It is relatively common knowledge that Euclid's axiomatization is not sufficient to prove all the things that Euclid wants to, and that there are other axiomatizations out there that strengthen Euclid's in order to make Euclidean geometry fully rigorous.
So now, imagine Euclid comes back from the dead and looks around at what the world has done with his geometry. There are many examples of texts that treat Euclidean geometry, but in some sense they are all different from what Euclid's Elements were - a theorem, a construction and a proof. It seems that the overwhelming majority of Euclidean geometry texts are Euclidean in the sense of the parallel postulate, but not Euclidean in the manner of exposition. In a lot of ways, analytic geometry has replaced the old straight-edge & compass method, and not without good reason, but I can't help but feel that something is lost by ditching this method, if nothing but the hands-on approach.
So the question is this: Is there a text in Euclidean geometry which has a sufficient axiomatization to prove (and shows to the reader) all the propositions in The Elements, and is written in the same style?
For clarity, I am seeking a text which captures the two things about Euclid that make him stand out - the axiomatic approach as well as the constructive straightedge/compass method. I am not sure if a contemporary text has been written that uses solely these tools, but if one existed, that would be the most ideal.