![]() ![]() It is further assumed that the reader is at least acquainted with high school geometry and trigonometry (e.g. Deductive proof is furthermore not presented here only as a means of verification, but also as a means of explanation, further discovery and systematization. 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. 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. The reader should therefore preferably always have paper and pencil handy. 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. Instead, this book attempts to actively involve the reader in the heuristic processes of conjecturing, discovering, formulating, classifying, defining, refuting, proving, etc. 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. Extensive attention is also given to the classification of the quadrilaterals from the symmetry of a side-angle duality. ![]() 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. This is however not the case many interesting and beautiful geometric results have been discovered during the past 300 years. 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. Unfortunately many people seem to regard mathematics in general as a boring, dead subject with nothing new to discover. 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. 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. 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. 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. © 2009 Michael de Villiers (trading as Dynamic Mathematics Learning) All rights reserved. ![]() SEPTEMBER 2009 © 2009 Michael de Villiers (trading as Dynamic Mathematics Learning) All rights reserved. Michael de Villiers 1st DRAFT, JULY 1994, 2nd DRAFT, JAN 1996
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