TITLE: I'm teaching a college geometry course. What should I cover?
QUESTION [30 upvotes]: I've been asked to teach "Foundations of Geometry" at the University of South Carolina. Apparently, professors in the past have all done very different things, and I have a lot of choice in the matter. What course would you like to see?
Some constraints:

I want to say at least a little something about (1) the axiomatic approach (e.g., Euclid's axioms), (2) the modern approach (cool theorems such as Ceva's theorem, the nine point circle, etc., etc.), (3) the constructive approach, with straightedge and compass.
My students differ in background and motivation. Most of them are prospective high school math teachers. Typically they have seen some proofs, but not a lot; for example, they may have proved that the sum of two odd numbers is even. Most of the students won't yet have taken analysis, or algebra, or any other course obliging them to work really hard.
It has been a long time since I have seriously dealt with the subject, so I will definitely want a good book (or books) or other materials to follow.
I won't say anything about projective or other non-Euclidean geometry, because there's another course for that.

I asked a similar question on MathOverflow, but I was a little bit spooked by the answers. I'm sure that the books of Hartshorne and Hadamard are excellent, but I suspect these might be better for stronger students. (And the Hartshorne book discusses "geometry over fields", etc., and most of my students won't have taken abstract algebra.) Some respondents rolled their own solutions -- but are there really not good source materials out there?
My colleague taught a course out of Isaacs' book -- but he really went the extra mile (more like the extra ten miles). He said that the exercises were a bit too difficult, and that he was constantly having to give the students hints, and he slaved over writing up complete solutions. I believe this would not work as well for me as it would for him: he has a very approachable demeanor that I haven't (yet) been able to duplicate, and I'm afraid the students would be unlikely to come to my office hours no matter how much I encouraged them.
I've also looked at other books -- Coxeter and Greitzer is very cool, but perhaps more naturally suited to hotshot high school kids. Posamentier looks promising, a review copy is on its way. Clark's book seems to be somewhat off the beaten path (where I'm not quite sure I want to follow), but it looks very interesting and I ordered a review copy of that also. Other suggestions, big or small, would be very welcome. Thank you so much! --Frank

REPLY [0 votes]: I teach Algorithmic Geometry at the grade 11-12 level for students heading off into 4-year STEM college.  I strongly believe that the practice of mathematical problem-solving has undergone a radical transformation in the real world of industry and R&D due to software computing.  We teach our students introductory Java programming, combined with computer graphics, using 2D and 3D vector geometry as the math foundation.  Every problem encountered is solved first on paper via geometric sketching, in such a way that the sketch serves as a spec for writing a numerical algorithm.  The payoff from this approach is the reuse of finished algorithms to solve harder problems.  For example, students use their solution to 2D intersection of two overlapping circles when they get to 3D, to help solve the intersection of two overlapping spheres ( a 3D circle).
Given your student demographic, I would suggest maybe a one month module on geometric computing.  It would entail students writing their own software to represent 2D points, 2D distances between points, and 2D direction vectors (obtained from a pair of points).  Some graphics work with points, line segments, triangles and rectangles could be undertaken.  The goal is to get the students comfortable with programming simple interactive 2D graphics widgets, such as a mouse-editiable line segment, mouse-editable circle, and mouse-editable triangle.
A cool way to finish up is solving the circumcircle of a dynamic triangle algorithmically, and testing the algo graphically.  A lot of applied vector math will be learned in the process, as well as the 21st century mathematical skill of encapsulating your math thinking permanently, and reusably, in software you've designed, written & tested.  Since the creative work is all done on paper with exploratory sketching, you are nurturing deep problem-solving skills.  The added value of the algorithmic math approach is that students begin to see geometric  problem-solving as a numerical information processing game, with input information purposefully crunched into the desired output information.
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