81. Meta-Sites The Best Way To Get Good Links Is To Subscribe The These cover theorems about common tangents to circles, Napoleon s theorem on theincenters of equilateral triangles and a number of textbook problems as http://www.mccallie.org/myates/1metasites.htm

Site Map Home Meta-Sites The best way to get good links is to subscribe: The Math Forum Ask Dr. Math Problems of the Week Mathematics Library ... Topics Map for all things Geometry. from the Math Forum E-Books of Mathematics About 20 books, most really advanced and free for download. The Busy Educator's guide to the World Wide Web This is a search engine for mathematical topics. Research and reference information on the Web for students and professionals. Links to journals, reference sites, libraries, research institutes, universities, government sites, databases and other sources. Access to only the highest quality sites to eliminate unwanted hits. The National Curve Bank is a site with information on the major curves in mathematics, links, and the history to go along with each curve. Demos with a positive impact . A collection of visuals for much math taught from the precalculus level and up. DEMOS with POSITIVE IMPACT is a project to connect mathematics professors with effective teaching tools. As instructors we use a variety of techniques to try to get important ideas across to students. This project focuses on demonstrations that use some form of instructional technology. Saltire's Geometry Gallery is a collection of Java applets showing interesting geometry configurations. All applets are

82. Mathematics Education 3105 Napoleon s theorem. Construct equilateral triangles on the sides of an arbitrarytriangle. Find the centroid of each of these triangles and connect them. http://www.math.uncc.edu/~droyster/courses/fall96/maed3105/gsproject3.html

MAED 3105 - 091 FALL 1996 Dr. David C. Royster droyster@math.uncc.edu Napoleon's Theorem 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.

83. ENC Online: Curriculum Resources: Geometry Java Applet Gallery (ENC-032059, Over These explore more complex concepts such as Napoleon s theorem and cubic splineconstructions. The mechanisms category contains four applets illustrating http://www.enc.org/resources/records/0,1240,032059,00.shtm

Skip Navigation You Are Here ENC Home Curriculum Resources Search the Site More Options Don't lose access to ENC's web site! Beginning in August, goENC.com will showcase the best of ENC Online combined with useful new tools to save you time. Take action todaypurchase a school subscription through goENC.com Classroom Calendar Digital Dozen ENC Focus ... Ask ENC Explore online lesson plans, student activities, and teacher learning tools. Search Browse Resource of the Day About Curriculum Resources Read articles about inquiry, equity, and other key topics for educators and parents. Create your learning plan, read the standards, and find tips for getting grants. Geometry Java applet gallery Grades: 9 10 11 12 Post-Sec. URL: http://www.saltire.com/gallery.html ENC#: ENC-032059 Publisher: Saltire Software Date: Similar Records Featured in Digital Dozen Subjects: Mathematics Algebra. Angles. Applied mathematics. Engineering. Geometry. Theorems. Resource Type: Learning tools and models; Math and science background information; Resource materials for math or science; Simulations and demonstrations. Media Type: Internet resource.

84. ENC Online: Curriculum Resources: Geometry From The Land Of The Incas (ENC-02690 Poncelet s theorem Napoleon s theorem Eyeball theorem Steiner s theorem Carnot stheorem Sangaku problem 1 An old Japanese theorem Sangaku problem 2 http://www.enc.org/resources/records/full/0,1240,026900,00.shtm

Skip Navigation You Are Here ENC Home Curriculum Resources Search the Site More Options Don't lose access to ENC's web site! Beginning in August, goENC.com will showcase the best of ENC Online combined with useful new tools to save you time. Take action todaypurchase a school subscription through goENC.com Classroom Calendar Digital Dozen ENC Focus ... Ask ENC Explore online lesson plans, student activities, and teacher learning tools. Search Browse Resource of the Day About Curriculum Resources Read articles about inquiry, equity, and other key topics for educators and parents. Create your learning plan, read the standards, and find tips for getting grants. Geometry from the land of the Incas Grades: URL: http://agutie.homestead.com/files/index.html ENC#: ENC-026900 Publisher: Antonio Gutierrez Date: Similar Records Subjects: Mathematics Geometry. Proofs. Shapes. Theorems. Integrated/interdisciplinary approaches Social sciences. Resource Type: Online textbooks and tutorials; Study guides and tutorials. Media Type: Internet resource. Geographic Focus: Peru.

