# 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