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81. Writing Assignment #4: Technology Applications
that are appropriate for an interactive approach, including applications of Menelaus and Ceva s theorem, Steiner s theorem, Napoleon s theorem, problems with

Illinois State University Mathematics Department

MAT 326: Technology Tools for Secondary School Mathematics Dr. Roger Day (

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.

82. Dynamic Geometry Module: Lesson 5
incenters and orthocenters!). Napoleon s theorem states that the centers X, Y, and Z form an equilateral triangle. Join these points
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

83. FlipCode Message Center - Any Nice Proofs Using Geometric Algebra - Random Bits
I m looking for a nice simple theorem (like Napoleon s theorem or something simple like that) but that can be proved with GA, preferably for any number of

84. Angles Orientés De Vecteurs Dans Le Plan
theorem - the Napoleon s theorem.
Angles orientés de vecteurs dans le plan
A partir d'une axiomatisation des angles orientés de vecteurs du plan euclidien,on donne des preuves classiques des théorèmes de cocyclicité, de Simson, de Napoléon et de l'orthocentre. Voir le rapport de recherche associé ( et le fichier README. Download (archive compatible with Coq V8.0) Author: Frédérique Guilhot ( Institution: INRIA Sophia Antipolis, projet Lemme Date: 15 janvier 2002 Keywords: Pcoq géométrie théorème démonstration angle cercle geometry theorem proof angle circle The README file of the contribution: This page was automatically generated from this description file

85. Australian Mathematics Trust
One notable example of the latter is the DouglasNeumann theorem, an extension of Napoleon s theorem discovered independently by Bernhard and the Fields
VALE BERNHARD NEUMANN 1909-2002 Emeritus Professor Bernhard Hermann Neumann, who provided the greatest inspirational influence in mathematics in Australia over a 40-year period, died in Canberra on 21 October 2002 not long after happily celebrating his 93rd birthday. He first visited Australia for three months in 1959, during sabbatical leave, and fell in love with the country. So when, late in 1960, he was invited to found a Department of Mathematics in the research-focussed Australian National University, he was receptive to the idea. Within days of his permanent arrival on 2 October 1962, he also became involved in activities supporting the teaching of mathematics in schools. Bernhard had a great influence in the founding and administration of the Australian Mathematics Trust. He became a mentor and source of inspiration to Peter O’Halloran (1931-1994) who, while on the staff of the Canberra College of Advanced Education (later the University of Canberra) during the period of the early 1970s to the early 1990s, is acknowledged as the Founder of the Trust. Peter gained great strength from Bernhard’s encouragement, not only while Bernhard held his position as head of mathematics in the Institute of Advanced Studies at the Australian National University, but also after Bernhard’s retirement. Bernhard took an active personal part in the Trust’s activities. He was the Inaugural Chairman of the Australian Mathematical Olympiad Committee, a position he held from 1980 to 1986. He was also the representative of the Canberra Mathematical Association (a sponsor of the Australian Mathematics Competition (AMC) for the Westpac Awards) on first the AMC Governing Board, and then on the Advisory Committee of the Trust. He was an active member of the Advisory Committee until his death.

86. Masteringti-92 Contents
117. Exploration 5 Geometry Napoleon s theorem 123. Exploration 6 Precalculus Polynomial Functions of Higher Degree 129. Exploration
Return to Gilmar's Home Page Return to "Our Books" Page Mastering the TI-92 Explorations from Algebra through Calculus by Nelson Rich, Judith Rose, and Lawrence G. Gilligan CONTENTS Preface vii PART 1: An Overview of the TI-92 . The Nine TI-92 Arenas 1 . The MODE Key 2 . Feeling at Home on the HOME Screen: Arithmetic Operations 5 . Algebra on the HOME Screen 6 . Some Editing Tips 10 . Graphing Functions and Displaying Tables 11 . The Zoom Graph Menu 19 .1 ZoomBox 20 .2 ZoomIn and ZoomOut 21 .3 ZoomDec 23 .4 ZoomSqr 23 .5 ZoomStd 25 .6 ZoomTrig 25 .7 ZoomInt 25 .8 ZoomFit 26 . The Math Graph Menu 26 .1 Value 27 .2 Zero, Maximum, and Minimum 27 .3 Intersection 29 .4 Derivatives and Tangent 30 .5 Integration 31 .6 Inflection 32 . Additional Types of Graphs 33 .1 Graphing Parametric Equations 33 .2 Polar Graphing 35 .3 Sequence Graphing 37 .4 3D Graphing 41 . The Calculus Menu 43 .1 differentiate 44 .2 integrate 46 .3 limit( 47 .4 sum and product 49 .5 fMin( and fMax( 49 .6 arcLen( 51 .7 taylor( 51 .8 nDeriv( 53 . Matrices and Systems of Equations 54 .1 Matrix Operations 55 .2 Systems of Linear Equations 58 . Statistics 59 1 Single Variable Statistics 60 2 Frequency Distributions 62 3 Two-Variable Statistics: Regression 65 . Geometry 66 1 The Pointer Menu 68 2 The Points and Lines Menu 68 3 The Curves and Polygons Menu 71 4 The Construction Menu 74 5 The Transformation Menu 85 6 The Measurement Menu 92 7 The Display Menu 97 PART 2: Explorations into the Mathematics Curricula Exploration #1: Algebra: Linear Equations 103 Exploration #2: Algebra: Equations of Lines 107

