Geometry Revisited

Anneli Lax New Mathematical Library

by H.S.M. Coxeter, S.L. Greitzer

Among the many beautiful and nontrivial theorems in geometry found in Geometry Revisited are the theorems of Ceva, Menelaus, Pappus, Desargues, Pascal, and Brianchon. A nice proof is given of Morley's remarkable theorem on angle trisectors. The transformational point of view is emphasized: reflections, rotations, translations, similarities, inversions, and affine and projective transformations. Many fascinating properties of circles, triangles, quadrilaterals, and conics are developed.

Genres: Mathematics, Geometry, Science, Nonfiction, Academic, Textbooks

193 pages, Paperback

First published January 1, 1967

About the author

Harold Scott MacDonald "Donald" Coxeter (1907-2003), CC, FRS, FRSC was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.

Reviews

[Beau · Rating 5 out of 5]

This book literally made me laugh out loud with sheer awe and delight at the theorems proved; there are so many beautiful, striking, and elegant results in this dense volume. Basic topics from Euclidean geometry covered include Ceva’s theorem, the Euler line, the nine-point circle, coaxal circles, Simson lines, Ptolemy’s theorem, the butterfly theorem, Morley’s theorem, Napoleon triangles, Varignon’s theorem, theorems of Menelaus, Pappus, and Desargues. Other chapters include using transformations of the plane (translation, rotation, reflection, spiral symmetry) to prove theorems; an introduction to inversive geometry—containing some of the most enjoyable material in the book including a wonderful proof that the nine-point circle is tangent to the incircle and three excircles of a given triangle; and an introduction to projective geometry.

The book is extremely well-written with an economy of language and notation and very readable, concise, and beautiful proofs. There are also bits of historical information and interesting quotations throughout. There are many excellent exercises ranging from trivial to extremely difficult; and there are outlines of solutions to all of them.

To be able to read this book you need to remember your high school geometry pretty well (as most of it is assumed), as well as some analytic geometry and trigonometry. This book would be an excellent resource for students preparing for difficult high school mathematical competitions, like olympiads and their precursors. Some of the exercises are probably of olympiad caliber.

[Michael · Rating 4 out of 5]

Cool approach on geometry, lots of impressive theorems. Some proofs are difficult to follow, for me that was inversion, which after high school, college, and then some more, I have STILL not understood.

But I think this book is an absolute essential for anyone studying math, especially for competition math folks (this is a book that I felt like I was actually better at problem solving and geometry in general after reading), and the proof techniques that can be learned from the problems are very useful for geometry if that is what you will pursue.

One of the most mind-blowing proofs in the book for me was the 3 jugs of water problem, that was solved using geometry techniques! If there's a proof you should read, definitely do that one.

[Andre Ruiz Loera · Rating 5 out of 5]

This book reads like a textbook that presents the proofs but often requires the reader to work their way to them rather than having every step spelled out. While it can be challenging for someone with only an analytical geometry background, the proofs are elegant and highly rewarding once understood. Actively engaging with the material in this way reinforces understanding and retention. With clear figures, exercises, and 150 pages of rigorous content, it’s an excellent introduction to proof-based mathematics. Recommended by Paul Zeitz, and it’s easy to see why.