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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:

  1. 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.]
  2. 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.]

Edit: when I say "circle" I should of course say "circle whose radius is a distance of two points", since, for example, a circle with transcendental radius in $\bar{Q}^2$ is empty.

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Have a look at Hartshorne's Geometry: Euclid and Beyond. He uses Hilbert's axioms for geometry and discusses (section 11) the following "circle-circle intersection axiom (E)":

Given two circles $\Gamma,\Delta$, if $\Delta$ contains at least one point inside $\Gamma$, and $\Delta$ contains at least one point outside $\Gamma$, then $\Gamma$ and $\Delta$ will meet.

Note that the hypothesis of this axiom is (at least arguably) a different way to say "unless they obviously don't" which is nevertheless based on betweenness and order rather than distance.

Hartshorne shows (section 12) that essentially all of Euclidean geometry can be developed in a plane satisfying the two-dimensional version of all of Hilbert's axioms, but with the completeness axiom replaced by the above axiom (E). This answers your second question: completeness is not needed, all you need is (E). Finally, in section 16 Hartshorne shows that your intuition about the algebraic version of (E) is correct: the cartesian plane over an ordered field satisfies (E) iff all positive numbers have square roots.

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I think that the Alexandrov's system is an answer.

Check "Minimal foundations of geometry" by Aleksandrov (Russian original and there is a translation in Siberian Math. J. 35 (1994), no. 6, 1057–1069).

He also wrote a textbook for school students based on this system.

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    $\begingroup$ Can you add anything about what this axiom system looks like, for the benefit of those who can't access the article or aren't sure whether they want to go to the effort of doing so? $\endgroup$ – Mike Shulman Feb 16 '18 at 6:06
  • $\begingroup$ @MikeShulman the system reminds Hilbert's system, but more intuitive and based on practice. For example equal angles are defines as corresponding angles of triangles with equal corresponding sides. $\endgroup$ – Anton Petrunin Feb 16 '18 at 22:44

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