85. June Lester - Mathematical Presentations A generalization of Napoleon s theorem to ngons. Geometry Workshop, Universityof Hamburg, Germany, July 1994. The Universe is 25 Billion Years Old, http://oldweb.cecm.sfu.ca/~jalester/WebCV/presentations.html

June Lester - Mathematical Presentations Invited talks Conference talks Invited talks Conformal Spaces. Geometry Seminar, Department of Mathematics, University of Toronto, Canada, February 1979 Cone Preserving Mappings. Workshop in Geometry and Algebra, Technical University of Munich, W. Germany, February 1980 Characterizations of Lorentz Transformations. Geometry Colloquium, Mathematics Institute, University of Hannover, W. Germany, June 1980 Characterizations of Spacetime Transformations. Mathematics Colloquium. York University, Toronto, Canada, February 1983 Characterization Theorems on Metric Vector Spaces. Geometry Seminar, Department of Mathematics, University of Toronto, Canada, September 1985 Some Characterizations of Euclidean Motions. Mathematics Colloquium, University of Oldenburg, W. Germany, November 1985 Transformations Preserving Null Line Sections of a Domain. Mathematics Colloquium, University of Duisburg, W. Germany, November 1985 Mappings Preserving Null Line Sections of a Domain.

86. June Lester- Mathematical Publications A generalization of Napoleon s theorem to ngons. CR Math. Soc. Canada 16 (1994)253 - 257. This work has spawned several other projects. http://oldweb.cecm.sfu.ca/~jalester/WebCV/publications.html

June Lester - Mathematical publications Matric vector spaces Geometric characterization problems Spacetime geometry Complex triangle and polygon geometry ... Misscellaneous topics (Note: some of the papers listed below appear in more than one section. There are 39 distinct papers.) Metric vector spaces A metric vector space is a vector space which has a (usually indefinite) scalar product. I first became fascinated with these spaces as a beginning master's student. Geometrically interesting in their own right (as Euclidean n-space or Minkowski spacetime, for example), they are also invaluable as coordinate spaces: it's quite extraordinary just how many classical geometries can be coordinatized by n-tuples subject to some indefinite scalar product. And looking at these geometries through their coordinate spaces often makes obvious the isomorphisms between different models of the same geometry, or even between different geometries: the same coordinate space implies the same or related geometries. On Null-Cone Preserving Mappings.

87. Dynamic Geometry Module: Lesson 5 Napoleon s theorem states that the centers X, Y, and Z form an equilateral triangle.Join these points and measure the sides of the resulting triangle to http://mtl.math.uiuc.edu/modules/dynamic/lessons/lesson5.html

Dynamic Geometry Module Lesson 5: The Center of Things Centers We Know and Love The medians of a triangle all intersect in a point (the centroid of the triangle). The same is true of the angle bisectors (the incenter), the altitudes (the orthocenter) and the perpendicular bisectors of the sides (the circumcenter). These are all examples of important "centers" for a triangle. Use Sketchpad to construct each of these centers. Napoleon's Configuration This is also referred to as the Toricelli Configuration. It consists of an equilateral triangle drawn outward from each side of a triangle. You can see an example in the accompanying sketch for this lesson ( See file ex5_1.gsp In this sketch, P Q , and R are the vertices of the equilateral triangles and X Y , and Z are their centroids (which also happen to be the circumcenters, incenters and orthocenters!). Napoleon's Theorem states that the centers X Y , and Z form an equilateral triangle. Join these points and measure the sides of the resulting triangle to verify this. Be sure to move the vertices around and see that this property holds in all cases. Now delete those segments and join each center to the vertex of the original triangle that is opposite it. That is, draw the lines

88. Writing Assignment #4: Technology Applications including applications of Menelaus and Ceva s theorem, Steiner s theorem,Napoleon s theorem, problems with the ninepoint circle, and re-creations of http://www.math.ilstu.edu/day/courses/old/326/wa04sample.html

