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**Downloadable Articles, Papers & Materials**

**Michael de Villiers/Dynamic Mathematics Learning**

(Picture taken by Rishi Hansraj at UDW, approx. 1994)

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**Last Update: 6 April 2024**

Updated: Math Ed & Math Articles, Dynamic Geometry Sketches

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**SKETCHPAD IS NOW FREE**, and a copy can be downloaded from FREE DOWNLOAD OF GEOMETER'S SKETCHPAD. Actvity Modules, *Sketchpad* sketches & other resources are also freely available at the preceding link.

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Bi-Annual Math e-Newsletter (Dec 2019)

Updated!!!Java Sketchpad, Cabri 3D, Cinderella & GeoGebra Sketches - interactive geometry on the web

Have a look at some of my geometry students' comments by clicking Feedback

For published reviews of my *Rethinking Proof with Sketchpad* book: Rethinking Proof Reviews

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**Most cited papers and books according to Google Scholar** (3 April 2024)

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**PODCAST**: Listen to the Podcast of my approx. 46 min *Interview* by Michelle Cirillo (University of Missouri) at ICME 13 in Hamburg in July 2016 about my career in Mathematics Education. (If problems are experienced playing it through completely in your browser, ensure that you have 'Allowed Flash Player' in your settings and preferences).

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**Free Articles/papers/talks**

Please note that most of the articles/items below are in pdf format.

The articles and materials below may be used for non-profit educational purposes only, provided that their source/origin is properly referenced at all times.

Though the items have been roughly classified into two broad categories of Mathematics Education and Mathematics, there is sometimes considerable overlap between the two categories.

They have also not been arranged in chronological or any other specific order.

Kindly please inform me of any dynamic sketches, links or downloads that are NOT working properly.

NEW*!!! Revisiting the Van Hiele theory. Invited Plenary Paper at the 2017 EIEM Conference, University of Lisbon, Portugal, organized by the Portuguese Society for Research in Mathematics Education. Video Recording on YouTube: click here.

NEW*!!! Making Democracy Fair: The mathematics of voting and apportionment, Grades 7-12, Co-author: Leslie Nielsen, complete book, 1st published by Key Curriculum Press, 1997, republished by Lulu Publishers, 2012, 169 pp, pdf, 13.6 MB.

NEW*!!! A Mathematical Look at "Voting Power" in *Pythagoras*, April 1994, pp. 33-36, pdf, 7.9 MB. The pdf also includes the "The voting power of the various parties at national and provincial level", in *Spectrum*, Aug 1995, pp. 36-37, analyzing the 1994 election result.

NEW!!! Intersecting Circles Investigation in *At Right Angles*, November 2022, pp. 67-73, pdf. Co-authors, James Metz & Brad Uy, Hawaii. Dynamic Sketch URL: click here.

NEW!!! A Forgotten Theorem for Triangle Similarity? in *Learning and Teaching Mathematics*, No. 33, 2022, pp. 16-19, pdf. Dynamic Sketch URL: click here.

NEW!!! The Value of using Signed Quantities in Geometry in *Learning and Teaching Mathematics*, No. 29, 2020, pp. 30-34, pdf. Dynamic Sketch URL: click here.

NEW!!! Rethinking Proof with Sketchpad, complete book, Key Curriculum Press/McGraw-Hill, 2012, pdf, 4.1 MB.

NEW!!! Proof and Proving in Mathematics Education: 19th ICMI Study, complete book, Springer, 2012, pdf, 7.3 MB. Co-Editor: Gila Hanna, Canada.

Does dynamic computer verification imply the end of proof? My Plenary Talk via Zoom, 26 October 2021, 2nd NaCoME Conference organized by the Mathematics Education Study Program of Sriwijaya University, Indonesia.

A Diagonal Property of a Rhombus Constructed from a Rectangle in *Learning & Teaching Mathematics*, Dec 2018, pp. 26-27, PDF; co-author Jim Metz, Hawaii. Dynamic Sketch URL: click here.

Tiling with a Trilateral Trapezium and Penrose Tiles Defines a "Trilateral Trapezium" and a "Triangular Kite" and then explores a tiling of respective special cases of these quadrilaterals, namely, a golden trilateral trapezium and its dual, a convex Penrose tile. *Learning & Teaching Mathematics*, March 2019, pp. 6-10, PDF.

Why does it Work? A Mathematical Explanation and Further Generalization of a Card Trick A card trick is mathematically explained using elementary school algebra and then generalized. *Learning & Teaching Mathematics*, Dec 2018, no. 24, pp. 30-31, PDF.

A cyclic Kepler quadrilateral and the Golden Ratio Defines a "Kepler quadrilateral" as a quadrilateral with sides in geometric progression of √φ, and proves an interesting property when it is cyclic. *At Right Angles*, March 2018, pp. 91-94, pdf. Dynamic Sketch URL: click here.

