Despite your attempt to obfuscate the issue with your wall of text, the simple fact of the matter is that you didn't give a single example of analytic geometry (i.e. coordinate geometry), which is based on the explicit use of a coordinate system.

> Use of formulae is outside the scope of pure geometry ala Euclidian style.

Like Heron's formula?

> Trigonometry using algebra is part of analytic geometry.

Yes, a part of analytic geometry. Distances and angles are also a part of analytic geometry.

> Explicit use of coordinate system is not essential.

Yes it is. That's literally what analytic geometry (i.e. coordinate geometry) means.