Dynamic Mathematics Explorations for Students

(Last Update: 5 January 2023)

Back Home

"A great discovery solves a great problem, but there is a grain of discovery in the solution of any problem. Your problem may be modest, but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery". - George Polya in How to Solve It (Princeton 1945).

"In real mathematics, the answers are not in the back of the book nor are the problems simply variations of preceding, carefully worked examples. In fact, in real mathematics, there is no book at all - no ready-made questions and problems; even those one has to invent oneself!"

............

Click on any of the above pictures for more information about these books and/or to purchase.

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.

*********

NOTE: Please WAIT while the applets at the links below load. If a security pop-up menu appears in your browser, please choose RUN/ALLOW to let the applets run properly. It is completely safe & can be trusted. If you have the very latest Java on your PC or Apple Mac, and experience problems with the applets loading, please go here for additional information on Java settings or downloading a bookmarklet that should resolve the issue.

Alternate Angles Sum Cyclic Hexagon and Alternate Sides Sum Circumscribed Hexagon Easy - Medium (Updated!*)

 

Anele Moli's Method: Constructing an equilateral triangle Easy

 

Area Formula for Quadrilateral in terms of its Diagonals Easy (NEW*!)

 

Area Ratios of Inscribed Polygons Medium-Hard

 

Bicentric Quadrilateral Construction Challenge Medium - Hard (NEW*!)

 

Bicentric Quadrilateral Property Medium

 

Bottema's Theorem aka Carnot's Theorem Easy-Medium (Updated!)*

 

Bride's Chair Concurrency & Locus Hard

 

British Mathematics Olympiad Concurrency Problems Medium (Updated!*)

 

Bug Escape Problem Easy

 

Circle Circumference Result Easy-Medium

 

Circle Concurrencies Easy-Medium (NEW*!)

 

Circumscribed Quadrilateral Side Divider theorem and Cyclic Quadrilateral Angle Divider theorem Easy-Medium (Updated*!)

 

Classifying, Exploring, Constructing & Defining Quadrilaterals Easy-Medium

 

Clough's Conjecture Medium-Hard

 

Collinear Conjecture Easy-Medium

 

Concurrent Angle Bisectors of Quadrilateral Easy (NEW!*)

 

Conic by Half-turn Medium

 

Constant perimeter triangle formed by tangents to circle Easy-Medium

 

Construct a Parallel-Hexagon with GEONexT Medium

 

Cosine-Sine Angle Rule Easy

 

Crossed Quadrilateral Properties Medium

 

Cyclic Crossed Pentagon Problem Easy

 

Cyclic Quadrilateral Arcs Theorem Hard

 

Cyclic Quadrilateral Difference of Squares Theorem Hard (NEW*!)

 

Cyclic Quadrilateral Rectangle Result Medium (Updated!)

 

Cyclic Kepler Quadrilateral and Golden Ratio. Easy

 

Desargues' Theorem Easy-Medium

 

Descartes' Geometric Solving of Some Quadratics Easy-Medium

 

Diagonal Division Ratios in a Quadrilateral Hard (NEW!)

 

A Diagonal Property of a Rhombus constructed from a Rectangle Easy (NEW!)

 

Dirk Laurie Tribute Problem Medium (NEW!*)

 

Double Angle Triangle Medium

 

Dynamic Art: A Parquet Deformation Tiling An art project idea (Updated!*)

 

Equi-inclined Lines Problem Easy (NEW!*)

 

The Equi-partitioning Point of a Quadrilateral Easy (NEW!*)

 

Euclid 1-43 Parallelogram Area Theorem Easy-Medium

 

Extangential Quadrilateral Medium (NEW!*)

 

Finding the Median and Centroid of a Triangle by Experimentation: A gravity simulation Use to explore. (Updated!)*

 

Finding the centroid of a triangle with different weights at the vertices: A gravity simulation Use to explore. (Updated!)*

 

Forgotten Similarity Theorem Easy (NEW!*)

 

A Fundamental Theorem of Similarity Easy-Medium (NEW!*)

 

Further properties of a particular hexagon Easy-Medium (NEW!*)

 

Generalizing perpendicular bisectors, angle bisectors, medians and altitudes to 3D Medium to hard.

 

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

 

Generalizing Viviani's theorem to 3D Easy to Medium.

 

GEONexT - online dynamic mathematics with full construction facility Use to explore.

 

Geometry Loci Doodling with a Cyclic Quadrilateral. Medium - Hard. (NEW!*)

 

Gielis Super Shape Formula Medium. (Updated!)

 

Golden Quadrilaterals (generalizing a 'golden rectangle' Medium. (NEW!*)

 

Haag Hexagon and its generalization to a Haag Polygon Medium.

