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

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 887385 Fun Facts viewed since 20 July 1999. Francis Edward Su All Fun Facts There are currently 165 Fun Fact files. The database is constantly growing, thanks to contributions from colleagues, Fun Facts yet to be entered, and submissions by you ! You can also see a list of recently added Fun Facts e is irrational All Horses are the Same Color ... Zero to the Zero Power Didn't find what you were looking for? How about submitting a fun fact of your own? Harvey Mudd College Mathematics Department Fun Facts

22. DC MetaData For: Napoleon's Theorem With Weights In N-Space napoleon's theorem with Weights in nSpace by H. Martini, B. Weißbach. Keywordsnapoleon's theorem, Torricelli's configuration. Upload 1998-08-24-08-24. 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.

23. 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

24. 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, 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.

25. 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/Fischbein/napoleon.triangle/Napoleon/nap

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.

26. Essay 3 Napoleon's Theorem napoleon's theorem goes as follows Given any arbitrary triangle ABC, constructequilateral 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.

27. Napoleon's Theorem napoleon's theorem. This is a theorem attributed by legend to NapoleonBonaparte. It is Century. Statement of napoleon's theorem. For http://www.math.washington.edu/~king/coursedir/m444a02/class/11-25-napoleon.html

Napoleon's Theorem This is a theorem attributed by legend to Napoleon Bonaparte.  It is rather doubtful that the Emperor actually discovered this theorem, but it is true that he was interested in mathematics.  He established such institutions as the Ecole Polytechnique with a view to training military engineers, but these institutions benefited mathematics greatly. French mathematicians made many important discoveries at the turn of the Eighteenth to the Nineteenth Century. Statement of Napoleon's Theorem For any triangle ABC, build equilateral triangles on the sides.  (More precisely, for a side such as AB, construct an equilateral triangle ABC', with C and C' on opposite sides of line AB; do the same for the other two sides.). Then if the centers of the equilateral triangles are X, Y, Z, the triangle XYZ is equilateral.

28. Math 444 Aut 2002 Week 9 Assignment 9 Due Monday 12/02; Statement of napoleon's theorem for Assignment 9.Wallpaper Groups concepts of transformation group, symmetry group, basic unit http://www.math.washington.edu/~king/coursedir/m444a02/wk09.html

Math 444/487 Geometry Week 9 Monday 11/25 - Friday 11/29 444 Home Page Week 9/30-10/04 Week 10/07-11 Week 10/14-18 ... Week 12/09-13 Monday 11/25 444 Assignment 8A and Assignment 8B due. Assignment 9 Due Monday 12/02 Statement of Napoleon's Theorem for Assignment 9. Wallpaper Groups - concepts of transformation group, symmetry group, basic unit (fundamental domain) of a transformation group. Web Resources : Some very good material on symmetry in general and wallpaper groups in particulat is on the web. The Geometry Center Symmetry page has detailed explanations and software. David Joyce's Wallpaper Groups Java software Escher Web Sketch 16 of the 17 groups in traditional Japanese patterns Some animated patterns Wednesday 11/27 444 Relations among tetrahedra, cubes and octahedra Cube decomposes into 3 congruent pyramids. From this and a bit more it follows that the volume of any pyramid is (1/3) * base area * height. Symmetries of tetrahedra, cubes and octahedra

29. NRICH | December 1998 | Napoleon's Theorem (Dec 98) skip to content, December 98 napoleon's theorem (Dec 98). http://www.nrich.maths.org.uk/mathsf/journalf/dec98/mcprob5.html

Napoleon's Theorem (Dec 98) NRICH Prime NRICH Club ... Get Printable Page March 03 Magazine Site Update News Events Problems Solutions ... Games Archive Problems Solutions Articles Inspirations ... Interactivities Web board Ask NRICH Asked NRICH NRICH Club Register Tough Nuts About Help! ... Where is NRICH? Associated Projects Maths Thesaurus MOTIVATE EuroMaths Millennium Maths ... Project Display maths using fonts images Help Back Issues Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Bernard's Bag(P) - solutions(P) Penta Probs(P) - solutions(P) Let Me Try(P) - solutions(P) Kid's Mag(P) Play Games(P) Staff Room(P) 6 Problems - solutions 15+Challenges - solutions Articles Games LOGOland Editorial News Geometry-Euclidean Properties of Shapes Polygons Problem 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? Click here for an interactive version of this problem. This week's interactive Java problem uses JavaSketchpad . Users may find that they need to update their browser. This applet works with or Microsoft Internet Explorer 4 Solution Triangle ABC has equilateral triagles drawn on its edges. Points