An Example of Constructive Defining: From a Golden Rectangle to Golden Quadrilaterals and beyond Investigates different possible definitions for various 'golden' quadrilaterals that involve some aspect of the golden ratio. *At Right Angles*, Vol 6, no 1, March 2017, pp. 64-69; Vol 6 No 2, August 2017, 1.9 MB, pdf. Dynamic Sketch URL: click here.

Reflecting on a 2nd Round 2013 SA Mathematics Olympiad Problem Presents a few different solutions as well as a more general underlying theorem. *Learning & Teaching Mathematics*, Dec 2013, No. 14, pp. 34-35, 193 KB, PDF. Dynamic Sketch URL: click here.

A Multiple Solution Task: a SA Mathematics Olympiad Problem Explores some different solutions for a Round 1, South African Math Olympiad Problem from 2016. *Learning & Teaching Mathematics*, June. 2016, No. 20, pp. 18-20, 258 KB, pdf. Dynamic Sketch URL: click here.

Order of Operations: The Math and the Myth Discusses why a well-defined order of arithmetic operations is needed, namely, to avoid ambiguity in interpreting a given number sentence. *AMESA News*, Nov. 2015, Vol. 57, pp. 9-11, 180 KB, PDF.

I have a dream ... Crossed Quadrilaterals: A Missed Lakatosian Opportunity? Initially presents some prose in the form of a critical satire of the still dominant practice in mathematics education to exclude crossed quadrilaterals from the set of quadrilaterals. The accompanying, follow-up paper briefly discusses this practice as a missed opportunity to engage students in a genuine Lakatosian experience. *Philosophy of Mathematics Education Journal, July 2015.
*, 451 KB, pdf. Dynamic Sketch URL: click here

Slaying a Geometrical Monster: Finding the Area of a Crossed Quadrilateral. Discusses how to find the area of a crossed quadrilateral and to prove Varignon's theorem in the case when the quadrilateral becomes crossed. *Scottish Mathematical Council Journal*, 44 (Dec 2014), pp. 71-74; *Learning and Teaching Mathematics*, No. 18, June 2015, pp. 23-28, 332 KB, pdf. Dynamic Sketch URL: Varignon's Theorem: the crossed case.

Conjecturing, refuting and proving within the context of dynamic geometry Two convincing geometry conjectures of points appearing to be collinear are given that easily pass a normal 'drag-test' with dynamic geometry software. Counterexamples only became evident after dilating the figures by a very large scale factor of about 100:1. *Learning & Teaching Mathematics*, Dec 2014, No. 17, pp. 20-26 pp, 795 KB, pdf. Co-author: Nic Heideman. Dynamic Sketches URL: Investigating incentres of some iterated triangles.

Over and over again: two geometric iterations with triangles Presents and proves two interesting geometric iterations that converge to equilateral triangles. Suitable dynamic geometry exploration for Grade 10-12 learners and up. *Learning & Teaching Mathematics*, July 2014, No. 16, pp. 40-45, 426 KB, pdf. Dynamic Sketches URL: Over and over again: two geometric iterations.

Learners' conceptualisation of the Sine function during an introductory activity using Sketchpad at Grade 10 level Discusses research that revealed some intuitions as well as misconceptions about the sine function. Presented by Janeeshla Jugmohan at the Educational Society of South Africa (EASA) Congress, Univ. of KwaZulu-Natal, 13-16 January 2009.

A Trapezium Theorem Generalized Generalizes the so-called Midpoint Trapezium Theorem to a simple inequality for any quadrilateral. Suitable exploration for Grade 9-10 learners and up. *At Right Angles*, Nov 2013, Vol. 2, No. 3, pp. 53-56, 213 KB, pdf. Dynamic Sketch URL: Midpoint Trapezium Theorem Generalized.

A dynamic approach to quadrilateral definitions By Rajendren Govender and Michael de Villiers. This study found that the evaluation of different definitions for a rhombus by accurate construction and measurement in a dynamic geometry context, enabled prospective mathematics students to achieve a better understanding of necessary and sufficient conditions, as well as the ability to more readily find counter-examples, and to recognise and improve on uneconomical definitions. *Pythagoras*, 2004, No. 58, pp. 34-45, 238 KB, PDF.

Competencies in using Sketchpad in Geometry Teaching and Learning: Experiences of preservice teachers By Mdutshekelwa Ndlovu, Dirk Wessels and Michael de Villiers. Describes some qualitative research used to investigate preservice teachers' emerging competencies. *African Journal of Research in Mathematics, Science and Technology Education*, 2013, Vol. 17, No. 3, 231-243, 479 KB, PDF.

