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I will be teaching a mid-level undergraduate course in Euclidean geometry this fall. Has anyone taught such a course, who can recommend a good textbook?

My students will mostly be future high school math teachers, who have some exposure to proofs and rigor but not extensively so. I hope to cover material such as constructions, semi-advanced theorems (Ceva's theorem, the nine point circle, etc., etc.), and the axiomatic approach. I won't do any projective geometry, or anything similar, as that is covered by a followup course here.

I am hoping to choose a book which covers a variety of approaches (so something short is unlikely to be suitable) and which is suitable for students with uneven preparation (i.e. whose ability to write proofs is shaky at the beginning).

Thank you!

EDIT: Cross posted a related question to Math.SE.

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What is the point of your course? My view of the high school geometry course is that only the most elementary geometry is important, and the main point of that course is the axiomatic logic. I have seen attempts to give high school geometry teachers access to advanced and exciting developments in geometry, and while conveying excitement and beauty is good, if advanced geometry displaces the logic in a high school course that is probably counterproductive. – Douglas Zare Jul 10 '12 at 20:37
@Douglas: Needless to say, an interesting and important question. I am just beginning to think about this course (it's not due to start for six weeks yet) and I don't have a strong opinion yet. I figured I would start by looking at a couple books and seeing what it was they are trying to accomplish. – Frank Thorne Jul 10 '12 at 20:52
What background in mathematics do your students have? In particular, have they done any proofed based mathematics courses? You need to be realistic about how much you'll actually be able to teach them. – Brian Borchers Jul 11 '12 at 3:49
I think most of them have written proofs of things like the sum of two odd numbers is even, and the like, but their background is still pretty limited. Most of them won't have had analysis or abstract algebra yet. – Frank Thorne Jul 11 '12 at 17:43

10 Answers 10

Frank, I sympathize with your dilemma which I faced for many years and finally found a solution that I am very happy with. Geometry is a multifaceted subject with many beautiful and fascinating topics to explore. The question is what is right for undergraduate students; particularly preservice teachers. I think there are two important objectives.

(1) I agree wholeheartedly with your first respondent, Douglas, who said that the primary purpose of a geometry course is to immerse students in a logical development of the subject from axioms. Axiomatic geometry was studied for 2000 years by anyone seeking a thorough education because it is an exercise in building facts from given information, something we all need to be able to do. Unfortunately the axiomatic approach was phased out of most of our secondary curricula in the seventies.

(2) Since your students, like most of mine, are future teachers, you want a book that covers the topics they will actually be teaching but at a more advanced level. Brendan and others emphasized this point.

There is a serious problem finding a book that does both (1) and (2).

There are two ways to fulfill requirement (1). You can use a book based on Euclid's axioms. Euclid's work was a great landmark in the history of western thought, but it is severely out of date today because it was written before we really understood axiomatic systems, before we had Dedekind's real number continuum to measure lengths, and before we had Lebesgue's theory of measure as a basis for measuring areas. The alternative is to use a version of Hilbert's axioms (e.g., Moore's or Birkhoff's). These modern approaches are mathematically sound and complete, overcoming the problems of Euclid. But they are not useful for our students. The approach is highly abstract, beginning with very rudimentary axioms about points, lines and betweenness, and building a thorough but tedious foundation before getting into the substance required of (2). If you do this at a pace that students can absorb, you have no chance of getting to most of the requirements of (2) in a single semester.

Frustrated by these two alternatives, I recently developed a new and modern axiom system from which students can and develop the standard topics required by (2) in a semester course. The text was refined through feedback from users of early drafts for several years before it was published by the AMS in the MSRI-MCL series, and it has just become available. It is only \$39 for students, \$32 for AMS members, and free for instructors who teach from it. See

or go to the AMS Bookstore and find the Math Circles Library.

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Thanks so much David. What I feel I'd like to accomplish is slightly difficult, but your motivation is very, very understandable. I ordered a review copy just now. – Frank Thorne Jul 18 '12 at 14:30
I agree with your statement about Hilbert, but you should not blame Birkhoff. His approach is not tedious at all. – Anton Petrunin Dec 9 '13 at 19:30

Hartshorne's Geometry: Euclid and Beyond (Springer Undergraduate Texts in Mathematics). I think it's a very instructive book and seems to be suitable for your purposes. He presents various geometrical constructions, Hilbert's Axioms (incidence, betweenness, congruence etc. ), geometry over fields, rigid motions, and so forth.

