A book that constructs analytic geometry from synthetic geometry In a math class I once had, the teacher constructed analytic geometry from synthetic geometry axioms. That is, given an Euclidean plane of points, he chose two perpendicular lines, along with two points, one on each line, one unit away from the two lines' intersection point. Then he gave each point on that plane an ordered pair of real numbers that measured their signed distance from those perpendicular lines. He then proved the usual theorems of analytic geometry. So for example, instead of defining lines with linear equations or defining distance by the distance formula, he proved that lines are solution sets of linear equations, and that the distance obeys the distance formula, along with lots of other results. Anyway, I would be very interested in some book that uses this approach.
 A: L'Enseignement de la géométrie by Choquet, Géométrie euclidienne plane by Doneddu, and Elementarnaya geometriya by Pogorelov (in Russian, with a Spanish translation titled Geometría elemental) all develop plane Euclidean geometry on very rigorous lines.
Choquet's book is closest to what you're asking for in making something of a bee-line for vector spaces with a scalar product.
Doneddu proposes a very natural and really interesting axiom system based on rigid motions. These axioms capture in a very intuitive way the content of the congruence axioms of most other systems.
Pogorelov's book is interesting in that, even though it's supposedly aimed at teachers, it is actually written in a way that addresses a pupil. It was sort of a "proof of concept" that such a rigorous approach could be taken with children, and though it would be too difficult as a school textbook itself, it served as the prototype for Pogorelov's later "Geometriya", which was widely used in schools in the U.S.S.R. starting in the early 1980s.
In English there are fewer options. There are books on the foundations of geometry that are aimed at quite a high level of mathematical maturity and demand a lot of effort. Perhaps something close to what you want is the textbook Basic Geometry by Beatley and Birkhoff, but their axiom system is ugly and assumes a lot from the start. I also think they fudge a bit in some places by appealing to a vague notion of "continuity," though I've forgotten the details.
If you know some rudimentary French or Spanish, it would be worth considering one of the first three books I mentioned.
