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## Dynamic Geometry Sketches

(Interactive dynamic geometry sketches with mainly WebSketchpad and some others like Cabri 3D, Cinderella & GeoGebra)

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
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: 4 March 2024)

#### .sketch_canvas { border: medium solid lightgray; display: inline-block; } 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.

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Explorations for Students (Updated)

Affine Equivalence of Cubic Polynomials (Updated!*)

Affine Invariance of Conics (Updated!*)

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

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)

Eight Point Conic for Quadrilateral (Updated!*)

Euler & Nagel Lines Analogy (NEW!*)

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

Flashback to the Past - a 1949 Matric Geometry Exam Question

Generalizing a theorem of Arsalan Wares (Updated!*)

Haruki's Theorem (Three Overlapping Circles) (Updated!*)

Investigating incentres of some iterated triangles

A cyclic Kepler quadrilateral and the golden ratio

Klingens' Intersecting Circles Theorem NEW!*

Para-Hexagon Concurrency Theorem
(A generalization of concurrency of triangle medians) (Updated!*)

Power Lines of a Triangle (Updated!*)

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

Some Parallelo-hexagon Area Ratios (Updated!*)

Sylvie's Theorem (Updated!*)

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