Equality is not always best! Describes a classroom discussion which revealed the surprising discovery that the circumcentre is perhaps not the 'best choice' for building a water reservoir for 3 villages lying in the shape of an obtuse triangle, despite the circumcentre being equi-distant from all 3 villages. A 'better' solution is then presented. *Learning & Teaching Mathematics*, no. 14, June 2013. pp. 17-21, 397 KB, PDF

What is the function? Models the height of water being poured into a tilted container in a video clip with a simple differential equation to find the function. *Learning & Teaching Mathematics*, no. 12, June 2012. pp. 16-18, 418 KB, pdf. URL 1: click here & URL 2: click here.

An Example of the Explanatory and Discovery Functions of Proof My ICME 12 invited, regular paper discusses a variation of Viviani's theorem, originally suggested by a high school student, Clough, and how reflecting - looking back - on different proofs of the result leads to various different generalizations. *Pythagoras*, 33(3), Dec 2012. 8 pp, 639 KB, pdf. Dynamic Sketch URL: click here.

An Alternative Approach to Proof in Dynamic Geometry Chapter in 1998 Erlbaum book *Designing Learning Environments for developing Understanding of Geometry and Space*, edited by Richard Lehrer and Daniel Chazan. It critiques the traditional approach to proof which focusses mainly on the verification function of proof, and argues for an alternative, more meaningful introduction to proof, specifically in the context of dynamic geometry, by focussing on the explanatory function of proof. 25 pp, 5.3 MB, PDF

An instrumental approach to modelling the derivative in Sketchpad Discusses a hypothetical learning trajectory for modelling the derivative and related concepts in introductory calculus with dynamic mathematics software, and which was implemented with first-year non-mathematics major undergraduate science students, Ndlovu, M., Wessels, D., & De Villiers, M. (2011). Pythagoras, 2011, 3.8 MB, PDF

Simply Symmetric Briefly discusses and illustrates the value of symmetry in choice of definitions in geometry and in problem solving, in *Mathematics Teaching* & *Learning and Teaching Math*, 2011, 303 KB, pdf, Dynamic Sketch URL: click here.

Worksheet: Order of operations Worksheet and Teacher Notes to develop understanding of the need for 'order of operations' in order to avoid 'ambiguity', Pythagoras, 1992, 1.9 MB, PDF

2010 Van Hiele plenary paper now available in Portuguese & Croatian

A Comparative Study of two Van Hiele Testing Instruments Poster paper presented by a then student of mine, Eddie Smith, at PME-13, Paris 1989

Some Reflections on the Van Hiele theory. Invited paper and video made at the Program Post-Graduate Studies in Mathematics Education at the Pontifical Catholic University of Sao Paulo, Brazil in April 2010, as well as invited plenary at the Croatian Mathematical Conference in Zagreb, Croatia in June 2010

Border Patterns, Tessellations & Problem Solving with Transformations. Invited paper AMESA Congress, 2007, 1.1 MB, PDF

Research Evidence on Hierarchical Thinking, Teaching Strategies and the Van Hiele Theory: Some critical comments. RUMEUS Report, pp. 36, 1987, 1.5 MB, PDF

ICMI-19 Discussion Document: Proof and Proving in Mathematics Education. pp. 12, 2007/8, 136 KB, PDF

Problemsolving and Proving via Generalization Learning & Teaching Mathematics, no. 5. pp. 19-25, 2008, 900 KB, PDF

Mathematical Applications, Modeling and Technology Invited Plenary at Annual Congress of AMESA, July 2007, AMESA Proceedings, 1.2 MB, PDF

Some pitfalls of dynamic geometry Discusses some pitfalls of using dynamic geometry in teaching geometry. Feb 2007, Learning & Teaching Mathematics, 276 KB, PDF

Recycling cyclic polygons dynamically Includes a download link to zipped Sketchpad sketches & deals with testing by construction, & proving the converse of a well-known cyclic quad theorem, & then considering some generalizations. May 2006, Mathematics in School, 64 KB, PDF.

Rol en Functie van het Bewys in de Dynamische Meetkunde (A somewhat modified and translated version into Dutch of my "Role & function of proof in Dynamic Geometry", pdf, 300 KB, EUCLIDES, Feb 2006, 81(4), pp. 184-188).

Mathematical Modeling and Proof (pdf, 24 KB, AMESA '04 paper. Co-author: Vimolan Mudaly).

Using dynamic geometry to expand mathematics teachers' understanding of proof 2004, IJMEST 35 (5), pp. 703-724, 176 KB.

Clough's Conjecture - A Sketchpad Investigation (Workshop worksheets on a variation of Viviani's theorem presented at AMESA 2004, Potchefstroom & NCTM 2004, Philadelphia). Dynamic Sketch URL: click here.

Rethinking Proof with Sketchpad - in Hebrew (pdf, 2003 in ALEH, 30, May, 19-26. Original English versions are further down below)

A further Pythagorean Variation on a Fibonacci theme (Mathematics in School, 2002, pdf)

All cubic polynomials are point symmetric (Learning & Teaching Mathematics, 2004, pdf). Dynamic Sketch URL: click here.

