**Spectrum**

Volume: 49;
2006;
153 pp;
Hardcover

MSC: Primary 51;
Secondary 00; 14

**Print ISBN: 978-0-88385-553-9
Product Code: SPEC/49**

List Price: $49.00

AMS Member Price: $36.75

MAA Member Price: $36.75

**Electronic ISBN: 978-1-4704-5858-4
Product Code: SPEC/49.E**

List Price: $49.00

AMS Member Price: $36.75

MAA Member Price: $36.75

# 99 Points of Intersection: Examples, Pictures, Proofs

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*Hans Walser*

MAA Press: An Imprint of the American Mathematical Society

Translated from the original German version by Peter Hilton and Jean Pedersen. The 99 Points of Intersection presented here were collected during a yearlong search for surprising concurrence of lines. For each example, we find compelling evidence for the sometimes-startling fact that in a geometric figure, three straight lines, or sometimes circles, pass through one and the same point. Of course, we are familiar with some examples of this from basic elementary geometry — the intersection of medians, altitudes, angle bisectors, and perpendicular bisectors of sides of a triangle. Here there are many more examples — some for figures other than triangles; some where even more than three straight lines pass through a common point. The main part of the book presents 99 Points of Intersection purely visually. They are developed in a sequence of figures, many without caption or verbal commentary. In addition, the book contains general thoughts on and examples of the points of intersection as well as some typical methods of proving their existence. Many of the examples shown were inspired by questions and suggestions made by students and high school teachers. Several of those examples have not only a geometrical aspect but also an intriguing aesthetic aspect. The book addresses high school students and students at the undergraduate level, as well as their teachers, but will appeal to anyone interested in geometry.

#### Reviews & Endorsements

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 Geomeetry Software.

-- Mary Coupland, Australian Mathematics Teacher

# Table of Contents

## 99 Points of Intersection: Examples, Pictures, Proofs

- Cover Cover11
- Copyright ii3
- Title iii4
- Author's Foreword vii8
- Foreword to the English Edition ix10
- Author's Note to the English Edition xi12
- Contents xiii14
- 1 What's it all about? 116
- 2 The 99 points of intersection 1732
- 3 The background 117132
- 3.1 The four classical points of intersection 117132
- 3.2 Proof strategies 118133
- 3.3 Central projection 122137
- 3.4 Ceva's Theorem 125140
- 3.5 Jacobi's Theorem 132147
- 3.6 Remarks on selected points of intersection 137152

- References 147162
- Index 151166
- About the Author 153168
- Back cover Back Cover1173