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99 Points of Intersection: Examples--Pictures--Proofs.

99 Points of Intersection: Examples--Pictures--Proofs

Hans Walser (Translated from the original German by Peter Hilton and Jean Pedersen) Published by The Mathematical Association of America, Spectrum Series. 2006 Hardback, 153 pp., ISBN 0-88385-553-4 US$48.50

The Spectrum Series of the MAA was established to publish a broad range of books likely to appeal to students and teachers of mathematics, mathematical amateurs, and researchers. It includes reprints and revisions of works that would otherwise not be accessible. 99 Points of Intersection fits nicely into that description as it is a translation of a German text.

Everyone knows that the medians of a triangle are concurrent, as are the altitudes and the perpendicular bisectors of the sides. If you also know that the three points of intersection just described are themselves collinear, and if you find facts like that intriguing, then you are likely to enjoy this book. It provides many more examples of concurrency of lines and circles constructed on geometric figures and on graphs in Cartesian and polar co-ordinate systems. The author thanks, among others, his students and preservice teachers for finding many of these examples.

The book is in three chapters. Chapter 1 introduces the ideas of the book through some startling examples. For example, if the flowers with 5, 7, and 11 petals are superimposed, the points of intersection lie on a circle and three of them form an equilateral triangle.

More complex examples reveal links between the expansion of (cosq + i sinq), the graphs of Chebyshev polynomials, Lissajous figures and the Golden Section! Some optical effects are also explored.

Chapter 2 is the majority of the book and contains, page by page, the 99 "Points of intersection" diagrams, mostly without any text at all. The background to many of the diagrams is provided in Chapter 3. Here you will find proof strategies including proofs of Ceva's Theorem and Jacobi's Theorem and examples of how one result in concurrency can be used to prove others. In this chapter is a brief discussion about the role of Dynamic Geometry Software (DGS) such as Cabri Geometry and Geometer's Sketchpad in exploring geometric constructions and making and "proving" conjectures. Computer algebra systems (CAS) such as Maple or Mathematica are also mentioned and the author points out that some of the 99 "Points of intersection" have only been "proved" using a CAS to perform some of the calculations. But the challenge remains for someone else to provide more elegant geometrical proofs.

In conclusion, this book is highly recommended for geometers of all ages and a wide range of expertise. It would be an excellent resource for a course taught with Dynamic Geometry Software.

Mary Coupland
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Author:Coupland, Mary
Publication:Australian Mathematics Teacher
Article Type:Book review
Date:Sep 22, 2006
Words:444
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