**Dynamic Mathematics Explorations for Students**

**(Last Update: 23 Jan 2018)**

*"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.*

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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

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

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 *(Updated with more solutions and generalizations)**

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

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

Side trisection triangle concurrency Easy - Medium

Similar
Triangles Easy

Square Trigonometry Easy

Tangents
Easy

Thabit's Generalization of Pythagoras. 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

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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, 23 Jan 2018.

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