Illinois State University Mathematics Department MAT 326: Technology Tools for Secondary School Mathematics Dr. Roger Day ( day@math.ilstu.edu Technology Applications for the Classroom: A Sample Report Roger Day return to Writing Assignment #4 a) McGehee, Jean J. "Interactive Technology and Classic Geometry Problems." Mathematics Teacher 91 (March 1998): 204-208. b.i) dynamic geometry software Geometer's Sketchpad b.iii) The author compares two approaches, a traditional approach and an interactive approach, for using dynamic geometry software to explore the circle of Appolonius. She provides step-by-step instructions on both approaches that a Sketchpad user can follow. She claims that the differences in approaches focus on whether students are provided any opportunity to investigate, conjecture, and otherwise carry out some of the steps that a mathematician may actually undergo in attempting to solve a problem. The traditional approach results in a successful verification of the constant ratio in the circle of Appolonius, but allows little if any investigation by users as well as fostering little connection between the concepts involved and the construction carried out. The interactive approach allows users to first experiment and carry out many examples of the situation in order to discover the resultthe constant ratioas a result of the construction. This seemingly subtle difference, the author contends, spells the difference between students simply following and completing a procedure to focusing on the concept of the locus and using technology for exploration and discovery. The author provides suggestions of other classical geometry constructions that teachers might consider for similar interactive approaches. In so doing, students and teachers will experience more completely the kind of activities engaged in by mathematicians.

89. Federico Poloni's Personal Web Page ps pdf, A proof of Napoleon s theorem (given a triangle, erect equilateraltriangles on its sides then their centroids form another equilateral triangle) http://fph.altervista.org/math/index.shtml

Home Here I have published a few unsorted documents about "olympiad-like" maths. All of these are in Italian. E-mails which point out errors and typoes and/or contain useful comments will be appreciated: my current addresses are fphthirtyseven@ngi.it or f.polonithirtyseven@sns.it . Please remove the number 37 to get the real addresses, I prefer not to publish them on the internet as they are to avoid spam. Anyway, I hope at least one of them is still active when you read this :) . News: 14/7/2004: added cp-pp.p* 13/7/2004: new version of arnesi.p*, with two minor corrections 26/6/2004: added arnesi.p* 21/3/2004: new versions of bt and olisyl Please note that there are a few restrictions upon the usage of the following documents: You may download these papers, print them, read them, use them privately and store them (unmodified) in any media or format. Any other use (e.g. republicating them, printed or over the Internet, modifying or selling them, using them as teaching material in a course) is forbidden unless explicitly authorised by the author (i.e. ME!). Please write me a mail and ask my permission; actually, I am likely to say yes: I am adopting this restriction only because I want to keep track of how my work is used. You are *NOT* free to redistribute these papers on your site, nor to put "deep links" to them directly in your web pages. PLEASE LINK ONLY TO THIS PAGE and not directly to the .ps/.pdf files.

90. Publications On Napoleon s theorem, Ellipse, 1 (Summer 1993) 8. 4. “Algebraic Diet Plan,”Centroid, 22 (Spring 1995) 30. 5. Gaskets Galore, Mathematics and Computer http://www.apsu.edu/HOEHNl/publications.htm

Publications: Mathematics Magazine 1. "Solution to Problem #648," 40 (Sept. 1967) 228. 2. "Averages on the Move," 58 (May 1985) 151-156. Co-authored with Ivan Niven. 3. "Summations Involving Computer-Related Functions,"62 (June 1989) 191-196. Co-authored with Jim Ridenhour. 4. "Solutions of x^n + y^n = z^(n+1) ," 62 (Dec. 1989) 342. 5. "A New Proof of the Double Butterfly Theorem," 63 (Oct. 1990) 256-257. 6. "A Menelaus-Type Theorem for the Pentagram," 66 (Apr. 1993) 121-123. 7. "Mathematical Quickie #866," 70 (June 1997) 224 & 229. 8. "Mathematical Quickie #869," 70 (Oct. 1997) 299 & 307. 9. "Problem Proposal #1574," 72 (June 1999) 236. 10. "Problem Proposal #1635," 74 (Dec. 2001) 403. 11. "Extriangles and Excevians," 74 (Dec. 2001) 384 - 388. 12. "Problem Proposal #1653," 75 (Oct. 2002) 317. College Mathematics Journal 1. "Problem Proposal #114," 9 (Mar. 1978) 95. 2. "A Geometrical Interpretation of the Weighted Mean," 15 (Mar. 1984) 135-139.

