Napoleon S Theorem - The Geometer S Sketchpad Resource Center 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.dynamicgeometry.com/JavaSketchpad/Gallery/Geometry/Napoleons_Theorem.
Extractions: Home Getting Started General Resources Instructor Resources ... Gallery Napoleon's Theorem Considering Sketchpad Learning Sketchpad ... 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 Gallery Legal Notices
Napoleon S Theorem - Wikipedia, The Free Encyclopedia In mathematics, Napoleon s theorem states that if equilateral triangles are constructed on the sides of any triangle, either all outward, or all inward, http://en.wikipedia.org/wiki/Napoleon's_theorem
Extractions: 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
PlanetMath: Napoleon's Theorem Theorem 1 If equilateral triangles are erected externally on the three sides of any given triangle, then their centres are the vertices of an equilateral http://planetmath.org/encyclopedia/NapoleonsTheorem.html
Extractions: 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
Extractions: From Wikipedia, the free encyclopedia Jump to: navigation search No file by this name exists. No pages on the English Wikipedia link to this file. (Pages on other projects are not counted.) Retrieved from " http://en.wikipedia.org/wiki/File:Napoleon%26 Personal tools Namespaces Variants Views Actions Search Navigation Interaction Toolbox What links here Upload file Special pages About Wikipedia
Napoleon's Theorem: Equilateral Triangles Adjoining Sides Of A Triangle Napoleon's Theorem if equilateral triangles are constructed on the sides of any triangle (all outward or all inward), the centers of those equilateral triangles themselves http://2000clicks.com/MathHelp/GeometryTriangleEquilateralNapoleonsTheorem.aspx
Extractions: Twitter Math Help ... Equilateral Triangle Napoleon's Theorem Napoleon's theorem: if equilateral triangles are constructed on the sides of any triangle (all outward or all inward), the centers of those equilateral triangles themselves form an equilateral triangle. Proof: Beginning with an arbitrary triangle ABC, construct equilateral triangles a, b, c as shown. Construct a fourth equilateral triangle, d, as shown, with sides equal to those of triangle b. Triangle d is oriented the same as triangle b. (Proof: trace a path along the four line segments consisting of sides of triangles d, a, a, and b. Treating left turns as negative angles and right turns as positive angles, the turns, in sequence, are B-180, 120, and 60-B, which add up to zero.) If both the triangle ABC and the equilateral triangle b are rotated counterclockwise about c through an angle of 120 degrees, side AC will line up with a different side of equilateral triangle c, and so the image of triangle b is concident with triangle d, proving line segments dc and cb are equal in length. Similarly, rotating triangle ABC and equilateral triangle b clockwise through an angle of 120 degrees shows line segments ab and ad are equal in length. By constructing the dark blue line ac, we see it bisects angles a and c, which are each 120 degrees, so angles cab, abc, and bca are all 60 degrees, proving the theorem. Internet references
A Generalization Of Napoleon's Theorem - GeoGebra Динамична карти For an arbitrary triangle ABC and arbitrary triangles CBA 1 ~ B 1 AC ~ AC 1 B, external to ABC and similar to each other, any three corresponding points of these triangles make a http://www.math.bas.bg/~bantchev/ggb/gnap.html
Extractions: For an arbitrary triangle ABC and arbitrary triangles CBA ~ B AC ~ AC B, external to ABC and similar to each other, any three corresponding points of these triangles make a triangle which is similar to them. Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser ( Click here to install Java now GeoGebra
A Second Proof With Complex Numbers Napoleon s Theorem, by complex numbers using a criterion for a 60 degrees http://www.cut-the-knot.org/proofs/napoleon_complex2.shtml
Mathematics Education 3105 MAED 3105 091 FALL 1996. Dr. David C. Royster droyster@uncc.edu. Napoleon's Theorem http://education.uncc.edu/droyster/courses/fall96/maed3105/gsproject3.html
Extractions: droyster@uncc.edu Construct equilateral triangles on the sides of an arbitrary triangle. Find the centroid of each of these triangles and connect them. This triangle is also an equilateral triangle. If you construct the segments connecting each vertex of the original triangle with the most remote vertex of the equilateral triangle on the opposite side, you will find that they are concurrent - all intersecting in one point. Does this point have any special properties? Construct the inner Napoleon triangle by reflecting each centroid across its corresponding edge in the original triangle. Note that the sum of the areas of the inner triangle and the original triangle give you the area of the outer Napoleon triangle. Mark one of the centroids as a center and rotate the entire figure 120 about this center. You will eventually be able to fill the plane, because you can fill the plane with the base Napoleon equilateral triangle. The sum of the measures of the angles about C is 360, because you have the three small equilateral triangles, each with 60 measures, together with the three angles from the original triangle, adding another 180. Thus the sum is 360.
