Alternative model of Euclidean geometry I'm planning to teach high-school geometry. As usual, this will be by building from axioms. (The axioms used are AFAICT particular to the book I've been assigned, but they're some combination of Hilbert's, SMSG's, and God knows what.) I'm considering demonstrating that geometry's axioms need not have their usual model by presenting an alternative model of at least a few basic axioms. Can anyone recommend such a model? I'd need it to be accessible to high schoolers (so, for example, not this).
The rational plane $\mathbb{Q}^2$ is a model for Euclid's five axioms, and I would think (hope?) that it is accessible to high-schoolers. Many common geometric constructions don't work as expected for it; for example, here's an excerpt from Explanation and Proof in Mathematics, p.66:
A geometric model for euclidean geometry should be presented as follows:


Euclidean plane be the open unit disk
A point of E_plane be a point inside the unit circle and vice versa
A line of E_plane be part of circle inside the unit circle which passes through a fixed point on the unit circle.


It seems that any student can easily prove that sum of internal angles is equal to $180^\circ$ for triangles.

A.V.Farimani