Overview

One of the challenges many mathematics students face occurs after they complete their study of basic calculus and linear algebra, and they start taking courses where they are expected to write proofs. Historically, students have been learning to think mathematically and to write proofs by studying Euclidean geometry. In the author's opinion, geometry is still the best way to make the transition from elementary to advanced mathematics. The book begins with a thorough review of high school geometry, then goes on to discuss special points associated with triangles, circles and certain associated lines, Ceva's theorem, vector techniques of proof, and compass-and-straightedge constructions. There is also some emphasis on proving numerical formulas like the laws of sines, cosines, and tangents, Stewart's theorem, Ptolemy's theorem, and the area formula of Heron. An important difference of this book from the majority of modern college geometry texts is that it avoids axiomatics. The students using this book have had very little experience with formal mathematics. Instead, the focus of the course and the book is on interesting theorems and on the techniques that can be used to prove them. This makes the book suitable to second- or third-year mathematics majors and also to secondary mathematics education majors, allowing the students to learn how to write proofs of mathematical results and, at the end, showing them what mathematics is really all about.

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9780821847947

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. . . A very useful and needed book for the course . . . The tone is exactly on target . . . The proofs are always clear and the level of rigor is the right one: always clear, never hiding the issues, never condescending . . . Many geometry books emphasize axioms giving students the impression that mathematics is a perverse game which tries to make the obvious difficult. This impression is definitely the wrong one to come away with for a future high school teacher. Hence, I agree with the author wholeheartedly that 'separation axioms, ' 'Hilbert axioms, ' and other late work on the foundations of geometry should be given at most a passing mention in such a course . . . This book promises to be a useful tool to teach prospective high school mathematics teachers what mathematics is all about.

. . . A very useful and needed book for the course . . . The tone is exactly on target . . . The proofs are always clear and the level of rigor is the right one: always clear, never hiding the issues, never condescending . . . Many geometry books emphasize axioms giving students the impression that mathematics is a perverse game which tries to make the obvious difficult. This impression is definitely the wrong one to come away with for a future high school teacher. Hence, I agree with the author wholeheartedly that 'separation axioms, ' 'Hilbert axioms, ' and other late work on the foundations of geometry should be given at most a passing mention in such a course . . . This book promises to be a useful tool to teach prospective high school mathematics teachers what mathematics is all about. A brisk, fast-paced and well-written introduction to college geometry. . . It avoids getting bogged down in questions of axiomatics and is an excellent training course for the development of mathematical thinking and proof writing. I strongly endorse this book . . .

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