The Role and Function of a Hierarchical Classification of Quadrilaterals (For the Learning of Mathematics, 1994, pdf)

The role of experimentation in mathematics (AMESA 2003 paper)

A Sketchpad Investigation of "Arithmetic Sequence" Parabola (AMESA 2003 Workshop)

Developing understanding for different roles of proof in dynamic geometry (ProfMat 2002, pdf)

Para uma Compreensao dos Differentes Paeis La Demonstracao em Geometria Dinamica (ProfMat 2002, Portugal, pdf)

The Role and Function of Proof with Sketchpad (Preface to Rethinking Proof with GSP, 1999, pdf)

Constructive Evaluation of Definitions in a Sketchpad Context (Amesa 2002 paper, pdf file, co-author: Rajen Govender).

The Role and Function of Proof with Sketchpad ('99, pdf) (The traditional view that the only function of proof is that of verification is criticised, and it is argued that other important functions of proof such as explanation, discovery, systematization, etc. ought not to be neglected in the classroom).

The Role and Function of Proof in Mathematics ('90, pdf) (From *Pythagoras*, No. 24, Nov. 1990, pp. 17-24) (Original version of above article).

El Papel Y La Funcion de la Demostracion en Matematicas ('93, pdf) (Spanish translation of above article from *Epsilon*, No. 24, 1993, pp. 15-30)

Papel e Funcoes da Demostracao no Trabalho com o Sketchpad ('01, pdf) (Portuguese translation of '99 Proof article from *Educacao e Matematica*, No. 62, April 2001, pp. 31-36)

Children's Acceptance of Theorems in Geometry (pdf) (Some empirical data from a questionnaire regarding the grounds of children's acceptance of the truth of some geometric theorems. Poster paper presented at PME 16, 1992).

Why Proof in Geometry? (Mathematics in College, 1996, pdf file) (A short discussion of why proof is still necessary and useful in a convincing environment such as dynamic geometry).

Pupils' Needs for Conviction & Explanation within Geometry (pdf) (Revised and slightly expanded version of PME 15 paper, 1991)

Learner's Needs for Conviction and Explanation within the Context of Dynamic Geometry (Published in Pythagoras, 2000, co-author: Vimolan Mudaly, based on his empirical master's study related to student experiences of a Sketchpad activity).

To Teach Definitions or Teach to Define? (PME 1998 paper) (Distinguishes two different types of defining and presents some empirical data related to a teaching experiment that focussed on defining as a process).

The Role of Axiomatization in Mathematics & Mathematics Teaching (RUMEUS Report, 1986, pdf) (Two different types of axiomatization are analysed in relation to their functions in mathematics and mathematics education).

Alternative Instructional Strategies for Geometry Education: A Theoretical Study (The theoretical background to the University of Stellenbosch Experiment in Mathematic Education (USEME) project in 1977/1978 in the Cape Province).

A Fibonacci Generalization: A Lakatosian Example (PME '97 paper - pdf file) (Describes a generalization of the Fibonacci series which followed a Lakatosian path, and includes some philosophical comments).

The Future of Secondary School Geometry (SOSI '96 Plenary, pdf) (Outlines recent developments in geometry research, the Van Hiele theory and the impact of dynamic geometry on the possible future of secondary school geometry).

El Futuro de la Geometria en Secundaria ('99, pdf) (Spanish translation by Martin Acosta of article "The Future of Secondary School Geometry" from *Proof Newsletter*, Nov/Dec 1999)

Computer Verification vs Algebraic Explanation (Published in Pythagoras, 1993, pdf) (A simple junior high school example to illustrate the difference between verification and explanation).

Olympic Mathematics: Is it fair? (Published in Pythagoras, 1996, pdf) (Discusses the Women's 100 m. final at the 1996 Atlanta Olympic Games and whether one or two gold medals should've been awarded).

Transformations: A Golden Thread in School Mathematics (Published in Spectrum, 1993, pdf) (Discusses and illustrates various examples of the application of transformations such as isometries, similarities and affinities to the study of the symmetry of border patterns, tessellations, graphs and the production of alternative proofs and definitions). (Apologies for poor quality of reproduction & some typos)

The Role of Technology in Mathematical Modelling (Published in Pythagoras, 1994, pdf) (Various examples of the use of technology such as graphing calculators, *Sketchpad* and symbolic algebra software in the modelling and solution of some real world problems are given).

NEW!!! Conway's Circle Theorem as a Special Case of a More
General Side Divider Theorem *Learning & Teaching Mathematics*, June 2023, No. 34, pp. 28-30, PDF. A dynamic geometry sketch is available online: click here to access.

NEW!!! Dao Than Oai's generalization of Napoleon's theorem *Global Journal of Advanced Research on Classical and Modern Geometries*, Vol.12, (2023), Issue 1, pp.158-168, pdf. Co-authors: Hans Humenberger, Austria; Berthold Schuppar, Germany. Dynamic Sketch URL: click here.

