41. Napoleon's Theorem And The Fermat Point Napoleon s theorem and the Fermat Point. This page has a proof of Napoleon stheorem and also proofs of the main properties of the special ines and circles 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'.  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:   The rotation Y maps A to C.  The rotation X maps C to B.  So if we define S = X Y , then S(A) = X (Y (A)) = X (C) = B.  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.  Thus YXD is an equilateral triangle. But also Z (A) = B, since Z

42. The Geometer's Sketchpad® - JavaSketchpad: Napoleon's Theorem Napoleon s theorem. This JavaSketch is based on one of the activities in the Students explore Napoleon s theorem, which states that the triangle NPQ is http://www.keypress.com/sketchpad/javasketchpad/gallery/pages/napoleons_theorem.

Getting Started Product Information How to Order Curriculum Modules ... gallery napoleons theorem Napoleon's Theorem This JavaSketch is based on one of the activities in the Sketchpad curriculum module Exploring Geometry . The first portion of the activity has students construct the figure below: an arbitrary triangle, with equilateral triangles on each side, and segments connecting the centers of the three equilateral triangles. The new segments form a third triangle, here triangle NPQ, which is the outer Napoleon triangle of triangle ABC. Drag the vertices of the original triangle ABC and observe the triangle formed by the centers of the equilateral triangles. Students explore Napoleon's Theorem, which states that the triangle NPQ is equilateral. Sorry, this page requires a Java-compatible web browser. Return to the JavaSketchpad Gallery Portions of the work on JavaSketchpad

43. Mudd Math Fun Facts: Napoleon's Theorem The Math Behind the Fact This theorem is credited to Napoleon, who was fond ofmathematics, though many doubt that he knew enough math to discover it! 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 181937315.184686 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...

44. Napoleon's Theorem Napoleon s theorem. by Kala Fischbein and Tammy Brooks Napoleon s Triangleappears to be congruent to the original equilateral triangle ABC by the SSS http://jwilson.coe.uga.edu/emt725/Class/Brooks/Napoleon/napoleon.html

Napoleon's Theorem 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.

45. Napoleon's Theorem Please Help,. david. Hi david,. The result is called Napoleon s theorem. There aredozens of elementary proofs; these can be found in Geometry books that http://mathcentral.uregina.ca/QQ/database/QQ.09.03/david5.html

Quandaries and Queries David Calculus, 12th grade HOw do i prove this : FOr any triangle, if you make 3 equillateral triangles using the sides of the the original triangle, the central points of the 3 tringles another triangle that is equillateral. Please Help, david Hi david, The result is called Napoleon's theorem. There are dozens of elementary proofs; these can be found in Geometry books that cover geometry beyond the basic theorems of Euclid. For example Coxeter's Geometry Revisited, or his Introduction to Geometry. You can also look on the Cut the Knot Web Site. Chris and Penny Go to Math Central

46. Napthm Napoleon s theorem is the name popularly given to a theorem which states that if The early history of Napoleon s theorem and the Fermat points F, http://www.pballew.net/napthm.html

Napoleon's Thm and the Napoleon Points Napoleon's Theorem is the name popularly given to a theorem which states that if equilateral triangles are constructed on the three legs of any triangle, the centers of the three new triangles will also form an equilateral triangle. In the figure the original triangle is labeled A, B, C, and the centers of the three equilateral triangles are A', B', C'. If the segments from A to A', B to B', and C to C' are drawn they always intersect in a single point, called the First Napoleon Point. If the three equilateral triangles are drawn interior to the original triangle, the centers will still form an equilateral triangle, but the segments connecting the centers with the opposite vertices of the original triangle meet in a (usually) different point, called the 2nd Napoleon Point. Although it is known that Napoleon had a keen interest in geometry, math historians seem unable to find evidence he really discovered the theorem. Here is a letter on the subject from Antreas P. Hatzipolakis, a real living Greek mathematician, to the Geometry Forum. The early history of Napoleon's theorem and the Fermat points F, F' (which are also called isogonic centers of ABC) is summarized in Mackey [21], who traces the fact that LMN and L'M'N' are equilateral to 1825 to one Dr. W. Rutherford [27] and remarks that the result is probably older.

47. A Generalization Of Napoleon's Theorem The famous Napoleon theorem is stated by Coxeter and Greitzer as follows This sounds even more surprising than the Napoleon s theorem itself. 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-Hélène 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'état and became head of France. As emperor, he instituted a number of juridical, economical and educational reforms and placed men such as

48. CMB - An Analogue Of Napoleon's Theorem In The Hyperbolic Plane Previous Abstract Previous Page, An Analogue of Napoleon s theorem in theHyperbolic Plane, Next Page Next Abstract http://www.journals.cms.math.ca/cgi-bin/vault/view/mckay7737

CMB (2001) Vol 44 No 3 / pp. 292-297 An Analogue of Napoleon's Theorem in the Hyperbolic Plane Angela McKay Abstract TeX format There is a theorem, usually attributed to Napoleon, which states that if one takes any triangle in the Euclidean Plane, constructs equilateral triangles on each of its sides, and connects the midpoints of the three equilateral triangles, one will obtain an equilateral triangle. We consider an analogue of this problem in the hyperbolic plane. For download Keywords none Language English Category Primary: 37D40 Secondary: none