A Geometric Proof Of Napoleon’s Theorem A Geometric Proof Of Napoleon’s Theorem . Chrissy Folsom. June 8, 2000. Math 495b http://www.math.ucla.edu/~tat/MicroTeach/napoleon.ppt
Napoleon's Theorem, A Generalization Generalization of Napoleon s Theorem On each side of a given (arbitrary http://www.cut-the-knot.org/Generalization/napoleon.shtml
No. 2550: Napoleons Theorem Could Napoleon have proved Napoleon’s Theorem? Today, did he, or didn’t he? The University of Houston’s College of Engineering presents this series about the machines that http://www.uh.edu/engines/epi2550.htm
Extractions: NAPOLEONS THEOREM by Andrew Boyd Click here for audio of Episode 2550 machines that make our civilization run, and the people whose ingenuity created them. N A lot of fuel for the debate was provided by an off-hand comment of two twentieth century mathematicians. One was the famed geometer Donald Coxeter. In the textbook, Geometry Revisited , he and co-author Samuel Greitzer write, the possibility of Napoleon knowing enough geometry [to prove the result] is as questionable as the possibility that he knew enough English to compose the famous palindrome ABLE WAS I ERE I SAW ELBA. Coxeter and Greitzer didnt just challenge the claim that Napoleon was first. They didnt think he was capable of solving it at all. Its quite an insult coming from the English born Coxeter. certainly could have. Napoleons Theorem requires logical thinking but little more. Most proofs of it are understandable by a good high school student. What led Coxeter and Greitzer to disparage Napoleons abilities isnt clear, though it may have been just a poor effort at humor. The palindrome ABLE WAS I ERE I SAW ELBA is fabled to have been uttered by Napoleon, who at one time was exiled to Elba. But was Napoleon the first to discover the result that bears his name? Thats not at all clear. Napoleon might never have actually discovered the steps in the proof. The result simply could have been named in his honor by someone seeking to curry favor.
Annie's Sketchpad Materials: Napoleon's Theorem 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 http://mathforum.org/~annie/gsp.handouts/napoleon/
Extractions: The Math Forum Annie's Sketchpad Activites Printable Version (no Java) 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.
Spreadsheets In Education (eJSiE) Website which focuses on the use of spreadsheets for the purposes of education, at both a school and university level. http://epublications.bond.edu.au/ejsie/
Extractions: Skip to main content My Account Help About ... EJSIE SIE is an electronic journal devoted to the publication of quality refereed articles concerned with studies of the role that spreadsheets can play in education. Our aim is to provide a focus for advances in our understanding of the role that spreadsheets can play in constructivist educational contexts. Spreadsheets in Education (eJSiE) is a free facility for authors to publish suitable, peer reviewed articles and for anyone to view and download articles. Please feel free to browse the site and to contact us if you have constructive feedback regarding eJSiE. ISSN 1448-6156 PDF Spreadsheet Implementations for Solving Boundary-Value Problems in Electromagnetics
Napoleon's Theorem Napoleon's Theorem. First try and explain why (prove) it is true yourself. But if you get stuck, have a look at my book, which contains a discovery of the result and a guided http://math.kennesaw.edu/~mdevilli/napole1.html
Extractions: 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 First try and explain why (prove) it is true yourself. But if you get stuck, have a look at my book, which contains a discovery of the result and a guided proof (as well as in the Teacher Notes, proofs of the generalizations below) Rethinking Proof with Sketchpad . Or alternatively, consult my book available in printed form or PDF download at Some Adventures in Euclidean Geometry Then explore the generalizations and variations below, and also try to explain (prove) why they are true. Generalizations of Napoleon's Theorem Related Variations of Napoleon's Theorem Converses of Napoleon's Theorem
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 http://www.maa.org/editorial/knot/Napolegon.html
Extractions: by Alex Bogomolny 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).
Napoleon's Theorem A thread from the Geometry Forum newsgroup archive Napoleon's Theorem Back to geometry.precollege Search topics All Newsgroups. Napoleon's Theorem by Keith Grove on 11/02/94. http://www.mathforum.com/~sarah/HTMLthreads/articletocs/napoleons.theorem.tufte.
A Generalization Of Napoleon's Theorem The latter illustrates the most general reformulation of the Napoleon s Theorem. The three similar triangles may be of various shapes and, in addition, http://mathsforeurope.digibel.be/Napoleon2.html
Extractions: He was born on the island of Corsica and died in exile on the island of Saint-Hlne 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'tat and became head of France. As emperor, he instituted a number of juridical, economical and educational reforms and placed men such as
Nrich.maths.org :: Mathematics Enrichment :: Napoleon's Theorem Triangle ABC has equilateral triangles drawn on its edges. Points P, Q and R are the centres of the equilateral triangles. What can you prove about the triangle PQR? http://nrich.maths.org/public/viewer.php?obj_id=1944