 

Hexagon Explorations Easy

 

Hippocrates' Theorem Easy (Updated!*)

 

Hypercube-Cross Polytope Duality Easy

 

IMO 2014 Problem 4 (Geometry) Medium-Hard (Updated!*)

 

Interior Angle Sum of Polygons: a General Formula Easy-Medium (Updated!*)

 

Interesting Collinearity Medium-Hard (NEW!)

 

Interesting property of the number e Medium

 

Intersecting Circles Easy (NEW!)

 

Investigating circumcentres of iterated median triangles Medium
Investigating incentres of some iterated triangles Medium-Hard

 

Japanese Circumscribed Quadrilateral Theorem Hard

 

Jannie's Father's Farm Problem Easy - Medium

 

Kinky Lines Easy

 

Klingen's Theorem of two intersecting circles or two adjacent isosceles triangles Medium - Hard (Updated!*)

 

The Lux problem Medium - Hard (NEW!*)

 

Matric Geometry Exam Question - 1949 Easy - Medium

 

Maximum area of quadrilateral problem Easy

 

Midpoint Trapezium (Trapezoid) Theorem Generalized Easy-Medium

 

Miscellaneous Transformations (of geometric figures & graphs) Easy - Medium

 

Napoleon Variation Problem Medium

 

Nine-point centre (anticentre or Euler centre) & Maltitudes of a Cyclic Quadrilateral Hard (NEW!*)

 

Orthocentre Quadrilateral of a Quadrilateral Hard (NEW!*)

 

Over and over: two geometric iterations with triangles Medium - Hard

 

Parabola Locus Theorem and Parabola Circumcircle Theorem Easy

 

Parallelogram Distances Easy (NEW!*)

 

Parallelogram Trio Medium

 

Paul Yiu's problem and a generalization Medium (Updated!*)

 

Perimeter of Hexagon inscribed in Triangle Easy

 

Pi in other geometries Medium

 

Platonic Solids plus Truncated Platonic Solids Easy

 

Point & Line Reflections in Concurrent Lines Easy (NEW!*)

 

Pompe's Hexagon Theorem Medium-Hard (NEW!*)

 

Pythagorean Quadrilateral 1 Medium

 

Pythagorean Quadrilateral 2 Easy

 

Pythagorean Quadrilateral 3 Easy

 

Quadrilateral Inequality involving Perimeter & Diagonals Medium

 

Quarter-Circle Investigation, Explanation & Generalization Easy (NEW!*)

 

The quasi-circumcentre and quasi-incentre of a quadrilateral Medium (Updated!*)

 

Reauleaux Triangle (a curve of constant width) - a generalization Medium

 

Rectangle Angle Trisection and Rhombus Angle Trisection Easy (Updated!*)

 

SA Mathematics Olympiad Problem 2016, Round 1, Question 20 Medium - Hard

 

SA Mathematics Olympiad Problem 2016, Round 2, Question 20 Medium - Hard

 

SA Mathematics Olympiad Problem 2022, Round 2, Question 25 Medium - Hard (NEW!*)

 

Side trisection triangle concurrency Easy - Medium

 

Similar Triangles Easy (Updated!*)

 

Similar Parallelograms: A Generalization of a Golden Rectangle property Easy

 

Six Point Cevian Circle Hard (NEW!*)

 

Square Trigonometry Easy

 

Tangents Easy

 

Tangential Quadrilateral Converse. Medium (NEW!*)

 

Tangential Quadrilateral - Pitot's Theorem. Easy (NEW!*)

 

Tangential Quadrilateral - Theorem of Gusic & Mladinic. Easy - Medium (NEW!*)

 

Tangential Hexagon Incircles Application. Medium (NEW!*)

 

Thabit's Generalization of Pythagoras. Easy

 

Theorem of Arsalan Wares Easy-Medium (Updated!*)

 

Three Circle Geometry Theorem Proofs by Transformations Easy

 

Toshio Seimiya Theorem: A Hexagon Concurrency result Hard

 

Trapezoid Explorations Medium

 

Trapezoid Explorations Medium

 

Triangle Circumcircle-Incentre Result Medium

 

Triangle Incentre-Circumcentre Collinearity Easy-Medium

 

Two circles result Easy

 

Two different centroids (balancing points) of a quadrilateral Easy-Hard (Updated!)*

 

What function? Modeling the height of water against time in a tilted container Medium

 

What number comes next? Easy (Updated!*)

***

External Links

South African Mathematics Olympiad: Question Papers & Solutions - Junior & Senior (from 1997)

Alex Bogomolny's Cut the Knot site: Geometry Problems, Applets & much more (incl other topics)

E-Classroom Free educational worksheets for teachers & parents

Gerrit Stols' School Math(s) site: Interactive Math, Books, Math Tools, Resources, Links & more

Mathematical Digest: The mathematical magazine for South African mathematics learners and teachers

Math Forum Electronic Newsletter

Logic, Strategy & Geometry Games

Last modified by Michael de Villiers, 22 February 2023.

Please kindly consider making donations to help update & maintain this site.