49. CMB - An Analogue Of Napoleon's Theorem In The Hyperbolic Plane Résumé précédent Page précédente, An Analogue of Napoleon s theorem in theHyperbolic Plane, Page suivante Résumé suivant http://www.journals.cms.math.ca/cgi-bin/vault/view/mckay7737?lang=fr

50. Beiträge Zur Algebra Und Geometrie / Contributions To Algebra And Geometry, Vol 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

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

51. Journal For Geometry And Graphics, Vol. 5, No. 1, Pp. 13-22, 2001 The Harmonic Analysis of Polygons and Napoleon s theorem From this point ofview Napoleon s theorem and its generalization, the socalled theorem of http://www.emis.de/journals/JGG/5.1/2.html

Journal for Geometry and Graphics, Vol. 5, No. 1, pp. 13-22 (2001) The Harmonic Analysis of Polygons and Napoleon's Theorem 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

52. Affine The outer vertices of the triangles define another regular ngon. Note that whenthe number of sides is equal to three this is Napoleon s theorem. http://www.angelfire.com/mn3/anisohedral/affine.html

setAdGroup('67.18.104.18'); var cm_role = "live" var cm_host = "angelfire.lycos.com" var cm_taxid = "/memberembedded" Search: Lycos Angelfire Free Games Share This Page Report Abuse Edit your Site ... Next 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. Sorry, this page requires a Java-compatible web browser. affine Return to applets. You can reach me by email.

53. Napoleon S Theorem Napoleon s theorem Draw three equilateral triangles using each side of the Napoleon s theorem states that the three centroids form the vertices of an http://www.math.psu.edu/dlittle/java/Geometry/Euclidean/napolean.html

54. DC MetaData For: Napoleon's Theorem With Weights In N-Space Abstract The famous theorem of Napoleon was recently extended to higher dimensions.With the help of weighted vertices of an nsimplex T in http://www.math.uni-magdeburg.de/preprints/shadows/98-20report.html

Napoleon's Theorem with Weights in n-Space by Preprint series: 98-20, Preprints MSC 51N10 Affine analytic geometry 51N20 Euclidean analytic geometry Abstract The famous theorem of Napoleon was recently extended to higher dimensions. With the help of weighted vertices of an n-simplex T in E n , n >= 2, we present a weighted version of this generalized theorem, leading to a natural configuration of (n-1)-speres corresponding with T by an almost arbitrarily chosen point. Besides the Euclidean point of view, also affine aspects of the theorem become clear, and in addition a critical discussion on the role of the Fermat-Tooicelli point in this framework is given. Keywords: Napoleon's Theorem, Torricelli's configuration Upload: Update: The author(s) agree, that this abstract may be stored as full text and distributed as such by abstracting services.

55. DC MetaData For: Zur Übertragung Des Satzes Von Napoleon Auf N-Simplexe Abstract We give changes of the wellknown ``Napoleon s theorem in such amanner that certain n-simplices take the place of the considered trian- http://www.math.uni-magdeburg.de/preprints/shadows/97-09report.html

by Preprint series: 97-09, Preprints MSC 51N20 Euclidean analytic geometry 51M05 Euclidean geometries (general) and generalizations Abstract We give changes of the well-known ``Napoleon's theorem'' in such a manner that certain n-simplices take the place of the considered trian- gle. For n>= 3 the given simplex are assigned two different ``Napoleon- simplices'' with remarkable properties. Upload: Update: The author(s) agree, that this abstract may be stored as full text and distributed as such by abstracting services.

56. Teachers Notes-Bernhard Neumann suitable for a popular talk – I found that somebody had called it Napoleon stheorem. Keywords. equilateral triangles geometry mathematics theorem http://www.science.org.au/scientists/notesbn.htm

Australian Academy of Science Science education Interviews with Australian scientists Teachers notes Teachers notes to accompany an interview with mathematician, Professor Bernhard Neumann (1909-2002) You can either print out these notes or use them online. If using them online, use the links below to take you to the relevant sections. Introduction Summary of career Extract from interview and focus questions Activities ... Keywords Professor Bernhard Neumann Introduction Professor Bernhard Neumann was interviewed in 1998 for the Interviews with Australian scientists series. By viewing the interviews in this series, or reading the transcripts and extracts, your students can begin to appreciate Australia's contribution to the growth of scientific knowledge. Information on how to order a copy of the video of Professor Neumann's interview is available at www.science.org.au/scientists , together with a full list of videotaped interviews, transcripts and teachers notes. The following summary of Neumann's career sets the context for the extract chosen for these teachers notes. The extract covers the geometrical ideas behind a talk that Neumann gave to general audiences. Use the focus questions that accompany the extract to promote discussion among your students. Summary of career Bernhard Neumann was born in Germany in 1909. He showed a precocious mathematical talent as a youngster, teaching himself calculus by the age of 12 and in Year 10 inventing three-dimensional analytical geometry. He earned a D Phil from Friedrich-Wilhelms Universität in Berlin in 1932, one of the youngest ever to receive this award in mathematics from Berlin.

