Challenging Problems in Geometry

Rating:
Summary: Great book on geometry.
Review: 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!

Rating:
Summary: Superb book
Review: 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.