TITLE: Why are we justified in using the real numbers to do geometry?
QUESTION [43 upvotes]: Context: I'm taking a course in geometry (we see affine, projective, inversive, etc, geometries) in which our basic structure is a vector space, usually $\mathbb{R}^2$. It is very convenient, and also very useful, since I can then use geometry whenever I have a vector space at hand. 
However, some of that structure is superfluous, and I'm afraid that we can prove things that are not true in the more modest axiomatic geometry (say in axiomatic euclidian geometry versus the similar geometry in $\mathbb{R}^3$).
My questions are thus, in the context of plane geometry in particular:

Can we deduce, from some axiomatic geometries, an algebraic structure?
Are some axiomatic geometries equivalent, in some way, to their more algebraic counterparts?

(Note that by « more algebraic » geometry, I mean geometry in a vector space. The « more algebraic » counterpart of axiomatic euclidian geometry would be geometry in $R^2$ with the usual lines and points, and where we might restrict in some way the figures that we can build.)
I think it is useful to know when the two approaches intersect, first to be able to use the more powerful tools of algebra while doing axiomatic geometry, and second to aim for greater generality.
Another use for this type of considerations could be in the modelisation of geometry in a computer (for example in an application like Geogebra). Even though exact symbolic calculations are possible, an axiomatic formulation could be of use and maybe more economical, or otherwise we might prefer to do calculations rather than keep track of the axiomatic formulation. One of the two approaches is probably better for the computer, thus the need to be able to switch between them.

REPLY [4 votes]: Euclid certainly did not use real numbers.  He showed that the diagonal and side of a square have no "common measure", i.e. there is no segment that can be laid end-to-end an integer number of times to get the length of the side and also to get the length of the diagonal, and he knew how to define what it means to say the ratio of lengths of segment $A$ to segment $B$ is the same as the ratio of lengths of segment $C$ to segment $D$ when the latter two are the side and diagonal of a square.
And as mentioned elsewhere, Hilbert also developed Euclidean geometry while not just not using real numbers, but while making a point of avoiding them.