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**Dynamic Mathematics Explorations for Students**

**(Last Update: 10 March 2024)**

"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

*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.

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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-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!**

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.

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!)**

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 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 *(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

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 *(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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