I am impregnated with the idea that pure mathematics is when you do mathematics in such a way that you begin with some set of axioms and definitions, and then you develop such things by means of logic.
So I am searching for a book or notes or a website where this is done in the case of analytic geometry. This "thing" must have at the beginning of each concept their formal definitions and the theorems, corollaries, lemmas, etc.
Can someone help me with something like this?
On
Try "Elementary Geometry From an Advanced Standpoint" by Edwin Moise.
It is not purely axiomatic but it is a rigorous development of the field and probably in the spirit of what you seek.
On
This is the exact purpose of the book Linear Algebra and Geometry by Dieudonné.
He starts with a set of assumptions about the real number system. Then he gives axioms for the Euclidean plane and Euclidean space, which amount to assuming given a two- or three-dimensional affine space whose associated vector space is endowed with an inner product.
The rest of the book is devoted to establishing analytic foundations for the usual geometric concepts.
The exercises have full solutions at the back of the book.
Analytical Geometry can be seen as a combination of calculus/analysis, algebra and geometry. There exist literature on Analytic Geometry where the focus on these three subjects vary.
The following books might be helpful (all full PDFs available by Google Scholar Search):
The last one I like very much, since it is more technical in terms of algorithms and one can play with the commands directly and get visual output using Wolfram. Please check the Webportal WolframAlpha as well.
An extensive list of Analytic Geometry resources is provided by this web page of MAA (Mathematical Association of America)