Customer Reviews

Challenging Problems in Geometry (Dover Books on Mathematics)

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Geometry problems are my favorite sort of math problems to do, because many geometry problems require, literally looking at the problem in a different way; a slight twist on the facts that you are using and the problem becomes much easier. It's usually a simple, yet ingenious insight that often solves the problem.

To that end, this book does not disappoint. I highly recommend this book, for it contains such problems, and at the end of the first section of problems, I had developed a sort of intuition for Euclidean 'way' of thinking. I am far from finishing this book, but I think it would take me a few years to do so.

The book is broken down into several chapters. The first chapter contains the problems, the next are the solutions, the next are hints to the problem, and finally an appendix of useful theorems and formulas. The useful theorems are mostly the results of Euclid's Book 1 and 3, and the immediate consequences of those theorems, e.g., the sum of the angles of a convex quadrilateral is 360.

The hint chapter may be too helpful for it usually outlines the steps you need. I would have preferred several hint chapters that are progressively more helpful. The solution section may show more than one solution to a problem. There were a few times my solution was not found in the back of the book, but that's not a fault of the book, but a delight if you can come up with an original solution!

The problem chapter is broken down into what I would call fundamentals and advanced sections. There are over 200 problems.

The fundamental section is further broken down into parts, either by method, e.g., similar triangles/pythagorean's theorem, or theme, e.g., problems concerning 'circles' and problems concerning 'areas'. Many the problems can be solved in different ways. The first section of problems can be done with a purely Euclidean style approach. But lots of problems require a *little* algebra, mainly to economize on thought, e.g., a variable place holder for proportions, and a simple formula or two, and of course Euclid's theorems. Each section is not isolated, they sort of build on the first part of this section.

The advanced section has a part containing a 'mixture' of techniques to use, and again themes which may not be familiar to the beginner, e.g., Simson lines, and Ceva's theorem.

The problems are of proof, or finding the measure of a line, angle, area, or finding the algebraic formula for a collection of objects. So far, I have not encountered a single construction problem. Some of these problems may be quite easy to solve, and some can be quite hard! For instance, one of the problems asks you to prove Heron's formula. The Euclidean proof takes several pages, and I would say is beyond that for a math olympiad. Most problems, are of course, not this hard.

You may have a tendency to want to 'angle-chase' or plug and play a formula. Such thinking will cause you to go mad! You'll endlessly try to some up combinations of angles, and construct new ones. Luckily, I broke that habit, and there are enough of these problems for you to break the habit in order to keep your sanity. Find the elegant solution, if you can, and most of these problems have them. And when you do -as George Polya said in "How to Solve It"- you'll see the solution `at a glance'. (It is more rewarding and more difficult, to do away with algebra, and think `purely' geometrically. It's an intuitive appreciation for the problem, and you can hold a longer argument chain in your head. Then, You'll begin to appreciate the qualitative style of thinking that is Euclidean. It's impossible, however, for many cases.)

Also, you will need to have another geometry book handy. There were one or two definitions that were unfamiliary to me, and I could not find them anywhere defined in the book. It would be nice on the next edition if they gave definitions of some of the terms. Dont' be alarmed, they were not technical terms, and more along the lines of 'what is a median?'

Finally, these problems are a good starting point for your own investigations into geometry. By varying a problem found in the 'Geometric Potpourri', I was able to finally figure out how to construct a pentagon, which has been stumping me for many years.

To round out your geometry skills, you will also want to do construction problems. I recommend the book 'Geometric Constructions' by George E. Martin, it is text book; so it contains more than just problems, but the problems also require ingenious solutions. (I hope to review this book.)

Mr. Posamantier, please print the next volume!! And for those who obtain this book, happy solving!

This book is a great one. Invaluable as a supplement to a basic geometry textbook. It includes approximately 200 problems dealing with congruence and parallelism, circles, area relationships, collinearity and concurrency and many other subjects. Detailed solutions and hints are provided for all problems, and specific answers for most. I highly recommend this book to anyone looking for a great book at an affordable price. Buy it. You won't regret it.

Posamentiers' book is a little bit unbalanced. It contains around 200 problems with solutions. The easy problems are just what you would expect in the exercises sections of an introduction to Euclidean geometry like Kiselev's Geometry / Book I. Planimetry. The the harder problems you will find as classical theorems and examples in more advanced texts like Altshiller Courts' College Geometry: An Introduction to the Modern Geometry of the Triangle and the Circle (Dover Books on Mathematics). A more balanced text would have contained more intermediate problems and the harder ones would have been more "original".

The methods used are purely synthetic, no analytic geometry. The book is aimed at an advanced high school level audience. Prerequisite is the stuff you find in a book like kiselev I mentioned above.

If you need a book to train your geometric problem solving abilities I think that Altshiller Courts' book is a better choice although there are no solutions to the exercises in that book. But what use are the solutions? Problems should be solved and not looked up!

For many problems, especially the hard ones, several solutions are provided. To me this is what makes the book attractive.

I liked this book very much. I solved every single problem in the book with two students that I tutor for International Math Olympiads and carefully read the hints and solutions proposed at the end of the book. They really teach how to "attack" geometry problems using simple stuff like angle chasing, drawing parallel lines etc. I cannot recommend this book more to the readers with some mathematical sophistication. I even have a suggestion for parents that have some sort of mathematical background (engineers, bankers, doctors, etc.): If you want to spend quality time with your children, get this book and enjoy solving the problems together. I cannot imagine a more amusing pastime. I am looking forward to seeing new titles from the authors.

In a very well organized fashion the authors have amassed a fabulous collection of geometry problems that are quite challenging indeed. Although most high schoolers will likely have difficulty it can be used wisely as a study tool since they have sections for hints and finally a solutions section. It should be noted that other solutions are possible but the ones given are very easy to follow. A fun time for all geometry teachers (like me) and good students.

This book is suitable for anyone who want to learn advanced high school geometry or want to participate in any mathematical contest and have to learn much though not really new on it. The problems are set from the quite easy to advanced, with many of them are quite simple so you don't get frustrated because you can't do any of the problems. It covers the basic and common problems in math contest.

Unfortunately, it don't have good set of theorems. Sure, it's written on the back but they're not easy to navigate. Also, many of them are not really basic even for contest geometry. Like its title, 'Challenging' is sure quite challenging, since many of the problems, though not really hard, need deeper thinking and you need to draw the 'help line' to solve the problem. The hints written on the back is not really helpful.

Overall, for those who want to improve skills, participate in contest, or teacher who wants to make a surprise test, this book is a really good source of information.

This books presents a lot of problems to solve. One challenge after another, you can find several problems and better still if you get stuck, you have hints that really help in solving the problems, and one, sometime two or even three different ways of solving it.
As written at Plato's academy, for geometers only, you have to be willing to spend sometime doing geometry.

This is a great book! But, don't buy it if you don't know the basics well. You'll get frustrated easily and leave geometry. For beginning this is what I recommend: [...]

Firstly, don't try a problem for 10 minutes and then just look at the solution. This way, you'll gain nothing from the book. You HAVE to try these problems for at least an hour (some may even take up to 2-3 hours)if you can't solve them. You can, however, break this up into pieces. For example, sit for about 15-25 minutes with one line of attack and then if couldn't solve it, then come back when you have time and give a try with another line of attack. And, let me tell you, when you do solve a problem after trying it for 1-2 hours, it'll be motivation enough to move on to the next problem.

I especially like this book because I can hide it under my desk at school :)

Also recommended is Geometry Revisited (buy it with this if you can).