NEW!!! An extension of the six-point circle theorem for a generalised Van Aubel configuration *The Mathematical Gazette*, Vol. 106, No. 567, Nov. 2022, pp. 400-407, pdf. Co-author: Dario Pellegrinetti, Germany/Italy. Dynamic Sketch URL: click here.

NEW!!! Jha and Savaran's generalisation of Napoleon's theorem *Global Journal of Advanced Research on Classical and Modern Geometries*, Vol.11, (2022), Issue 2, pp.190-197, pdf. Co-authors: Hans Humenberger, Austria; Berthold Schuppar, Germany. Dynamic Sketch URL: click here.

NEW*!!! Forgotten Properties of the Van Aubel & Bride’s Chair Configurations *International Journal of Geometry*, Vol. 10 (2021), No. 3, pp. 5 - 10, pdf. Co-author: Dario Pellegrinetti, Germany/Italy.

*NEW!!! An interesting theorem related to a hexagon with opposite sides that are parallel *International Journal of Mathematical Education in Science & Technology*, 2021, Volume 52, Issue 1, pp. 155-163, PDF. A dynamic geometry sketch is available online: click here to access.

NEW!!! Another Bicentric Quadrilateral Construction *Learning & Teaching Mathematics*, Dec 2021, No. 31, pp. 40-41, pdf. Dynamic Sketch URL: click here to access.

NEW!!! Ghosts of a Problem Past *At Right Angles*, March 2021, pp. 105-111, pdf. An old, challenging problem in a different guise is revisited & multiple solutions are given. Co-author: Hans Humenberger, Austria. A dynamic geometry sketch is available online: click here to access.

*NEW!!! Proof as a means of discovery: Area parallelogram partition theorem *International Journal of Mathematical Education in Science & Technology*, 2020, Vol. 51, Issue 3, pp. 451-455, PDF. A dynamic geometry sketch is available online: click here to access.

An Interesting Collinearity *Learning & Teaching Mathematics*, July 2019. No. 26, pp. 28-30, PDF. Co-author: Piet Human, South Africa. An application of Desargues and Pappus. Dynamic Sketch URL: click here.

A
surprising 3D result involving a hexagon *Mathematical Gazette*, July 2018. 2018, Vol. 102, No. 554, pp. 328 - 330, PDF. Co-author: Heinz Schumann, Germany. A link to a YouTube video giving a 3D illustrations of the result is available online: click here to access.

A Merry-Go-Round the Triangle Illustrates the value of problem-posing, e.g. how asking a 'what-if' question leads to an interesting result. *Learning & Teaching Mathematics*, June 2018, No. 24, pp. 17-20, 446 KB, PDF. Co-author: John Silvester, UK. Dynamic Sketch URL: click here.

*NEW!!! An explanatory, transformation geometry proof of a classic treasure-hunt problem and its generalization *International Journal of Mathematical Education in Science & Technology*, 2016, PDF. A dynamic geometry sketch is available online: click here to access.

A Multiple Solution Task: Another SA Mathematics Olympiad Problem Explores some different solutions for a Round 2, South African Math Olympiad Problem from 2016. *Learning & Teaching Mathematics*, June 2017, No. 22, pp. 42-46, 464 KB, pdf. Dynamic Sketch URL: click here.

Enrichment for the Gifted: Generalizing Some Geometrical Theorems & Objects (pdf, 385 KB, 2016 in *Learning and Teaching Mathematics* (Dec)). This ICME 13 presented paper discusses the importance of encouraging generalization in mathematics as well as engaging learners in constructive defining. A dynamic geometry sketch related to the generalization of a rectangle to a hexagon is available online at Dynamic sketch.)

Flashback to the Past: a 1949 Matric Geometry Question (pdf, 713 KB, 2015 in *At Right Angles* (Nov) and *Learning and Teaching Mathematics* (Dec)). This paper explores a 1949 Grade 12 Geometry Exam question, by looking at a generalization and some variations. Dynamic Sketch URL: click here.)

A trisection concurrency: a variation on a median theme (pdf, 881 KB, *Learning and Teaching Mathematics*, Dec 2013, pp. 36-40). Co-author: Shunmugam Pillay. Or alternatively, download and read in iBook format (2 MB). This result deals with an interesting concurrency found experimentally when the sides of a triangle were trisected, its proof and further generalization. Dynamic Sketch URL: click here.)

A variation of Miquel's theorem and its generalization (pdf, 233 KB, *Mathematical Gazette*, July 2014, pp. 334-339. This note describes a little known 'converse-like' variation of Miquel's theorem and its generalization to higher polygons. Dynamic Sketches URL: click here.)

