Hi -
Geometry in the style of Euclid is called "synthetic" in distinction to using coordinates à la Dedekind, but I haven't heard coordinate geometry being called "analytic" and I'm not sure what Euclid was "putting together".
I have always heard "analytic geometry" used to mean coordinate geometry. From the Wikipedia article Analytic geometry <https://en.wikipedia.org/wiki/Analytic_geometry>: In mathematics <https://en.wikipedia.org/wiki/Mathematics>, *analytic
geometry*, also known as *coordinate geometry* or *Cartesian geometry*, is the study of geometry <https://en.wikipedia.org/wiki/Geometry> using a coordinate system <https://en.wikipedia.org/wiki/Coordinate_system>. This contrasts with synthetic geometry <https://en.wikipedia.org/wiki/Synthetic_geometry>.
Best, jb