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**Dynamic Geometry Sketches**

**(Interactive dynamic geometry sketches with JavaSketchpad**

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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.

**Michael de Villiers**

**(Last Update: 30 January 2024)**

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

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*.

15% off Print Books till 25 February 2022 **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.

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**Explorations for Students ** *( Updated)*

**Affine Equivalence of Cubic Polynomials** *(Updated!*)*

**Affine Invariance of Conics** *(Updated!*)*

**An associated Van Aubel result & further generalizations***NEW!**

**Area Parallelogram Partition Theorem: Another Example of the Discovery Function of Proof***NEW!**

Area Ratios of Inscribed Polygons - another generalization of Varignon's theorem Medium-Hard **(NEW*!)**

**Area Ratios of Octagons/Parallelograms: IMTS problem***Updated!**)

**Balancing Quadrilaterals Theorem: A 'Van Aubel-like' result*** **(Updated!*)*

(An application of centroids - centres of mass)

**Bradley's Theorem, its Generalization & an Analogue Theorem*** **(NEW!*)*

**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** *(Updated!*)*

**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** *(NEW!*)*

**Conway's Circle Theorem as special case of Side Divider Theorem** *NEW!**

**Cyclic Quadrilateral Incentre-Rectangle**

**Cyclic Quadrilateral Rectangle Result** *(Updated!)*

**Dao Than Oai's generalization of Napoleon's theorem - geometric proofs & further generalizations****(NEW!*)**

**De Villiers points and hyperbola of a triangle****(Updated!*)**

**Du Plooy's generalization of the Theorem of Pythagoras **

**Eight Point Conic for Quadrilateral****(Updated!*)**

**360 Encyclopedia of Triangle Centres****(NEW!*)**

**Euler & Nagel Lines Analogy****(NEW!*)**

**Fermat-Torricelli Point Generalizations **

**Feynman's Triangle and some generalizations ** **(Updated!*)**

**A Fibonacci Generalization - Kendal's theorem** **(NEW!*)**

**Flashback to the Past - a 1949 Matric Geometry Exam Question **

**Generalizations of Cross's (Vecten's) Theorem ** *(Updated!*)*

**Generalization of the Cyclic Quadrilateral Angle Sum theorem (Updated!*)
Generalization of the Tangential Quadrilateral Side Sum theorem**

**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** **(Updated!*)**

**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)** *(Updated!*)*

**A Hierarchical Classification of Quadrilaterals **

(Inclusive Classification of Quadrilaterals) *(Updated!*)*

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

**Investigating incentres of some iterated triangles **

**Jha and Savaran's generalisation of Napoleon's theorem****(NEW!*)**

**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** *(Updated!*)*

** **

**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** *(Updated!*)*

** **

**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** *(Updated!*)*

** **

**Spieker conic and Nagel line generalization**

**Sylvie's Theorem** *(Updated!*)*

**Trisection concurrency: a variation on a median theme**

**Van Aubel's Theorem and some Generalizations** *(Updated!*)*

**An associated result of the Van Aubel configuration and some generalizations** *(NEW!*)*

**A variation of Miquel's theorem and its generalization** *(Updated!*)*

**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

*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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