91. Euclidean-taxicab-contrast This theorem is sometimes called Napoleon s theorem. Remark Taxicab Geometry isone of an infinite family of geometries discovered by the mathematician http://www.york.cuny.edu/~malk/mycourses/math244/euclidean-taxicab-contrast.html

Contrasting the Euclidean and Taxicab Planes prepared by: Joseph Malkevitch Department of Mathematics and Computing York College (CUNY) Jamaica, New York 11451 email: joeyc@cunyvm.cuny.edu web page: www.york.cuny.edu/~malk Let A = (0, 6), B = (4, 0), and C = (8, 10) 1. Find the Euclidean lengths of the AB, BC, and CA. 2. Find the Taxicab lengths of AB, BC, and CA. 3. Find the midpoint of the each side of the triangle in a. the Euclidean plane b. the Taxicab plane 4. a. In the Euclidean plane, find the equations of the perpendicular bisectors of the sides of the triangle. Find the coordinates of the point X (points) where these lines meet. Can you you find the equation of a circle which passes through all of the vertices of the triangle? If there is such a circle what is its radius? What is the Taxicab distance between X and the vertices of the triangle? b. Determine with respect to Taxicab distance: i. The set of points U which are equidistant from A and B. ii. The set of points V which are equidistant from B and C. iii. The set of point W which are equidistant from points C and A. What can you say about the point (points) that are in common to U, V, and W? Can you find a Taxicab circle which passes through all three of vertices of this triangle? If there is such a circle what are the coordinates of its center and what is the distance of the vertices of the triangle from this center? What is the radius of this circle if it exists? Comment: In the Euclidean plane a circle which passes through the vertices of a triangle is called the

92. College Mathematics Journal, The: Some Recent Discoveries In Elementary Geometry This article also includes a generalization of Ceva s theorem to three dimensions,a new proof of Napoleon s theorem, and results on dissections of planar http://www.findarticles.com/p/articles/mi_qa3773/is_199811/ai_n8811577

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93. Proceedings Of The American Mathematical Society Napoleon in isolation. Author(s) Danny Calegari Journal Proc. Amer. Napoleon s theorem in elementary geometry describes how certain linear operations http://www.ams.org/proc/2001-129-10/S0002-9939-01-05915-9/home.html

Journals Home Search Author Info Subscribe ... Help ISSN 1088-6826 (e) ISSN 0002-9939 (p) Previous issue Table of contents Next issue Articles in press ... All issues Napoleon in isolation Author(s): Danny Calegari Journal: Proc. Amer. Math. Soc. MSC (2000): Primary 57M50, 57M25 Posted: April 2, 2001 Retrieve article in: PDF DVI TeX PostScript ... Additional information Abstract: Napoleon's theorem in elementary geometry describes how certain linear operations on plane polygons of arbitrary shape always produce regular polygons. More generally, certain triangulations of a polygon that tiles admit deformations which keep fixed the symmetry group of the tiling. This gives rise to isolation phenomena in cusped hyperbolic -manifolds, where hyperbolic Dehn surgeries on some collection of cusps leave the geometric structure at some other collection of cusps unchanged. References: D. Calegari, A note on strong geometric isolation in -orbifolds , Bull. Aust. Math. Soc.

94. Fourier Polygons NAPOLEON S theorem; Lighting Design by Accident; The Fourier Transform;Fourier Polygons. Download PDF Download Full Issue (Compressed file of PDFs) http://doi.ieeecomputersociety.org/10.1109/38.736472

Search: Advanced Search Home Digital Library Site Map ... January/February 1999 (Vol. 19, No. 1)   pp. 84-91 Fourier Polygons Andrew Glassner Full Article Text: DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/38.736472 Back to Top Additional Information Citation:   Andrew Glassner. "Fourier Polygons," IEEE Computer Graphics and Applications , vol. 19,  no. 1,  pp. 84-91,  January/February  1999. Abstract Contents: Abstract Citation Free access to Abstracts Selected PDFs Electronic subscribers log in to Access HTML/PDFs of full text articles Download full issue (ZIP of PDFs) Subscription information Get a Web account Usage of this product signifies your acceptance of the

95. Untitled Document 5 John E. Wetzel, Converses of Napoleon s theorem, Am. Math. Monthly 4 (1992),339351. 6 Aureliano Faifofer, Elementi di geometria, Venezia 1911. http://matematica.uni-bocconi.it/betti/note.htm

Bibliografia [1] Frank Morley, On the metric geometry of the plane n-line , Trans. Am. Math. Soc. 1 [2] Harold S.M. Coxeter, Introduction to geometry [3] Cletus O. Oakley e Justine C. Baker, The Morley trisector theorem , Am. Math. Monthly [4] David Gale, Mathematical entertainments, The Mathematical Intelligencer [5] John E. Wetzel, Converses of Napoleon's theorem, Am. Math. Monthly [6] Aureliano Faifofer, Elementi di geometria , Venezia 1911. Great problems of elementary mathematics , Dover 1965.