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+1, when I was a freshman our professor in geometry closely followed this book and I really liked it. The first proofs are easy to grasp even for people who have never seen proofs before and it really helped me understand how to work with axioms. – Huy Dec 7 '13 at 20:58

I was teaching a similar course 2 years ago and also had a problem to find right book.

The course I had to teach had to cover axiomatic approach and give some feeling of Lobachevsky geometry.

There are many books for school students, they are not exactly suitable for undergraduates. I ended up at writing my lecture notes. It is based on Birkhoff's axioms (=minimalism with no cheating). (This year I will teach it again, maybe after that the lecture notes become better.)

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You can never go wrong with J. Hadamard's classic Lessons in Geometry, recently translated by AMS. This translation covers only half of Hadamard's book, namely plane geometry.

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Seconding Liviu's reply about Hadamard's Lecons! This used to be one of my favorite texts (although I, too, didn't care much about the stereometric part).

As far as those "semi-advanced theorems" go, there are lots of sources for them nowadays. In no particular order:

H. S. M. Coxeter, Samuel L. Greitzer, Geometry Revisited

Nathan Altshiller-Court, College Geometry.

Roger A. Johnson, Advanced Euclidean Geometry.

Ross Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry.

Coxeter/Greitzer is the most well-known of these, I think for good reasons. Altshiller-Court is pretty comprehensive (far more than you'll need unless the whole course is supposed to be about these semi-advanced theorem). Johnson is interesting (it includes some beautiful things that few remember nowadays) but aged (some proofs need a serious amount of work to be made correct by 20th century standards). Honsberger is no more systematic than its name ("Episodes") would suggest, but it is very readable.

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I would recommend Alfred Posamentier's Advanced Euclidean Geometry (Key College Press, 2002). It covers much of the same topics as Geometry Revisited by Coxeter/Greitzer and Episodes... by Honsberger, and it also presents accompanying technology (namely, Sketchpad applications) that allow the students to play around with the results. That is, it gives the students more opportunity to learn how to think geometrically.

I love all of the texts mentioned (Altshiller-Court, Coxeter, Coxeter/Greitzer, Honsberger,...), but their approach to the material is very different from what undergraduates would be used to. And very different than what they will be teaching.

You might find yourself spending a lot of time helping them process the text material into concepts they would find more natural, particularly, if any one of these were the primary text of the course. This may be more work than you would have originally desired: not only teaching the mathematical content but also how to translate mathematical texts.

Then again... this is a good thing for a high school teacher to know how to do...

Either way, I strongly recommend that you look at the Common Core Standards for geometry (at and familiarize yourself with what content these future teachers will expect to be teaching. From there, you can get a good idea of what type of thinking and content knowledge you feel that someone would need to teach this material excellently.

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You can try Michelle Audin's Geometry, which seems to fit your demand.

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There are a lot of good books, most of which have already been recommended in previous answers.

I find the book of Coxeter & Greitzer the most appropriate for a semester course, and I have used it as a main reference in designing an undergraduate class on the subject. If one wants to suggest literature for more challenging problems in Euclidean Geometry (IMO level problem), there is a big list of great books, as for example

T. Andreescu, 106 Geometry Problems From the AwesomeMath Summer Program.

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I really like Isaacs' "Geometry for College Students", which I have taught from several times. It is proof-focussed but not pedantic. It's also pretty cheap ($62).

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When did USD 62 become "pretty cheap" for a 216 pages text? – darij grinberg Jul 11 '12 at 17:00
That's the book that was recommended to me here. The previous instructor really liked the book, but he also said that there was a serious lack of easy exercises, and that the weaker students were unable to get through the course without a lot of hand-holding. – Frank Thorne Jul 11 '12 at 17:41

I haven't used it myself, but there is Patrick D. Barry's Geometry with Trigonometry (Horwood Publishing, Chichester, England, 2001). From the blurb:

This book addresses a neglected mathematical area where basic geometry underpins undergraduate and graduate courses [...] This text emphasises a systematic and complete build-up of material, moving from pure geometrical reasoning aided by algebra to a blend of analytic geometry and vector methods with trigonometry, always with a view to efficiency.

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