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(Last Update: 10 March 2024)

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"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 research 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 research mathematics, there is seldom a book at all - no ready-made questions and problems; even those one often has to invent oneself!" - Michael de Villiers

Math of Voting PDF...Rethinking Proof cover...Adventure Geometry...Boolean Algebra...Elementary Logic

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NEW! Problem solving and logical thinking games for young children Contact dynamic learn @mweb. co. za (without the spaces) to order.


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


Anele Moli's Method: Constructing an equilateral triangle Easy


Another concurrency related to the Fermat point of a triangle Medium (NEW*!)


Another Construction of a Parallel-hexagon with Opposite Sides in Same Ratio (and a Golden Ratio parallel-hexagon) Easy (NEW*!)


Another parallelogram area ratio Easy (NEW*!)


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


Area Ratios of Inscribed Polygons Medium-Hard (Updated*!)


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 (Updated!*)


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-turnMedium


Constant perimeter triangle formed by tangents to circle Easy-Medium


Construct a Parallel-Hexagon with GEONexT Medium


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


Cosine-Sine AngleRule Easy


Crossed Quadrilateral Properties Medium Updated!*


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-medium (NEW!*)


Euclid 1-43 Parallelogram Area Theorem Easy-Medium


Euler line proof Easy (NEW!*)


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!)*


Finsler-Hadwiger Theorem plus Gamow-Bottema's Invariant Point Medium-Hard (NEW!*)


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. (Updated!*)


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 ExplorationsEasy (Updated!*)


Hippocrates' Theorem Easy (Updated!*)


Hypercube-CrossPolytope 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 trianglesMedium - Hard (Updated!*)


The Lux problemMedium - Hard (NEW!*)


Matric Geometry Exam Question - 1949 Easy - Medium (Updated!*)


Maximum area of quadrilateral problem Easy


Midpoint Trapezium (Trapezoid) Theorem GeneralizedEasy-Medium


Minimum Area of Miquel Circles Centres Triangle Medium (NEW!*)


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


Napoleon Variation Problem Medium


Maximising the Area of the 3rd Pedal Triangle in Neuberg's theoremHard (NEW!*)


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!*)


Parallelo-hexagon with Obtuse Angles Easy (NEW!*)


Parallelogram Trio Medium


Parallel-pentagon and the Golden Ratio Medium - Hard (NEW!*)


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!*)


PythagoreanQuadrilateral 1 Medium


PythagoreanQuadrilateral 2 Easy


PythagoreanQuadrilateral 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




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 Area Formula in terms of angles, r & R Easy (NEW!*)


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


Number Patterns: 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, 2 April 2024.

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