Quasi-circumcenters and a Generalization of the Quasi-Euler Line to a Hexagon (pdf, 45 KB, *Forum Geometricorum*, Vol 14(2014), pp. 233-236. This short note generalizes the quasi-Euler line of a quadrilateral to a hexagon involving its quasi-circumcenter, its quasi-orthocenter and its lamina centroid. Dynamic Sketch URL: click here.)

An Investigation of Some Properties of the General Haag Polygon (pdf, 553 KB, *Mathematics in School*, 43(3), May 2014, pp. 15-18. Investigates the generalization of some angle and concurrency properties of the tessellating Haag hexagon. Dynamic Sketches URL: click here.)

Generalizing a theorem of Arsalan Wares (pdf, 279 KB, *The Scottish Mathematical Council Journal*, no. 42, 2012, pp. 41-43. Presents some generalizations of Wares' theorem - involves similar triangles. Dynamic Sketches URL: click here.)

3D Generalizations of Viviani's theorem (pdf, 385 KB, *The Mathematical Gazette*, Vol, 97, no. 540, Nov 2013, pp. 441-445. Presents some 3D generalizations of Viviani's theorem. Dynamic geometry sketches with *Cabri 3D* (for Windows only - download required plug-in) are available at Dynamic 3D.)

LTM Feb 2006 Cover Diagram: A semi-regular side-gon (pdf, 287 KB, *Learning & Teaching Mathematics*, no. 12, June 2012, pp. 34-35. Discusses some mathematical properties of the semi-regular octagon on the 2006 cover of the LTM journal). Dynamic Sketch URL: click here.

Another proof of De Opgave 2011 (pdf, 164 KB, *Euclides*, Dec 2012. Provides another proof of a geometry problem posed by Dick Klingens in 2011, & shows that it is also a special case of a similar rectangle generalization of Van Aubel's theorem). Dynamic Sketch URL: click here.

Relations between the sides and diagonals of a set of hexagons (pdf, 586 KB, *Mathematical Gazette*, July 2012. Explores the analogous case for parallelo-hexagons of the parallelogram theorem that the sum of the squares of the sides of a parallelogram is equal to the sum of the squares of its diagonals). Dynamic Sketch URL: click here.

Generalizing a problem of Sylvester (pdf, 258 KB, *Mathematical Gazette*, March 2012. Generalizes Sylvester's result that the resultant of three equal forces *OA*, *OB* and *OC* acting on the circumcentre *O* of a triangle *ABC*, is the force represented by *OH*, where *H* is the orthocentre of the triangle). Dynamic Sketch URL: click here.

Equi-angled cyclic & equilateral circumscribed polygons (pdf, 565 KB, *Mathematical Gazette*, 2011. Generalizes *rectangles* & *rhombi* to higher polygons). Dynamic Sketch URL: click here.

Feedback: Equi-angled cyclic & equilateral circumscribed polygons (pdf, *Mathematical Gazette*, 2011. Generalizes *isosceles trapezia* & *kites* to higher polygons). Dynamic Sketch URL: click here.

From the Fermat points to the De Villiers points of a triangle (pdf, 1.2 MB, AMESA Congress July 2009 paper. Describes the discovery and proof of the so-called De Villiers points of a triangle, tracing its origin from the Fermat points of a triangle). Dynamic Sketch URL: click here.

Vanaf die Fermat punte na die De Villiers punte van 'n driehoek (pdf, 344 KB, *SA Tydskrif vir Natuurwetenskap & Tegnologie*, Sept 2010. Beskryf die ontdekking en bewys van die sogenaamde De Villiers punte van 'n driehoek, en trek die oorsprong terug na die Fermat punte van 'n driehoek). Dynamic Sketch URL: click here.

Some hexagon area ratios: Problem solving by related example (pdf, 324 KB, Mathematics in School, 2010. Describes the discovery and proof of an interesting area ratio result for a parallelo-hexagon). Dynamic Sketch URL: click here.

Generalizing the Nagel line to Circumscribed Polygons by Analogy & Constructive Defining (pdf, 548 KB, Pythagoras, 2008. First discusses the genetic approach & the importance of the history of math, reasoning by analogy, and the role of constructive defining before presenting the result). Dynamic Sketch URL: click here.

LTM Cover Problem (pdf, 248 KB, Learning & Teaching Mathematics Journal, 2009. Gives a proof of the result, and two additional problems). Dynamic Sketch URL: click here.

A Further Generalization of Turnbull's Theorem (pdf, 464 KB, IJMEST, 1993. Generalizes the familiar "opposite angle supplementary" theorem for cyclic quadrilaterals to other cyclic polygons as well as a dual for tangential polygons). Dynamic Sketch URL: click here.

Solving a Locus Problem via Generalization (pdf, 20 KB, Reflections, Spring 2008, pp. 20-21. Applies circle geometry and similarity to solving a locus problem).

A Question of Balance: An Application of Centroids (pdf, 416 KB, Mathematical Gazette, Nov 2007, pp. 525-528, and March 2008. Applies similarity and centroids to proving a Van Aubel type result about similar quadrilaterals on the sides of a quadrilateral). Dynamic Sketch URL: click here.

