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 such as the hierarchical classification of quadrilaterals, interior angle sum of polygons, etc. 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
Prof Extraordinaire, Research Unit for Mathematics Education (RUMEUS)
University of Stellenbosch, South Africa
[Retired, Dept of Mathematics & Computer Science Education
University of KwaZulu-Natal, South Africa]

(Last Update: 17 June 2017)

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

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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


Flashback to the Past - a 1949 Matric Geometry Exam Question


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


Haruki's Theorem (Three Overlapping Circles)


A Hierarchical Classification of Quadrilaterals
(Inclusive Classification of Quadrilaterals)


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


Investigating incentres of some iterated triangles


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 Circle and Spieker Circle Duality (NEW!)


Nine-point conic and Euler line generalization (Updated!)


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) (Updated!)


Pirate Treasure Hunt and a Generalization


Power Lines of a Triangle


Quadrilateral Similar-Triangles Collinearity Theorem


The quasi-circumcentre and quasi-incentre of a quadrilateral


The quasi-Euler line of a quadrilateral and a hexagon


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 (Updated!)


Sylvie's Theorem


Trisection concurrency: a variation on a median theme


Van Aubel's Theorem and some Generalizations


A variation of Miquel's theorem and its generalization


Some unproved conjectures


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 & Geometry 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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