30. NRICH | December 1998 | Napoleon's Theorem (Dec 98) skip to content December 98 napoleon's theorem (Dec 98) http://www.nrich.maths.org.uk/mathsf/journalf/dec98/mcprob5_printable.shtml

Napoleon's Theorem (Dec 98) Geometry-Euclidean Properties of Shapes Polygons Problem 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? Click here for an interactive version of this problem. This week's interactive Java problem uses JavaSketchpad . Users may find that they need to update their browser. This applet works with or Microsoft Internet Explorer 4 Solution Triangle ABC has equilateral triagles drawn on its edges. Points P Q and R are the centres of the equilateral triangles. Experimentation with the interactive diagram leads to the conjecture that PQR is an equilateral triangle. This can be proved using vectors or complex numbers. In the following w e p i so that 1 + w w = 0. Also multiplying a complex number by w rotates it by 60 degrees. Referring to the given diagram let A B be represented by the complex numbers a b . The third vertex of the triangle with base AB is represented by the complex number b w a b ). Therefore

31. Glossary Of Mathematical Terms point; Napier Bones; napoleon's theorem napoleon's theorem, a generalization;napoleon's theorem by Plane Tesselation; napoleon's theorem http://www.cut-the-knot.org/glossary/ntop.shtml

CTK Exchange Front Page Movie shortcuts Personal info ... Recommend this site Glossary A B C D ... Z N 9-Point Circle 9-point Circle as a Locus of Concurrency Nagel point Napier Bones ... Neuberg's Theorem , Synthetic proof New-States Paradox Newton's line Newton's Theorem Nim, the game of ... Alexander Bogomolny G o o g l e Web Search Latest on CTK Exchange embedding applets Posted by Stuart Carduner 1 messages 09:05 AM, Feb-19-03 math problem no answer in sight Posted by Dmom 4 messages 10:40 AM, Feb-28-03 anticentre+argand diagrams Posted by Taka 0 messages 07:02 PM, Mar-15-03 Tiling hyperspheres Posted by Graham C 4 messages 10:38 AM, Mar-12-03 archive says feb instead of march Posted by Frank Anzalone 1 messages 01:15 AM, Mar-06-03 Probability Posted by Ralph scalitzia 1 messages 07:03 PM, Mar-15-03

32. Napoleon's Propeller (2), Of course, (2) could be used to derive napoleon's theorem. napoleon's theoremis equivalent to the Asymmetric Propeller's theorem! How small is the world! http://www.cut-the-knot.org/ctk/NapoleonPropeller.shtml

CTK Exchange Front Page Movie shortcuts Personal info ... Recommend this site 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. 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

33. Crocodile Clips Lesson Plan Crocodile Clips Homepage Mathematics products. Lesson Plan napoleon's theorem(LP0148). Author John Buckley. Learn and explore napoleon's theorem. Resources. http://www.crocodile-clips.com/gpv70/LP/mathematics/LP0148/LP0148.htm

Mathematics products Lesson Plan: Napoleon's Theorem (LP0148) Author: John Buckley Published: th September 2002 Age group: Student activity: Simulation files: Learning objectives Allow investigation of the relationship between equilateral triangles constructed on the sides of an arbitrary triangle. Learn and explore Napoleon's theorem. Resources Software [Crocodile Mathematics 1.2 (or later)] Example 1 [ "Constructing Napoleon's theorem"] Example 2 [ "More from Napoleon's Theorem"] Procedure Ensure Crocodile Mathematics 1.2 (or later) is installed and the simulation files and are accessible Introduce and explain the learning objectives listed above. Complete lesson activity Discuss the activity and the related points listed below. Classroom discussion points Do you think you could investigate Napoleon's theorem using algebra rather than a graphical method? What is another mathematical name for the pattern produced by arranging a number of Napoleon's theorem constructions together? Assessment Completion of lesson activity lessons@crocodile-clips.com

