Some Adventures In Euclidean Geometry

220 Pages·2009·3.08 MB·English

by  Michael de Villiers

Some Adventures
in
Euclidean Geometry

Michael de Villiers

1st DRAFT, JULY 1994, 2nd DRAFT, JAN 1996
SEPTEMBER 2009
© 2009 Michael de Villiers (trading as Dynamic Mathematics Learning)
All rights reserved.
ISBN 978-0-557-10295-2
“Life without geometry is pointless.”

© 2009 Michael de Villiers (trading as Dynamic Mathematics Learning)
All rights reserved.
ISBN 978-0-557-10295-2
Preface
"... the spirit of mathematics ... is active rather than contemplative - a spirit of
disciplined search for adventures of the intellect." - Alfred Adler (1984:7)
To the author also, mathematics is an exciting never-ending adventure, always full of lovely
and beautiful surprises. It is like wandering in an uncharted jungle where one never knows
what sparkling brooks, cascading waterfalls, exotic plants and strange animals may lurk just
around the next corner. If one is prepared to keep an open mind, asking questions and
continually exploring, mathematics provides an inexhaustible source of inspiration and
stimulation; there is always something new to discover, or at the very least, new ways of
looking at old results. Of course, like in any jungle, there is also danger in various forms, and
one has to constantly guard in one's explorations against false conclusions and conjectures.
Unfortunately many people seem to regard mathematics in general as a boring, dead subject
with nothing new to discover. In particular with regard to geometry, they believe that the old
Greeks and other ancient civilizations have already discovered all there is to discover in
geometry about 2000 years ago. This is however not the case; many interesting and beautiful
geometric results have been discovered during the past 300 years. Apart from dealing with
some such examples like the Euler line, the theorems of Ceva, Napoleon, Morley, Miquel,
Varignon, etc., this book will also present some generalizations of these, and other results
which, as far as the author knows, are original and have not been published elsewhere before.
Extensive attention is also given to the classification of the quadrilaterals from the symmetry
of a side-angle duality.
This book does not follow a traditional mathematics textbook approach by starting from
carefully defined axioms and definitions, and monotonously churning out one after the other,
Theorem 1, Theorem 2, etc. Instead, this book attempts to actively involve the reader in the
heuristic processes of conjecturing, discovering, formulating, classifying, defining, refuting,
proving, etc. Mathematics is not a spectator, but a participator discipline; one simply cannot
sit on the sideline and watch other's play; one must get involved to appreciate and enjoy it.
The reader should therefore preferably always have paper and pencil handy. It should also be
noted that later chapters build on the preceding exercises; so it is advisable to work through
the chapters and exercises in sequence.
Exploration on computer by construction and measurement with Sketchpad, or other dynamic
geometry programs like Cabri, Cinderella, etc., is strongly encouraged throughout, although
not essential. Deductive proof is furthermore not presented here only as a means of
verification, but also as a means of explanation, further discovery and systematization.
It is further assumed that the reader is at least acquainted with high school geometry and
trigonometry (e.g. congruency, similarity, circles, concurrency, sine & cosine rules, etc.)
Some of the exercises would however be accessible to junior secondary students as well.
Some references for further background reading are provided in the bibliography. Most of the
content comes out of the author's on-and-off explorations and experiences in geometry over
the past fifteen years or so. The book starts in the style of Lakatos' Proofs and Refutations
with the fictional dramatization of an actual classroom episode some years back. The reader
is then led through various explorations and extensions of high school geometry. The book is
therefore aimed mainly at gifted high school pupils, secondary mathematics teachers who are
looking for enrichment material and the undergraduate training of prospective mathematics
teachers.
Note
Although a list of quadrilateral definitions are provided at the back for referencing purposes,
it is expected of readers to formulate their own definitions for the various quadrilaterals as
they go along, and to keep and update their own lists.
Michael de Villiers
Mathematics Education
University of KwaZulu-Natal
Private Bag X03
3605 ASHWOOD
South Africa
E-mail: [email protected] (w) or [email protected] (h)
Downloadable papers: http://mysite.mweb.co.za/residents/profmd/homepage4.html
Dynamic Geometry Sketches: http://math.kennesaw.edu/~mdevilli/JavaGSPLinks.htm
August 2009 (1st draft, July 1994; 2nd draft 1996)

* Sketchpad is available at $70 (single), $250 (10 lab pack), $1000 (50-User site license) or $1500
(Unlimited User site license) from Key Curriculum Press, Key Curriculum Press, 1150 65th Street,
Emeryville, CA 94608, USA. Phone US: Monday through Friday from 6 a.m. to 5 p.m. PT 800-995-
MATH (6284). http://www.keypress.com/x1571.xml
In Southern Africa, please contact Dynamic Mathematics Learning at [email protected] or
at: 0836561396.
Contents
Preface i i
Chapter 1 A classroom episode 1
Questions and Problems 1 11
Chapter 2 Defining and Classifying 13
Descriptive defining 13
Constructive defining 17
The relationship between classifying and defining 19
The role and function of a hierarchical classification 22
Questions and Problems 2 25
Chapter 3 Mathematical discovery and proof 29
Discovery or creation? 29
Some important mental attitudes 30
The logic of mathematical discovery and proof 31
The role and function of quasi-empirical methods 31
Proof as a means of explanation and discovery 38
Making conjectures 43
Questions and Problems 3 45
Chapter 4 An interesting duality 52
Duality in triangles 53
Duality in quadrilaterals 55
Utilizing duality in the discovery of new results 56
Questions and Problems 4 61
Chapter 5 Some generalizations of Pythagoras 69
Generalizing to n dimensions 71
Shear similarity 72
Generalizing to right polygons 73
Chapter 6 Generalizing Varignon's theorem 76
A first generalization 76
A second generalization 79
A counter-example 81
Considering converses 82
An analogous result 84
Some reflections 85
Further questions 86
Chapter 7 Generalizing some geometrical gems 89
Some "what-if" questions 90
Generalizing the first result 90
Generalizing the second result 91
Generalizing the third result 93
A dual for the first result and its generalization 97
Further questions 98
Chapter 8 Dual generalizations of Von Aubel's theorem 99
Proofs 101
Questions and Problems 8 106
Some further "what-if" questions 106
Epilogue 107
Solutions 1 108
Solutions 2 111
Solutions 2 (continued) 128
Solutions 3 130
Solutions 3 (continued) 137
Solutions 4 149
Solutions 4 (continued) 173
Solutions 8 203
Summary: quadrilateral definitions 206
Glossary 208
Bibliography 210