87. Gov's Web
1. Properties of Parallel Lines 2. Equations of Lines 3. The Burning Tent Problem 4. Angle Bisectors in a Triangle 5. Napoleon s theorem 6. Properties of
Governor Janklow's Faculty Awards for
Teaching with Technology
Report on Activities and Accomplishments
William Haigh

Professor of Mathematics
Northern State University
Major Activities
Aberdeen Area Workplaces Visited A total of eight Aberdeen area workplaces (Table 1) were visited to observe the computer technology used and, where appropriate, incorporated into university mathematics classes (Table 2) . The computer technology used at these worksites included: (1) spreadsheets; (2) programming in various languages; (3) statistical analysis packages; (4) Internet; (5) software to solve systems of differential equations; (6) topographic map packages; and (7) graphing packages.
Table 1. Aberdeen Area Workplaces Visited Horton Vehicle Components Inc. Northwestern Public Service Student Loan Finance Corporation Dalager Engineering K.O. Lee Company Hub City Inc. National Weather Service
Table 2. Classroom Activities Developed

88. Center For Technology And Teacher Education: Content Areas: Mathematics
Napoleon s theorem Explorations Draw a triangle. On the edges of the triangle, construct equilateral triangles. Find the centroids




Social Studies
... Related Links
Geometry and the Geometer's Sketchpad Links
101 Project Ideas for The Geometer's Sketchpad
This booklet from Key Curriculum Press is full of exciting project ideas for use in the classroom or at home. The projects are designed for users with varying degrees of Sketchpad experience and cover a wide range of subject areas (Art/Animation, Triangles, Real World Modeling, Calculus, Transformations and Tessellations, Trigonometry, Fractals, and many more).
Algebra and Calculus Sketches
Allows the user to explore equations for lines, parabolas, and tangents using Geometer's Sketchpad.
Betweenness Theorem
If point B lies between points A and C, the sum of the distances from A to B and from B to C will equal the distance from A to C. This site explores a geometric representation of this theorem.

89. Some Illustrations Of Theorems In Geometry
The Euler Line Morley s theorem The outer internal angle trisectors of a triangle intersect to form an equilateral triangle. Napoleon s theorem.
Some illustrations of theorems in Geometry
Pythagoras' Theorem.
The square on the hypotenuse of a right triangle is equal to the sum of the squares on the other two sides.
Euclid's proof:
Another proof:
The medians of a triangle are concurrent at the centroid.
The altitudes of a triangle are concurrent at the orthocentre.
The internal bisectors of the angles of a triangle are concurrent at the centre of the incircle.
The perpendicular bisectors of the sides of a triangle are concurrent at the centre of the circumcircle.
Pappus' Theorem:
Pascal's Theorem: The Simson Line The Nine-Point Circle The Euler Line Morley's Theorem The "outer" internal angle trisectors of a triangle intersect to form an equilateral triangle. Napoleon's Theorem

90. The Gateway To Educational Materials
Fridjof Nanton, Tricky Sam Nantucket Nantucket Island Naomi Hupert Napoleon Napoleon Napoleon Iii Napoleon Point Napoleon S theorem Napoleonic Apologist


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