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(Last Update: 17 June 2017)

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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 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!"

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Alternate Angles Sum Cyclic Hexagon and Alternate Sides Sum Circumscribed Hexagon Medium

 

Anele Moli's Method: Constructing an equilateral triangle Easy

 

Area Ratios of Inscribed Polygons Medium-Hard

 

Bicentric Quadrilateral Property Medium

 

Bottema's Theorem Easy-Medium

 

Bride's Chair Concurrency & Locus Hard

 

British Mathematics Olympiad Concurrency Problems Medium

 

Bug Escape Problem Easy (NEW!)

 

Circle Circumference Result Easy-Medium

 

Circum Cyclic Quadrilateral Result and Cyclic Circum Quadrilateral Result Medium

 

Classifying, Exploring, Constructing & Defining Quadrilaterals Easy-Medium

 

Clough's Conjecture Medium-Hard

 

Collinear Conjecture Easy-Medium

 

Conic by Half-turn Medium

 

Construct a Parallel-Hexagon with GEONexT Medium

 

Cosine-Sine Angle Rule Easy (Updated)

 

Crossed Quadrilateral Properties Medium

 

Cyclic Crossed Pentagon Problem Easy

 

Cyclic Quadrilateral Arcs Theorem Hard

 

Cyclic Quadrilateral Rectangle Result Medium

 

Desargues' Theorem Easy-Medium

 

Descartes' Geometric Solving of Some Quadratics Easy-Medium

 

Double Angle Triangle Medium

 

Dynamic Art: A Parquet Deformation Tiling An art project idea

 

Euclid 1-43 Parallelogram Area Theorem Easy-Medium

 

Finding the Median and Centroid of a Triangle by Experimentation: A gravity simulation Use to explore.

 

Finding the centroid of a triangle with different weights at the vertices: A gravity simulation Use to explore.

 

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.

 

Gielis Super Shape Formula Medium.

 

Haag Hexagon and its generalization to a Haag Polygon Medium.

 

Hexagon Explorations Easy

 

Hippocrates' Theorem Easy

 

Hypercube-Cross Polytope Duality Easy

 

IMO 2014 Problem 4 (Geometry) Medium-Hard

 

Interior Angle Sum of Polygons: a General Formula Easy-Medium

 

Interesting property of the number e Medium

 

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

 

Japanese Circumscribed Quadrilateral Theorem Hard

 

Kinky Lines Easy

 

Jannie's Father's Farm Problem Easy - Medium

 

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

 

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

 

Parabola Locus Theorem and Parabola Circumcircle Theorem Easy

 

Parallelogram Trio Medium

 

Paul Yiu's problem and a generalization Medium

 

Perimeter of Hexagon inscribed in Triangle Easy

 

Pi in other geometries Medium

 

Platonic Solids plus Truncated Platonic Solids Easy

 

Pythagorean Quadrilateral 1 Medium

 

Pythagorean Quadrilateral 2 Easy

 

Pythagorean Quadrilateral 3 Easy

 

Quadrilateral Inequality involving Perimeter & Diagonals Medium

 

The quasi-circumcentre and quasi-incentre of a quadrilateral Medium

 

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

 

Rectangle Angle Trisection and Rhombus Angle Trisection Easy

 

SA Mathematics Olympiad Problem 2016, Round 1, Question 20 Medium - Hard (Updated with solution!)

 

SA Mathematics Olympiad Problem 2016, Round 2, Question 20 Medium - Hard (NEW!)

 

Side trisection triangle concurrency Easy - Medium

 

Similar Triangles Easy

 

Square Trigonometry Easy

 

Tangents Easy

 

Theorem of two intersecting circles or two adjacent isosceles triangles Medium - Hard

 

Theorem of Arsalan Wares Easy-Medium

 

Three Circle Geometry Theorem Proofs by Transformations Easy

 

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

 

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

 

What number comes next? Easy

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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, 17 June 2017.

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