An example of the discovery function of proof (pdf, 32 KB, Mathematics in School, Sept 2007, pp. 9-11. Generalizes a 9th grader's discovery for a triangle to a convex quadrilateral). Dynamic Sketch URL: click here.

A hexagon result & its generalization via proof (pdf, 88 KB, The Montana Mathematics Enthusiast, June 2007, Vol. 4, No. 2. Illustrates the discovery function of proof by generalizing a hexagon result about centroids to a generalization of Varignon's theorem). Dynamic Sketch URL: click here.

Further Reflection on a SA Mathematics Olympiad Problem (pdf, 40 KB, Learning & Teaching Mathematics, March 2007, pp. 25-27. Discusses underlying hidden theorem related to a special type of quadrilateral, and how this in turn can be viewed as a special case of Dandelin's Theorem.)

More on hexagons with opposite sides parallel (Contains link to a Sketchpad sketch, pdf, 120 KB, Mathematical Gazette, Nov 2006). Dynamic Sketch URL: click here.

A generalization of the Spieker circle and Nagel line (pdf, 100 KB, Pythagoras, July 2006). Dynamic Sketch URL: click here.

A generalization of the Nine-point circle and Euler line (Contains a further generalization of the Euler line and a synthetic proof, pdf, 40 KB, Pythagoras, Dec 2005). Dynamic Sketches URLs: click here or click here.

The nine-point conic: a rediscovery and proof by computer (Describes a generalization of nine-point circle and associated Euler line & proof by computer, pdf, 216 KB, IJMEST, Jan 2006). Dynamic Sketches URLs: click here or click here.

Feynman's Triangle: Some feedback and more (pdf, 16 KB, Math Gazette, March 2005). Dynamic Sketch URL: click here.

A dual to a British Mathematical Olympiad problem (pdf, 12 KB, March 2002, Math Gazette). Dynamic Sketch URL: click here.

Ezit's Rule: An Explanation & Proof (From LTM, Feb 2005, PDF, 20 KB)

An Extended Hierarchical Classification of Quadrilaterals (An excerpt from my Some Adventures In Euclidean Geometry booklet, 8 KB, pdf). Dynamic Sketch URL: click here.

Crocodiles and Polygons: A generalization of Viviani's theorem to equi-angled polygons. (Mathematics in School, March 2005, pdf). Dynamic Sketches URLs: click here or click here.

Stars: A Second Look (pdf, 1999 article in *Mathematics in School* as a short response to another article on the interior angle sum of star polygons). Dynamic Sketch URL: click here.

An interesting property of a quadrilateral circumscribed around a circle. (An excerpt from my book Some Adventures in Euclidean Geometry, 1996, pdf). Dynamic Sketch URL: click here.

'n Bewys deur matematiese induksie van 'n bekende bewering - in Afrikaans (pdf, 1988, Spectrum 26(3), August, p.38. (Will be rewritten in English at some stage & posted here. Alternatively, read Alwyn Olivier's classroom notes (1999), which deals (among others) with the same result, at: Circles, Regions and Chords).

A Fibonacci Generalization and its Dual (2000, IJMEST, pdf). Dynamic Sketch URL: click here.

Generalizations of maltitudes (IJMEST, 1999, pdf) (includes a generalization of Euler line to cyclic polygons). Dynamic Sketch URL: click here.

The Affine Equivalence of Cubic Polynomials (1992, IMSTUSNews; 2003, KZN Math Journal, pdf). Dynamic Sketch URL: click here.

How does the wind affect road-running achievement? (1991, The Physics Teacher, pdf)

From nested Miquel Triangles to Miquel Distances (2002, Mathematical Gazette, pdf). Dynamic Sketch URL: click here.

Dual Generalizations of Van Aubel's Theorem (1998, Mathematical Gazette, pdf). Dynamic Sketch URL: click here.

A Generalization of the Fermat-Torricelli Point (1995, Mathematical Gazette, pdf). Dynamic Sketch URL: click here.

A Further Generalization of the Fermat-Torricelli Point (1999, Mathematical Gazette, pdf). Dynamic Sketch URL: click here.

Generalizing Van Aubel using Duality (2000, Mathematics Magazine, pdf file). (Presents a summary of some dual generalizations of Van Aubel's quadrilateral theorem to similar rhombi, similar rectangles & similar parallelograms on the sides of a quadrilateral). Dynamic Sketch URL: click here.

A Dual to Kosnita's Theorem (Mathematics & Informatics Quarterly, 1996, Provides a dual to Kosnita's theorem and uses a generalization of the Fermat-Torriceli point to establish the result, as well as Kosnita's theorem). Dynamic Sketch URL: click here.

