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This site has interactive geometry 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, Fibonacci, 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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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]
Visit my homepage for downloadable articles Downloadable articles on Math & Math Ed
Side Divider Theorem for a Circumscribed/Tangential Quadrilateral
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.
Get up to 15% or more off Print Books at Lulu Publishers: Downloadable PDF papers/books on Geometry, Mathematics & Mathematics Education for sale
NEW! Problem solving and logical thinking games for young children Contact dynamic learn @mweb. co. za (without the spaces) to order.
DAILY PUZZLE from the FREE online problem solving tutoring system MyTutor.Chat - for more information click the preceding link.
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SKETCHPAD IS NOW FREE!. More info: FREE SKETCHPAD DOWNLOAD & INSTALLATION.
Explorations for Students
Affine Equivalence of Cubic Polynomials
Affine Invariance of Conics
An associated Van Aubel result & further generalizations
Area Parallelogram Partition Theorem: Another Example of the Discovery Function of Proof
Area Ratios of Inscribed Polygons - another generalization of Varignon's theorem Medium-Hard
Area Ratios of Octagons/Parallelograms: IMTS problem
Balancing Quadrilaterals Theorem: A 'Van Aubel-like' result
(An application of centroids - centres of mass)
Bradley's Theorem, its Generalization & an Analogue Theorem
British Mathematics Olympiad Problem and its Dual
(An application of perpendicular and angle bisectors of a triangle) (Updated!*)
Centroids of Triangles subdividing a Hexagon forming Parallelo-Hexagon
(A generalization of Varignon's Theorem to 2D and 3D)
Circumcentres (or Perpendicular Bisectors) of Circumscribed (tangential) Quadrilateral Theorem
Clough's Theorem (a variation of Viviani) and some Generalizations
Concurrency and Euler line locus result
Concurrency, collinearity and other properties of a particular hexagon
Conway's Circle Theorem as special case of Side Divider Theorem
Cyclic Quadrilateral Incentre-Rectangle
Cyclic Quadrilateral Rectangle Result
Dao Than Oai's generalization of Napoleon's theorem - geometric proofs & further generalizations
De Villiers points and hyperbola of a triangle
Du Plooy's generalization of the Theorem of Pythagoras
Eight Point Conic for Quadrilateral
360 Encyclopedia of Triangle Centres
Euler & Nagel Lines Analogy
Fermat-Torricelli Point Generalizations
Feynman's Triangle and some generalizations
A Fibonacci Generalization - Kendal's theorem
Flashback to the Past - a 1949 Matric Geometry Exam Question
Generalizations of Cross's (Vecten's) Theorem
Generalization of the Cyclic Quadrilateral Angle Sum theorem (Updated!*)
Generalization of the Tangential Quadrilateral Side Sum theorem (Updated!*)
A generalization of Neuberg's Theorem and the Simson line (Updated!*)
A Geometric Paradox Explained (Another variation of an IMTS problem) (NEW!*)
Gregory Ellipse Quadrilateral Generalization (NEW!*)
2D Generalizations ofViviani's Theorem (Updated!*)
3D Generalizations ofViviani's Theorem
2D Generalizations of a theorem of Sylvester (Updated!*)
Sylvester's theorem for a tetrahedron
Generalizing a theorem of Arsalan Wares (Updated!*)
Generalizing the Nagel line to Circumscribed Polygons by Analogy & Constructive Defining
A generalization of a Parallelogram Theorem to Parallelo-hexagons, and Douglas' Theorem (Updated!*)
Haag Hexagon and its generalization to a Haag Polygon (Updated!*)
Haag Hexagon - some extra properties (NEW!*)
Golden Quadrilaterals (generalizing a 'golden rectangle') (NEW!*)
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
Jha and Savaran's generalisation of Napoleon's theorem
A cyclic Kepler quadrilateral and the golden ratio
Klingens' Intersecting Circles Theorem
Locus Problem (fixed chords in circle)
A Merry-Go-Round the Triangle (Example of problem posing)
Napoleon's Theorem and some Generalizations, Variations & Converses
Nine-point (or Euler) centre & Maltitudes of Cyclic Quadrilateral
Nine Point Circle and Spieker Circle Duality
Nine-point conic and Euler line generalization
Para-Hexagon Concurrency Theorem
(A generalization of concurrency of triangle medians) (Updated!*)
Perpendicular Bisectors of the Sides of a Hexagon with Opposite Sides Parallel NEW!*
Pirate Treasure Hunt and a Generalization
Power Lines of a Triangle
Quadrilateral Similar Triangles Collinearity Theorem - a generalization of an equilic quadrilateral theorem (Updated!*)
The quasi-circumcentre and quasi-incentre of a quadrilateral (Updated!*)
The quasi-Euler line of a quadrilateral and a hexagon (Updated!*)
Semi-regular Angle-gons and Side-gons
(A generalization of rectangles and rhombi)
Some Parallelo-hexagon Area Ratios
Spieker conic and Nagel line generalization
Sylvie's Theorem
Trisection concurrency: a variation on a median theme
Van Aubel's Theorem and some Generalizations
An associated result of the Van Aubel configuration and some generalizations
A variation of Miquel's theorem and its generalization (Updated!*)
Vertical Line and Point Symmetries of Differentiable Functions (NEW!*)
Other Activities and Links
Cut-the-Knot - Interactive Mathematics Miscellany and Puzzles
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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