34. Crocodile Clips Lesson Activity Student Activity napoleon's theorem (LA0148). Author John Buckley. In this lessonwe will investigate this theorem. Constructing napoleon's theorem (LF0148a). http://www.crocodile-clips.com/gpv70/LP/mathematics/LP0148/LA0148.htm

Mathematics products Student Activity: Napoleon's Theorem (LA0148) Author: John Buckley Published: th September 2002 Lesson plan: Introduction The French emperor Napoleon (1769 - 1821) is attributed with the discovery of a theorem relating equilateral triangles constructed on the sides of an arbitrary triangle. In this lesson we will investigate this theorem. Constructing Napoleon's theorem (LF0148a) Open the Crocodile Mathematics simulation file Arrange each of the three equilateral triangles so that one edge is coincident with one edge of the blue triangle (put a different triangle on each edge of the blue triangle). Now drag on a scalene triangle and arrange it so that each point is coincident with the centre (pivot) of the three equilateral triangles. What do you notice about the scalene triangle, is there anything special about it? You can check the side lengths by hovering over them with the mouse. Drag on another equilateral. Is it possible to make it exactly coincident with the scalene you dragged on in step 2? Rivet the three coincident lines together and then resize the blue triangle. If you repeat step 2 does the relation you found still hold?

35. Index A Generalization of napoleon's theorem, napoleon's theorem Explorations.napoleon's theorem (Jessica D. Dwy), Interactive Geometry Problem. http://poncelet.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

36. Mathematisches Seminar: Geometrie Translate this page Galerie - Bildersammlung. Satz und Beweis napoleon's theorem with 2 Proofs.Geschichte Napoléon Ier, Empereur des Français / Napoleon Bonaparte. http://www.gris.uni-tuebingen.de/gris/grdev/java/geometry/doc/html/MainPage.html

mit oder ohne Frames Frank Hanisch G leichseitige Dreiecke - Der Satz von Napoleon hnliche Dreiecke PQR - Verallgemeinerung Fall 1 hnliche Dreiecke LMN - Verallgemeinerung Fall 2 P flasterung - Auspflasterung der Ebene G alerie - Bildersammlung Satz und Beweis: Napoleon's Theorem with 2 Proofs Geschichte: Napoleon Bonaparte Geometrie: Triangle Centers Mathematik: Math Forum Programm: JavaSketchpad

37. Geometry Problems Poncelet Morley Butterfly Geometry Problems. 1. Poncelet's Theorem. 2. napoleon's theorem. 3. EyeballTheorem. 4. Steiner's Theorem. Proof. Top 2. napoleon's theorem. Proof. Top http://agutie.homestead.com/files/Geoproblem_B.htm

Geometry Problems Poncelet's Theorem Napoleon's Theorem Eyeball Theorem Steiner's Theorem ... Sangaku Problem (An Old Japanese Theorem) Sangaku Problem 2 Sangaku Problem 3 Butterfly Theorem Langley Problem: 20° Isosceles Triangle ... Morley's Theorem 1. Poncelet's Theorem. Proof Home Top 2. Napoleon's Theorem. Proof Home Top 3. 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. Home Top 4. Steiner's Theorem. Proof Home Top 5. 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 Home Top 6. Sangaku Problem (An Old Japanese Theorem) Let a convex inscribed polygon be triangulated in any manner, and draw the incircle to each triangle so constructed. Then the sum of the inradii is a constant independent of the triangulation chosen.

38. Geometry Problems Index Geometry Problems. Poncelet's Theorem. napoleon's theorem. Eyeball Theorem.Steiner's Theorem. Carnot's Theorem. Sangaku Problem 1 An Old Japanese Theorem. http://agutie.homestead.com/files/Geoproblem_A.htm

Geometry Problems Poncelet's Theorem Napoleon's Theorem Eyeball Theorem ... Sangaku Problem 1 An Old Japanese Theorem Sangaku Problem 2 Sangaku Problem 3 Butterfly Theorem Langley Theorem ... Home

39. Napthm napoleon's theorem is the name popularly given to a theorem which states that ifequilateral triangles are constructed on the three legs of any triangle, the 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.

40. MATH WORDS, AND SOME OTHER WORDS OF INTEREST If you have suggestions or comments Email to Pat Ballew A B C D E F G H IJ K L M N O P Q R S T U V W X Y Z N Napoleon's Point; napoleon's theorem; http://www.pballew.net/etyind2.html

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