"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
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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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Alternate Angles Sum Cyclic Hexagon and Alternate Sides Sum Circumscribed Hexagon Easy - Medium
Anele Moli's Method: Constructing an equilateral triangle Easy
Another concurrency related to the Fermat point of a triangle Medium
Another Construction of a Parallel-hexagon with Opposite Sides in Same Ratio (and a Golden Ratio parallel-hexagon) Easy
Another parallelogram area ratio Easy
Area Formula for Quadrilateral in terms of its Diagonals Easy
Area Ratios of Inscribed Polygons Medium-Hard
Bicentric Quadrilateral Construction Challenge Medium - Hard
Bicentric Quadrilateral Property Medium
Bottema's Theorem aka Carnot's Theorem Easy-Medium
Bride's Chair Concurrency & Locus Hard
British Mathematics Olympiad Concurrency Problems Medium
Bug Escape Problem Easy
Circle Circumference Result Easy-Medium
Circle Concurrencies Easy-Medium
Circumscribed Quadrilateral Side Divider theorem and Cyclic Quadrilateral Angle Divider theorem Easy-Medium
Classifying, Exploring, Constructing & Defining Quadrilaterals Easy-Medium
Clough's Conjecture Medium-Hard
Collinear Conjecture Easy-Medium
Concurrent Angle Bisectors of Quadrilateral Easy
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
Crossed Quadrilateral Properties Medium
Cyclic Crossed Pentagon Problem Easy
Cyclic Quadrilateral Arcs Theorem Hard
Cyclic Quadrilateral Difference of Squares Theorem Hard
Cyclic Quadrilateral Rectangle Result Medium
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
A Diagonal Property of a Rhombus constructed from a Rectangle Easy
Dirk Laurie Tribute Problem Medium
Double Angle Triangle Medium
Dynamic Art: A Parquet Deformation Tiling An art project idea
Equi-inclined Lines Problem Easy
The Equi-partitioning Point of a Quadrilateral Easy-medium
Euclid 1-43 Parallelogram Area Theorem Easy-Medium
Euler line proof Easy
Extangential Quadrilateral 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.
Finsler-Hadwiger Theorem plus Gamow-Bottema's Invariant Point Medium-Hard
Forgotten Similarity Theorem Easy
A Fundamental Theorem of Similarity Easy-Medium
Further properties of a particular hexagon Easy-Medium
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
Hippocrates' Theorem Easy (Updated!*)
Hypercube-CrossPolytope Duality Easy
IMO 2014 Problem 4 (Geometry) Medium-Hard
Interior Angle Sum of Polygons: a General Formula Easy-Medium
Interesting Collinearity Medium-Hard
Interesting property of the number e Medium
Intersecting Circles Easy
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
The Lux problemMedium - Hard
Matric Geometry Exam Question - 1949 Easy - Medium
Maximum area of quadrilateral problem Easy
Midpoint Trapezium (Trapezoid) Theorem GeneralizedEasy-Medium
Minimum Area of Miquel Circles Centres Triangle Medium
Miscellaneous Transformations (of geometric figures & graphs) Easy - Medium
Napoleon Variation Problem Medium
Maximising the Area of the 3rd Pedal Triangle in Neuberg's theoremHard
Nine-point centre (anticentre or Euler centre) & Maltitudes of a Cyclic Quadrilateral Hard
Orthocentre Quadrilateral of a Quadrilateral Hard
Over and over: two geometric iterations with triangles Medium - Hard
Parabola Locus Theorem and Parabola Circumcircle Theorem Easy
Parallelogram Distances Easy
Parallelo-hexagon with Obtuse Angles Easy
Parallelogram Trio Medium
Parallel-pentagon and the Golden Ratio Medium - Hard
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
Pompe's Hexagon Theorem Medium-Hard
PythagoreanQuadrilateral 1 Medium
PythagoreanQuadrilateral 2 Easy
PythagoreanQuadrilateral 3 Easy
Quadrilateral Inequality involving Perimeter & Diagonals Medium
Quarter-Circle Investigation, Explanation & Generalization Easy
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
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
Side trisection triangle concurrency Easy - Medium
Similar Triangles Easy
Similar Parallelograms: A Generalization of a Golden Rectangle property Easy
Six Point Cevian Circle Hard (NEW!*)
Square Trigonometry Easy
TangentsEasy
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
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
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
Number Patterns: 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, 2 April 2024.
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