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**Dynamic Mathematics Explorations for Students**

**(Last Update: 5 January 2023)**

*"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!"*

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

15% off Print Books till 25 February 2022 **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.

*********

**NOTE**: Please WAIT while the applets at the links below load. If a security pop-up menu appears in your browser, please choose **RUN/ALLOW** to let the applets run properly. *It is completely safe & can be trusted*. If you have the very latest Java on your PC or Apple Mac, and experience problems with the applets loading, please go *here* for additional information on Java settings or downloading a bookmarklet that should resolve the issue.

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

Anele Moli's Method: Constructing an equilateral triangle Easy

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

Area Ratios of Inscribed Polygons Medium-Hard

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

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

Crossed Quadrilateral Properties Medium

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 **(NEW!*)**

Euclid 1-43 Parallelogram Area Theorem Easy-Medium

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!)***

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.

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 Explorations
Easy

Hippocrates' Theorem Easy **(Updated!*)**

Hypercube-Cross
Polytope 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 triangles
Medium - Hard **(Updated!*)**

The Lux problem
Medium - Hard **(NEW!*)**

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

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!*)**

Parallelogram
Trio Medium

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
Quadrilateral 1 Medium

Pythagorean
Quadrilateral 2 Easy

Pythagorean
Quadrilateral 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

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

Trapezoid Explorations
Medium

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 **(Updated!)***

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

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, 22 February 2023.

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