99 Points of Intersection: Examples, Pictures, Proofs

This book appears in the Spectrum Series of the Mathematical Association of America and was translated from the original German by Peter Hilton and Jean Pedersen.

The Spectrum Series was named to reflect its purpose: “to publish a broad range of books including biographies, accessible expositions of old and new mathematical ideas, reprints and revisions of excellent out-of-print books, popular works, and other monographs of high interest that will appeal to a broad range of readers, including students and teachers of mathematics, mathematical amateurs, and researchers.”

Although this text reflects a particular interest of the author, it satisfies the goals of the Spectrum Series, especially the ones about accessibility and very good exposition. Some of the points of intersection are very well known from elementary geometry (those regarding the medians, altitudes, angle bisectors, and perpendicular bisectors of a triangle), others are new or even surprising.

The author mentions that one of the requirements of this book was a purely visual presentation, which he accomplished with great skill. He suggests a search for “dynamic geometry software” on the web which will reveal many programs (such as Cabri Geometry II, Geometer’s Sketchpad, Cinderella, or Euklid) that are very helpful for discovering and testing points of intersection.

The book has three parts:

• Examples: containing general comments and a few examples;
• Pictures: containing a sequence of 99 pages, each with a picture of one point of intersection preceded by three “introductory” figures. This is the main part of the book, the “purely visual” one.
• Proofs: containing some general methods of proving the existence of such points of intersection, some “classical theorems” with proofs and some historical references.

Overall, this is a very accessible and well-written book that can be used by anybody (including students and teachers) interested in geometry, particularly in “points of intersection”.

Mihaela Poplicher is associate professor of mathematics at the University of Cincinnati. Her research interests include functional analysis, harmonic analysis, and complex analysis. She is also interested in the teaching of mathematics. Her email address is Mihaela.Poplicher@uc.edu.