[Yet another geometry question that involves no geometry exercises]
So after reading various sources on the internet, it seems like Analytic Geometry is an extension of Synthetic Geometry (or a more tool rich replacement). That is, all the theorems of Euclid are automatically true in the R^2 plane or space because the coordinate systems are simply just something "overlaid" (like grid marks) on the regular Euclidean plane or space.
The advantage of overlaying these grid marks is that we can use analysis and algebra to solve harder geometry problems, correct? And correct me again, anything we prove (where the fact doesn't involve the coordinates explicitly) using this extra machinery is automatically true without coordinate systems.
----- Now, slightly off topic, but something that's confusing me:
Are the Cartesian coordinates more "fundamental" then other coordinate systems? When someone says R^n do we mean the space or the space+coordinate system? Sometimes I read "Cartesian space" for R^n, but what about calling R^n "polar space" (would that be slightly more silly or way more silly or not silly at all) ?