A dual, and generalisations, of a Sharp result (KwaZulu-Natal Mathematics Journal, 2000, pdf file) (A side-angle dual to a problem in the Sharp Competition in the Mathematical Digest is formulated and proved). Dynamic Sketch URL: click here.

Overlapping Circles (KwaZulu-Natal Mathematics Journal, 2000, pdf file) (Uses Ceva's theorem and power lines to prove Haruki's theorem). Dynamic Sketch URL: click here.

Mathematical Treasure Hunting (KwaZulu-Natal Mathematics Journal, 1999, pdf file). A more recent version of this paper was published in IJMEST in 2016. (The power of transformation geometry is illustrated in not only the solution of an interesting, classical problem, but also in leading to a further generalization involving a spiral similarity). Dynamic Sketch URL: click here.

Place-kicking Locus in Rugby (Pythagoras, 1999, No. 49, Aug, 64-67, pdf file) (Examines the "best position" for a place kick in rugby and shows that the locus is a rectangular hyperbola).

A Generalization of an IMTS problem (KZN Mathematics Journal, 1999, pdf file) (A generalization of an International Mathematical Talent Search problem which involves ratios of areas). Dynamic Sketch URL: click here.

Game, Math, and Luck! (Pythagoras, 2001, No. 53, pp. 14-17, pdf file) (Provides a simple mathematical model, using probability, of the point scoring system in tennis & then analyzes the so-called 'no-add' or 'short-deuce' rule.)

Exploring Loci on Sketchpad (Pythagoras, 46/47, Dec 1998, 71-73, pdf file) (Three examples are given, including the classic approach to a parabola as the locus of all points equidistant from a point and a line).

Problem Posing Variations on Fermat's Last Theorem (KZN Mathematics Journal, Dec 1995, 1(1), 16-18, pdf file) (Some simple variations on Fermat's Last Theorem are posed).

An Interesting Duality in Geometry (AMESA Congress Proceedings, 1-5 July 1996, 345-350, pdf) (The side-angle duality in Euclidean geometry is briefly discussed in relation to quadrilaterals). Dynamic Sketch URL's: click here or click here.

The Affine Invariance & Line Symmetries of the Conics (Australian Senior Mathematics Journal, 1993, Vol 7, no 2, pp 32-50, pdf) (Please note converted from a fax) (This paper proves the affine invariance of the conics and derives formulae for the axes of symmetry of the general conics using transformation geometry). Dynamic Sketch URL: click here.

A Sketchpad Discovery involving Triangles and Quadrilaterals (KZN Mathematics Journal, 1998, pdf) (An interesting further extension of Varignon's theorem & suitable for a high school geometry investigation). Dynamic Sketch URL: click here.

Conjecture & Proof with Sketchpad: a Case Study (Pythagoras, 37, August 1995, pp. 30-32, pdf) (Another investigation suitable for high school geometry learners involving points on the incircle & circumcircle of an equilateral triangle).

All Parabolas Similar? Never! (Pythagoras, Aug 1994, 34, 26-30, pdf) (Provides a heuristic description of the author's surprising, personal (re)discovery, that all parabola are similar, i.e. any parabola can be translated and then enlarged or reduced to map onto any other). Dynamic Sketch URL: click here.

**More articles (not yet updated, links need to be changed, etc.)**

**Please click on the links below for other older articles & publications.**

SKETCHPAD COMPETITION SOLUTIONS

READER INVESTIGATIONS from KZN AMESA MATHEMATICS JOURNAL

**Sketchpad Sketches (not yet updated, links need to be changed, etc.)**

*Experience the POWER and BEAUTY of moving geometric figures; i.e. DYNAMIC GEOMETRY!*

Download FREE Sketchpad 3 sketches for investigations or demonstrations by clicking on SKETCHPAD 3 SKETCHES

Download FREE Sketchpad 4 sketches for investigations or demonstrations by clicking on SKETCHPAD 4 SKETCHES

Dynamic Geometry sketches for my paper THE ROLE OF PROOF IN INVESTIGATIVE, COMPUTER-BASED GEOMETRY: SOME PERSONAL REFLECTIONS in the MAA Book "Geometry Turned On: Dynamic Software in learning, Teaching, and Research", 1997, edited by James King & Doris Schattschneider (as well as other sketches & files from the book) can be found at Geometry Turned On .

**My Favorite Links**

ASSOCIATION FOR MATHEMATICS EDUCATION OF SOUTH AFRICA(AMESA)

MATHEMATICIANS BORN OR DIED TODAY and other history of math

GEOMETRY STEP BY STEP FROM THE LAND OF THE INCAS

FIBONACCI A site on FIBONACCI numbers and the GOLDEN SECTION is a useful resource for encouraging students into math.

GEOMETRY in ART & ARCHITECTURE

** Comprehensive List of Links to Math Sites**

(links still need to be fixed)

**Cartoon**

**Quote of the Month**

Contact me at: profmd @ mweb.co.za (use without spaces)