Dynamic Geometry Sketches

(Interactive dynamic geometry sketches with JavaSketchpad, Cabri 3D, Cinderella, GeoGebra & GEONexT)

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This site has interactive Java sketches which demonstrate several famous and beautiful theorems in mathematics such as Napoleon, Viviani, nine-point circle, Euler line, Neuberg, Van Aubel, Spieker circle, Fermat-Torricelli points, etc., as well as further generalizations of many of these. It also includes Student Explorations which contain mostly geometric problems, challenges, explorations, and classroom material of interest hopefully to both students and teachers of mathematics in school as well as undergraduate level at university. Some of the explorations are directly related to the school math curriculum such as the "classifying, exploring and defining quadrilaterals" activity, while others such as "Bottema's theorem" are intended as challenges to extend students and teachers a little beyond the usual confines of the school. Some challenges are more suitable for training of enthusiasts of mathematical competitions.

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Michael de Villiers
Dept of Mathematics & Computer Science Education
University of KwaZulu-Natal, South Africa

(Last Update: 7 October 2014)

Visit p. 4 of my homepage for downloadable articles Downloadable articles on Math & Math Ed

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Circumcyclicquad

A proof of the result above, a generalization as well as an interesting dual, and much more,

can be found in my book, available in bound form or downloadable PDF Some Adventures in Euclidean Geometry

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Explorations for Students (Updated)

 

Affine Equivalence of Cubic Polynomials

 

Affine Invariance of Conics

 

Area Ratios of Octagons/Parallelograms: IMTS problem

 

Balancing Quadrilaterals Theorem: A 'Van Aubel-like' result
An application of centroids (centres of mass)

 

British Mathematics Olympiad Problem and its Dual
An application of perpendicular and angle bisectors of a triangle

 

Circumcentres (or Perpendicular Bisectors) of Circumscribed Quadrilateral Theorem

 

Clough's Theorem (a variation of Viviani) and some Generalizations

 

Concurrency and Euler line locus result

 

Cyclic Quadrilateral Incentre-Rectangle

 

Cyclic Quadrilateral Rectangle Result

 

De Villiers points and hyperbola of a triangle

 

Du Plooy's generalization of the Theorem of Pythagoras

 

Eight Point Conic

 

Fermat-Torricelli Point Generalizations

 

Feynman's Triangle and some generalizations

 

Generalizations of Cross's (Vecten's) Theorem

 

Generalization of Euclid Book III, proposition 22 (cyclic quadrilateral theorem), and its dual

 

A generalization of Neuberg's Theorem and the Simson line

 

2D Generalizations of Viviani's Theorem
3D Generalizations of Viviani's Theorem

 

2D Generalizations of a theorem of Sylvester
Sylvester's theorem for a tetrahedron

 

Generalizing a theorem of Arsalan Wares

 

Generalizing the Nagel line to Circumscribed Polygons by Analogy & Constructive Defining

 

A generalization of a Parallelogram Theorem to Parallelo-hexagons, and Douglas' Theorem

 

Haag Hexagon and its generalization to a Haag Polygon (New!)

 

Haruki's Theorem (Three Overlapping Circles)

 

Hierarchical Quadrilateral Classification or Tree

 

Interior Angle Sum of Polygons (incl. Crossed): a General Formula

 

Locus Problem (fixed chords in circle)

 

Napoleon's Theorem and some Generalizations & Variations

 

Nickalls' Conic Theorem

 

Nine-point (or Euler) centre & Maltitudes of Cyclic Quadrilateral

 

Nine-point conic and Euler line generalization

 

Octagon Centroids forming Parallelo-octagon: A Varignon Generalization
A generalization of Varignon's Theorem to 2D and 3D

 

Para-Hexagon Concurrency Theorem
A generalization of concurrency of triangle medians

 

Pirate Treasure Hunt and a Generalization

 

Power Lines of a Triangle

 

Quadrilateral Similar-Triangles Collinearity Theorem

 

The quasi-Euler line of a quadrilateral and a hexagon (New!)

 

Semi-regular Angle-gons and Side-gons
A generalization of rectangles and rhombi

 

Similarity of Parabola

 

Some Parallelo-hexagon Area Ratios

 

Spieker conic and Nagel line generalization

 

Sylvie's Theorem

 

Van Aubel's Theorem and some Generalizations

 

Some unproved conjectures (Updated!)

 

Other Activities and Links

 

Cut-the-Knot - Interactive Mathematics Miscellany and Puzzles

 

Online Graphing Calculator

 

Solve Equations Numerically

 

Linear Equations Solver

 

Dynamically Explore Elliptic (Spherical) Geometry

 

Dynamically Explore Hyperbolic Geometry

 

Logic & Strategy Games

 

LOGO (Turtle) Geometry Online or download StarLogo turtle geometry from MIT

 

Mathematics Archives - Topics in Mathematics - Geometry

 

Math Forum Electronic Newsletter

 

The Importance of Geometric Literacy in Engineering & Design

 

Wolfram Alpha Online Computational Knowledge Engine

 

Homepage of Yutaka Nishiyama with several interesting mathematical applications

 

List of online resources for math teachers

 

 

By Michael de Villiers at: prof md @ mweb .co.za (use without spaces in between)

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