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.
Best Answer
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 Euclid's work was a great landmark in the history of western thought, but it is severely out of date today because it was written before we really understood axiomatic systems, before we had Dedekind's real number continuum to measure lengths, and before we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
https://bookstore.ams.org/mcl-9/
or go to the AMS Bookstore and find the Math Circles Library.