96. Fermat Point Of A Triangle (Did you expect me to talk about Napoleon s theorem? No, I would rather deal withdiagonals this time. The initial sketch is in fact the same but the story http://www.mtm.ufsc.br/~andsol/geom/fermat/fp-en.html

The Fermat Point of a Triangle If you prefer concise stories, take a look at p.83 of Geometry Revisited by H.S.M.Coxeter and S.L.Greitzer. Using squares you get a pleasing design but this time you would rather do without Pythagorean theorem and the cosine theorem. You want to hit new lands. Why not to try equilateral triangles? They seem to open like petals of a flower. And each of them forms a quadrilateral with the initial triangle. So there comes an idea: how would it look if one drew their diagonals? (Did you expect me to talk about Napoleon's theorem ? No, I would rather deal with diagonals this time. The initial sketch is in fact the same but the story is different.) Are the lines too thick or do the diagonals really meet in one point? Perhaps it takes a bit of time to find out. But it may be amusing to try. However, a suspicion arises. Does the supposed crossing point always lie inside of the triange? Some experiments with pencil and compass show that it might go for a walk. It is clear enough that the sketch changes completely when one of the angles reaches 120°. One thing at a time. While the meeting point is in question, we had better stay inside of the triangle. Therefore do agree, please, that we bar from our examinations the triangles having an angle of 120° or even more.

97. Abbeys Bookshop - Mathematical Chestnuts From Around The World The ninepoint circle and Coolidge s theorem, the De Longchamps point of atriangle, Cantor s theorem, and Napoleon s theorem; 23. http://www.abbeys.com.au/items.asp?id=107019

98. Index A Generalization of Napoleon s theorem Napoleon s theorem Explorations Napoleon s theorem (Jessica D. Dwy) Interactive Geometry Problem http://steiner.math.nthu.edu.tw/chuan/99s/

Geometric Constructions ±Ð®v¡G¥þ¥ô­« e-mail: jcchuan@math.nthu.edu.tw ¹q¸Ü¡G3029 ºô­¶¡Gponcelet.math.nthu.edu.tw/chuan/99s ¿ì¤½«Ç¡Gºî¦X¤TÀ] ¤W½Ò¦aÂI¡Gºî¦X¤TÀ]203«Ç¤Î¤T¼Ó¼Æ¾Ç¹Ï®ÑÀ]¤º Introduction to Maple V JavaSketchpad files Photos ¤§®a ... ¾Ç¥Íªº®a

99. PF JU - Katedra Matematiky - Doc. RNDr. Pavel Pech, CSc. - Publikacni Cinnost The harmonic analysis of polygons and Napoleon s theorem. Univ. S. Boh. Dept.Math. Rep. Ser. 6 (1998), 113. Area of a cyclic polygon. http://www.pf.jcu.cz/stru/katedry/m/pech/publ.phtml

Katedra matematiky Informace o studiu Prijimaci rizeni Dalkove studium - financni matematika ... doc. RNDr. Pavel Pech, CSc. Kvalifikacni prace: Integrabilita skoro tecnych struktur. Diplomova prace, MFF UK Praha, 1974, 25 stran. O nerovnostech prostorovych krivek a prostorovych n-uhelniku. Kandidatska disertacni prace, MFF UK Praha, 1991, 73 stran. Vztahy mezi nerovnostmi v n-uhelnicich. Habilitacni prace, PF JU C. Budejovice, 1994, 101 stran. Vysokoskolska skripta a ucebni texty: Priprava k prijimacim zkouskam z matematiky. Pedagogicka fakulta C. Budejovice, 1986, 33 stran. Analyticka geometrie linearnich utvaru. PF JU C. Budejovice, 1994, 158 stran, (spoluautor J. Strobl). Kuzelosecky. Ucebni text, Pedagogicka fakulta C. Budejovice, 2002, 98 stran. Puvodni vedecka sdeleni: Inequality between sides and diagonals of a space n-gon and its integral analog. Cas. pro pest. mat. 115 (1990), 343-350. Petrova veta. Sbornik 11. seminare odborne skupiny pro geometrii a pocitacovou grafiku. Bedrichov, JCMF, 1991, 6-12. A sharpening of a discrete analog of Wirtinger's and isoperimetric inequalities.

100. Napoleon-Like Properties Of Spherical Triangles -- From Mathematica Information In elementary Euclidean geometry this result is known as Napoleon s theorem.Consider the following generalization of this construction. Let d(.,. http://library.wolfram.com/infocenter/Articles/1610/

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