First of all, it's clear that any theorem of synthetic geometry can be proven in analytic geometry. But I'm not so sure that the opposite holds. Perhaps if our synthetic geometry had a notion of measure with the entire set of real numbers, we could use that to "mimic" the coordinate system.
However, if it didn't have any sort of measurement (i.e. Hilbert's axiomatization), I'm thinking that maybe you could take a line segment to be the "unit length", and somehow construct the rational lengths and some irrational lengths. The problem is, I don't know whether being able to give coordinate names to points actually gives us any power to carry out analytical proofs. Also, we won't even have the entire real line using this method, since there are numbers that aren't constructible. So essentially what I'm asking is: Is synthetic geometry logically complete? Moreover, is it possible to mimic analytic proofs in coordinates by constructing a coordinate system out of a synthetic geometry without measurement?