TITLE: Are there good mutually interpretable axioms for synthetic Euclidean and hyperbolic geometry?
QUESTION [14 upvotes]: There are familiar analytic equiconsistency proofs for Euclidean and hyperbolic geometry.  Those proofs are so robustly geometric that it seems like they must have synthetic analogues.
Looking into the literature, though, I wonder if I am too optimistic about this.  The most common rigorous axiomatizations for synthetic Euclidean and hyperbolic geometry, so far as I can tell, are Hilbert's from his Foundations of Geometry, with two changes.  They omit the non-elementary axiom of continuity, and they add something to assure that circles will actually have points of intersections with lines and other circles that they cross.  The only difference, in these axiomatizations, between Euclidean and hyperbolic geometry is in the axiom of parallels. But this synthetic hyperbolic geometry is incapable of some constructions that we take pretty much for granted both in Euclidean geometry and also (more to my point) in analytic hyperbolic geometry.   Notably $n$-section of lines, see trisection-of-a-hyperbolic-line-segment.  Maybe the analytic equiconsistency proofs really do not transfer well to synthetic geometry.
I know Greenberg's discussion of axiomatic issues in Greenberg. But I do not have all his references at hand.  Judging from the ones I do have (including Hartshorne) it seems likely that  his discussion of equiconsistency for the two geometries (pp.213-214) refers to analytic presentations of geometry.
Can I find mutually interpretable axioms for synthetic Euclidean and hyperbolic geometry?
Edit:  Erik Walsberg's comment about Tarski's axioms answers my title question, even though not in the way I had in mind when I wrote the text.  My text was ambiguous about "synthetic" methods, in just the way that Tarski What is elementary geometry? means when he says   "In colloquial language the term elementary geometry is used loosely [...with] no well determined meaning."   
Hilbert, Tarski, and Greenberg all show that the important logical distinction characterizing elementary methods is not between using or not using coordinates in some field.  It is between using or not using higher order notions like point-set continuity and limits.  First order algebraic considerations on fields (notably pythagorean fields) are already implicit in Euclid and  central to successful first-order elementary geometry.  
The work that led me to this question is about interpretation in first order logic, and not about compass (or horocompass) and straightedge or other such construction methods, and really not about avoiding coordinates.  So my title was true to my actual concern.  Some of my text concerning synthetic methods was less relevant (though those questions too intrigue me).  Walsberg's comment is really an answer both to the question as titled and to my actual concern, though he correctly saw I had another issue about methods also in mind.

REPLY [4 votes]: One should not forget Arthur Cayley's claim "And thus projective geometry is all geometry".
The treatise of Oswald Veblen and John Wesley Young, "projective geometry", follows that path. The first volume is dedicated to synthetic projective geometry. The second one builds both euclidean and non-euclidean geometry in that context, the first one by choosing a line (to be the line at infinity) and an involution on that line (to be the orthogonal relation on the set of directions), the second one by choosing a conic (to be the ideal points of the absolute circle at infinity). Then Veblen studies the group of transformations leaving invariant these datas, by pure synthetic methods. This is Klein original idea.
Both geometries are consistent because they are derived from projective geometry, and Veblen provides a set of axioms for both.
Nowadays, projective geometry is often built from affine geometry (the set of lines through a point), and that closes the loop if really you want to go from euclidean to non-euclidean geometry. But really, projective geometry should not be set aside, at least that's what the ancients say.
From the introduction of the second volume of Veblen and Young,
The ideal of such books should be not merely to prove every theorem rigorously but to prove it in such a fashion as to show in which spaces it is true and to which geometries it belongs.
The treatise is 990+ pages long and certainly achieves its goal. From the synthetic viewpoint, modern treatises such as Hartshorne and Greenberg certainly pale in comparison to the works of geometers from the beginning of the XX century.