Have you found the papers by Roger Alperin? He has some very nice articles, especially in regards to relating various construction systems -- in addition to origami and compass-straightedge constructions, there are a variety of others, like the Vieten constructions, Pythagorean constructions, and even some variants on the origami constructions. Basically, each additional axiom potentially provides an entirely new class of constructions (which may or may not actually enlarge the set of valid constructions). Some of these questions end up being very geometric in nature, and others have some fantastic ties to the structure of algebraic numbers.
In any case, you can find several papers of his via Google. I believe your specific question about intersecting two circles is answered in Section 6 of Alperin's "Mathematical Origami: Another View of Alhazen's Optical Problem."
Hope that helps.
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 Euclid's work was a great landmark in the history of western thought, but it is severely out of date today because it was written before we really understood axiomatic systems, before we had Dedekind's real number continuum to measure lengths, and before we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.
Best Answer
Have you taught this course before? After teaching it several times from Millman/Parker and other materials using Birkhoff's axioms, I suggest you consider using Euclid himself plus Hartshorne's guide, Geometry: Euclid and beyond, which uses a form of Hilbert's axioms.
The problem for me is that real numbers are much more sophisticated than Euclidean geometry, and the Birkhoff approach is thus a bit backwards except for experts like us who know what real numbers are.
When we covered as much of Millman/Parker as we could manage, the most enjoyable part for the class was the section on neutral geometry, which I learned recently was lifted bodily from Euclid Book I.
If you like assuming that every line in the plane is really the real numbers R, what about going the rest of the way and assuming the plane itself is R^2? Then you can use matrices to define rigid motions and do a lot that connects up to their calculus courses.
Moise is more succinct than the 500 pages suggests as I recall, and is an excellent text from a mathematician's standpoint, but very forbidding probably from a student's. I noticed Moise went from 1.4 to 1.9 pounds from 1st edition to third so maybe the first is also 25% shorter.
The old SMSG books in the 1960's were based on Birkhoff's approach, but are not short. They are also available free on the web.
I just looked at the old SMSG book and found the following circular sort of discussion of real numbers: "if you fill in all those other non rational points on the line, you have the real numbers."
Clint McCrory spent several years developing his own course using Birkhoff's approach at UGA, and made it very successful. Here is a link to his course page. The students loved his class at least in its evolved form after a couple years. they especially appreciated the GSP segment at the start. Apparently many students had little geometric intuition and used that to acquire some. Clint apparently never found an appropriate book to use though.
http://www.math.uga.edu/~clint/2008/geomF08/home.htm
After teaching this course myself from Greenberg, Millman/Parker, Clemens, supplemented by Moise, and the original works of Saccheri, my own Birkhoff axioms, I finally found Euclid and Hartshorne to be my favorite, by a large margin.
But the beauty of the topic is that there is no perfect choice. You will likely enjoy the search for your favorite too. There is a reason however that Euclid has the longevity it has.
In a nutshell, there are two equivalent concepts, similarity and area, that are treated in opposite order in Euclid and Birkhoff. Euclid's theory of equal content, via equidecomposability, in his Book I, allows him to use area to prove the fundamental principle of similarity in Book VI. Birkhoff assumes similarity as an axiom, and area is relatively easy using similarity, e.g. similarity allows one to show that the formula A = (1/2)BH for area of a triangle is independent of choice of base. Euclid himself uses similarity to deduce a general Pythagorean theorem in Prop. VI.31, whose proof many people prefer as "simpler" than Euclid's own area based proof of Pythagoras in Prop.I.47. The problem is that there has, to my knowledge, never been a civilization in which similarity or proportionality developed before the idea of decomposing and reassembling figures. Briefly, congruence, on which equidecomposability is based, is more fundamental than similarity. Hence, although logically either concept can be used to deduce the other, it seems to me at least that the more primitive concept should be placed first in a course.