57. From Israel@math.ubc.ca (Robert Israel) Subject Re NAPOLEON On The Napoleon s theorem I know about says the following (see Napoleon s theoremstates that the centers of the three outer Napoleon triangles form http://www.math.niu.edu/~rusin/known-math/00_incoming/napoleon

From: israel@math.ubc.ca (Robert Israel) Subject: Re: NAPOLEON on PBS TV; questions Date: 10 Nov 2000 00:17:28 GMT Newsgroups: sci.edu,soc.history,sci.math Summary: [missing] In article , guillaume agostini

58. 2005 Spring Meeting MD-DC-VA MAA Is it possible to prove Napoleon s theorem by tiling your bathroom floor?In this (very interactive!) afternoon session expect to be surprised. http://www.math.vt.edu/org/maa/spr05/

MD-DC-VA Section of the MAA Spring 2005 Meeting April 1-2 at the University of Virginia Charlottesville, VA Program Highlights Program Details Schedule and abstracts now available (3/22/05) Hotel Information *Deadlines approaching Driving Directions, Maps, and Parking Call for Papers Special Event - Undergraduate Conference Please visit this site if you are an undergraduate planning to attend and/or to present at the spring 2005 meeting. Registration Note: The deadline for paper submissions is March 16, 2005. Program Highlights On April 1-2, 2005, the University of Virginia will host the Spring 2005 MD-DC-VA Regional Meeting of the MAA. This spring's conference should be very exciting! Program highlights are now available, but you will want to return to our web site periodically for additional information regarding the conference. On Friday afternoon from 4-6 PM, James Tanton will be conducting a workshop activity titled, "Accessible - but surprisingly sophisticated - Math Activities for Students, Clubs, and Research." Workshop Abstract How can meanderings through the floor plan of a house prove Brouwer's Fixed Point Theorem? How can gnomes playing deadly hat games help with the study of error-correcting codes? How can braiding one's hair lead to concepts in abstract algebra and invariant theory? How can playing with weird "triangular number arithmetic" lead to Euler's Theorem in graph theory, summation formulae, more invariant theory, and new questions in these classic studies? Why does the long division algorithm one learns in fifth grade work and what does it tell us about base one-and-a-half? How is this linked to the famous "3n+1" conjecture? Is it possible to prove Napoleon's theorem by tiling your bathroom floor?

59. The Educational Encyclopedia, Mathematics Carnot s theorem, bounded distance, barycentric coordinates, Pythagoreantheorem, Napoleon s theorem, Ford s touching circles, Euclid s Fifth postulate, http://users.pandora.be/educypedia/education/mathematics.htm

EDUCYPEDIA The educational encyclopedia Home Electronics General Information technology ... Science Science Automotive Biology Biology-anatomy Biology-animals ... Space Social science Atlas - maps Countries Dinosaurs Environment ... Sitemap Mathematics Algebra Complex numbers Formulas Fractals ... Fourier General overview Geometry Integrals and differentials Logarithms and exponentials Matrices and determinants ... Trigonometry General overview Aplusmath this web site is developed to help students improve their math skills interactively, algebra, addition, subtraction, multiplication, division, fractions, geometry for kids Ask Dr. Math Ask Dr. Math a question using the Dr. Math Web form, or browse the archive Calculus tutorial Karl's calculus tutorial, limits, continuity, derivatives, applications of derivatives, exponentials and logarithms, trig functions (sine, cosine, etc.), methods of integration Cut the knot! algebra, geometry, arithmetic, proofs, butterfly theorem, chaos, conic sections, Cantor function, Ceva's theorem, Fermat point, cycloids, Collage Theorem, Carnot's theorem, bounded distance, barycentric coordinates, Pythagorean theorem, Napoleon's theorem, Ford's touching circles, Euclid's Fifth postulate, Non-Euclidean Geometry, Projective Geometry, Moebius Strip, Ptolemy's theorem, Sierpinski gasket, space filling curves, iterated function systems, Heron's formula, Euler's formula, Hausdorff distance, isoperimetric theorem, isoperimetric inequality, Shoemaker's Knife, Van Obel theorem, Apollonius problem, Pythagoras, arbelos, fractals, fractal dimension, chaos, Morley, Napoleon, barycentric, nine point circle, 9-point, 8-point, Miquel's point, shapes of constant width, curves of constant width, Kiepert's, Barbier's

60. Nrich.maths.org::Mathematics Enrichment::Napoleon's Theorem (Dec 98) The Nrich Maths Project Cambridge, England. Mathematics resources for children,parents and teachers to enrich learning. Published on the 1st of each month. http://nrich.maths.org/public/viewer.php?obj_id=1944&part=index&refpage=monthind

Page 3     41-60 of 101    Back | 1  | 2  | 3  | 4  | 5  | 6  | Next 20