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. So please use the problems to stimulate your mind - to ask 'what-if' questions, to specialize, to generalize, to vary, etc. Do not be scared to make conjectures of your own. Experiment and have fun inventing, discovering and proving your own results.

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

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

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.

*********

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

Anele Moli's Method: Constructing an equilateral triangle Easy (Updated*!)

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*!)

Another Property of an Opposite Side Quadrilateral Easy (NEW*!)

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

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

Attached Regular Pentagons form Congruent Equilateral Triangles Easy (NEW!*)

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

Bicentric Quadrilateral Property Medium

Some Properties of Bicentric Isosceles Trapezia & Kites Easy-Medium (NEW*!)

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

Bretschneider's Quadrilateral Area Formula & Brahmagupta's Formula Medium-Hard (NEW!)*

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-turn Medium

Constant perimeter triangle formed by tangents to circle Easy-Medium

Construct a Parallel-Hexagon with GEONexT Medium

Converse of Tangent-Secant Theorem (Euclid Book III, Proposition 36) Easy-Medium NEW!*

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

Cosine-Sine Angle Rule 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

Dào Thanh Oai's Perpendicular Lines Van Aubel Generalization Hard (NEW!*)

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.

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 (Updated!*)

Hindu Circle Dissection for finding Area of Circle Easy (NEW!*)

Hippocrates' Theorem Easy (Updated!*)

Hypercube-Cross Polytope Duality Easy

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

An extension of the IMO 2014 Problem 4 Medium (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!)

Invariant Product related to Triangles on sides of a Triangle plus Anghel's Hexagon Concurrency theorem Easy to hard (NEW!)

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

Japanese Circumscribed Quadrilateral Theorem Hard (Updated!)

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 (Updated!*)

Maximum area of quadrilateral problem Easy

Midpoint Trapezium (Trapezoid) Theorem Generalized Easy-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 theorem Hard (NEW!*)

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

Opposite Side Quadrilateral Properties by Kalogerakis Medium (NEW!*)

A Property of an Opposite Angle Quadrilateral Easy (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!*)

Pythagorean Quadrilaterals (including Orthodiagonal Quadrilaterals) Easy-Medium

Quadrilateral Inequality involving Perimeter & Diagonals Medium (Updated!*)

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!*)

Rugby Place-kicking Problem Easy - Medium (NEW!*)

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

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

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

Trapezium (Trapezoid) Visual Introduction Easy (Updated!*)

Trapezoid (Trapezium) Explorations Easy-Medium (Updated!*)

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

A Van Aubel like property of an Equidiagonal Quadrilateral Hard (NEW!)*

A Van Aubel like property of an Orthodiagonal Quadrilateral Hard (NEW!)*

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, 14 March 2025.

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