC2000): Full text of the article will be available in end of 2002. Previous Article Next Article Contents of this Number ELibM for the EMIS Electronic Edition

39. Napoleon's Theorem
Napoleon s theorem. Napoleon=proc() local A,B,C ItIsEquilateral( CET(A,B) , CET(B,C) , CET(C,A) )
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

40. Introduction To Shalosh B. Ekhad XIV's Geometry Textbook
Hence in order to understand the statement of Napoleon s theorem you only need to look up the definitions Ce, Center, CET, Circumcenter, DeSq and
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.

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