Euclidean geometry is a special case of the theory of Hilbert spaces; but in order to convince small children of basic facts, e.g. that the line segments from each of the vertices of a triangle to the midpoint of the opposite side are concurrent, I've found I need to resort to synthetic arguments.
Do you know of a comprehensive reference for synthetic euclidean geometry?
An good one is the old classic geometry book by Jacques Hadamard. The first volume covers plane geometry: Lessons in Geometry by Jacques Hadamard (published by AMS, 2008.)
There is also a companion book with the solutions to problems (AMS, 2010): Hadamard's Plane Geometry by Mark Saul.
I wonder if the second volume (solid geometry) is available in English?
As Andrew L has mentioned, Hartshorne's book Geometry: Euclid and Beyond is very good. If you really want a comprehensive book (for yourself, not for the children you teach) then Hartshorne is as comprehensive and synthetic as you can get. It also explains the connections with algebra very well.
Less comprehensive, but very insightful about Greek geometry and its history, is Benno Artmann's Euclid -- The Creation of Mathematics (Springer 1999).
"Geometry Revisited" by Coxeter, or "Introduction to Geometry" by the same author. Really, anything by him is good :)
There are a lot of good modern texts on Euclidean geometry now. Kiselev's Geometry is the best introduction out there-I was one of the first to order it from Professor Givental's company. A more sophisticated introduction is the classic by Greenberg,Harsthorne's book is also very good,but it's quite a bit harder. Sadly,the teaching of Euclidean geometry got lost somewhere along the way in this country.Along with reading,writing,vocabulary,thinking..............
A book that I like for synthetic euclidean geometry is Geometry Unbound by Kedlaya, even though it's not as comprehensive (and lacks diagrams), because it has a lot of (nice) problems, and it talks about most of the important stuff (it goes from the basics of triangle geometry all the way up to inversive and projective geometry).
It's not such a good reference as Geometry Revisited, but even though I think that for learning it's a great book because it teachs you the basics then makes you struggle with some (tough) problems (and it's free!).
Try Continuous Symmetry by Howe and Barker!
Howe and Barker take the Erlangen program very seriously and prove Euclidean geometry via a synthetic fashion guided by symmetries.
A very important book that concretely fleshes the Erlangen Programme which at times appears to be high-brow philosophy.
Jeremy Avigad, Edward Dean, and John Mumma have an article in the Review of Symbolic Logic, A Formal System for Euclid's Elements (arxiv), which (a) gives a sequent calculus for the arguments used in Euclid, and (b) furthermore formalizes which facts can be taken as "clear from the diagrams" and shows when they can be safely omitted from the Euclidean proof.
It is a lovely paper, which requires only the most modest familiarity with sequent calculus (even Wikipedia can supply sufficient background).
(I'm french so sorry for my poor English)
I think the best way to teach Euclidean space to children is to just make it simple:
Two parallel lines will never meet. Like on a map that you can show them.
Then the best way to explain a non-Euclidean space to children is with a Earth globe. You show them the meridians and the longitudes. How the meridians are meeting at the poles even if while watching outside, the Earth seems plane. The globe is the best thing to explain that to a child I think. Just put his finger on the North pole.
How could you expect a child to understand that a triangle doesn't have 180 degrees of corners in a non-Euclidean space ?
Maybe i'm too simplist for you but that's what I think.
There is also Elementary Geometry from an Advanced Standpoint by Moise.
