In the summer I will be teaching a course in (plane) Euclidean geometry to future high school teachers and I am looking for a suitable axiom system (unlike College (Euclidean) geometry textbook recommendations where books are discussed).
Since the students should do a lot of ruler-and-compass constructions, I would like the axiom system to make the possibility of these constructions as obvious as possible (much like Birkhoff’s system makes constructions with marked ruler and protractor natural). In particular, I would like to see easily (postulate?) that two circles intersect unless the triangle inequality says they don’t (I think the proof would usually involve introducing coordinates and using the intermediate value theorem more or less explicitly). On the other hand I don’t think I actually need completeness.
In algebraic terms I would like to axiomatize $ K^2$ where $ \mathbb{Q} < K < \mathbb{R}$ and $ K$ is quadratically closed.
So the questions are:
- Is there a good reference for such an axiom system? Is it safe to replace the completeness axiom by “two circles intersect unless they obviously don’t” in an appropriate system? [Of course, to say “they obviously don’t” I need the axiom system to be based on metric rather than in-betweenness, which is fine with me.]
- Is there any need for completeness when doing basic Euclidean geometry?
[One statement I can think of that does need completeness is: any finite set of points is contained in a unique minimal disc. But the proof uses completeness explicitly and thus is beyond high school teachers anyway.]
