Amol Sasane states on p. viii:

The aim of this book is to cover the basics of the wonderful subject of Planar Geometry, at high school level, requiring no prerequisites beyond arithmetic, and hopefully to convey the sense of joy which I had when I was taught geometry.

A review of *Plain Plane Geometry* must necessarily be based upon the degree to which Sasane fulfills this aim. I believe that Sasane’s book is an excellent gateway into planar geometry: it is accessible, systematic, and thorough. More importantly, it does communicate the beauty and charm of geometry.

The book follows the “Definition-Theorem-Proof-Exercise” format and begins with an introductory chapter on geometric figures. The fundamentals of geometry, from points to polygons, are rigorously defined. Chapter 2 is on congruent triangles, and includes topics such as congruency rules, geometric constructions, and concurrency. Quadrilaterals become the theme in chapter 3, along with topics such as the midpoint theorem, area, and Ceva’s theorem. In this and latter chapters, there are several digressions into the history of geometry along with multiple real-world applications. For example, Kepler’s second law is discussed and proved using nothing more than basic geometry. These digressions and applications show off the beauty of geometry and are quite enjoyable.

The difficulty increases in chapter 4. The focus is now on similar triangles, and the chapter includes topics such as similarity rules and Menelaus’ theorem. There are several memorable exercises, including one where readers can estimate the radius of the Earth with similar triangles. The difficulty again increases in chapter 5. The theme is circles, including topics such as tangent lines, Simson’s line, and Ptolemy’s theorem. Aspiring geometers may feel overwhelmed by the challenging final chapter, but it’s an excellent chapter nonetheless.

Sasane’s *Plain Plane Geometry* covers the fundamentals of planar geometry while also introducing additional topics. The exercises cultivate appreciation of geometry and encourage mathematical play, but additional drill exercises would have been nice to help readers master important concepts. Many of the exercises in the book have hints, and the exercises without hints aren’t difficult enough to merit one. The solutions, much like the book itself, are thorough and easy to follow. Given that the book is aimed at the pre-university and advanced high-school levels, some parts of the book, especially chapter 5, may prove difficult. Overall, the book is a fine resource for readers who are new to geometry as well as those who want to sharpen their geometric skills.

Jack Chen (jackchen5@hotmail.com) is a high school student who enjoys mathematics. He hopes to pursue mathematics in his post-secondary education. His current mathematical interests are in modular arithmetic and cryptography.