As someone who wanted to learn more nonroutine math, this book was perfect for that purpose. Most of the topics in this book were new to me, and they were all presented in a clear and elegant manner. This book is full of interesting and challenging exercises, which made reading it very enjoyable. Chapter 1 discusses the broken line method, a generalization of the triangle inequality that I had never heard of before. There were many diagrams throughout this chapter, and they served to complement what was covered in the text; these diagrams aided my understanding and intuition of the content covered. This was special mainly because some geometry textbooks do not even include diagrams, or include very little diagrams. Throughout my experience, I had seen assorted geometric inequalities similar in flavor to the content discussed in the book, and Chapter 1 covered the theory behind some of them very thoroughly. I believe the rigor of Chapter 1 really took my understanding of these types of problems from rudimentary to solid; not only are problem solving tricks discussed, but the motivation behind them is covered in a rather interactive way. Also, I always wanted to use more advanced techniques such as calculus and linear algebra on geometry problems, and Chapters 2 and 3 proved to be the perfect guide on how to apply techniques and related theory on geometric inequalities. The theory is discussed in a very motivated way and is written in a way that can help readers apply the content to problems. I really enjoyed reading through the examples, as they were chosen very appropriately given the content covered, and each of them incorporated some aspect of the theory discussed, which made me appreciated the power of the math covered in these two chapters. I also loved reading Chapter 6, which focuses on area inequalities. I think the discussion in several examples on how to translate area inequalities into algebra problems was written in a very informative and intuitive manner and it made the experience of reading through how to solve these problems very enlightening. I liked how conic sections and general convex regions were discussed, as they are topics that are not usually seen in competitions, but I believe having some sort of experience with them is beneficial. I particularly enjoyed reading through the section on the Brunn-Minkowski Inequality, as it introduced more "exotic" concepts and theory and demonstrated how these notions can be applied to solve advanced problems, such as one from the IMO; the solution to that problem, I believe, was made much more clear to me after reading about the theory that was presented before, and the solution presented in the book was very elegant. Lastly, I thought that the solutions were very helpful, not only because they frequently included diagrams, but also because a lot of the steps were explained in detail. Although some small steps were skipped, I found working out why those steps followed on my own to be beneficial for me in understanding the solution. From my experience, many solutions in math textbooks are rather vague (or the solutions are not included at all), and this textbook was definitely an exception, due to how informative each of the solutions were. Overall, this book helped me further my understanding of geometric inequalities because it covered a broad spectrum of topics, and went into detail on the theory behind each of those topics; the book showed effectively the power behind the techniques it introduced through the example, which helped when I worked through the problems at the end of each chapter. This book was fun to read, and that made it easy to pick up the math that it covered.
