Is every proposition on Cartesian geometry provable on synthetic Euclidean geometry?

Question

Obviously everything that is associated with coordinates can’t be analyzed within synthetic geometry. But existence, measure and incidence statements are provable; since Cartesian geometry is an Euclidean geometry, we know that every statement on Euclidean geometry is provable — but is the reverse true? For example, the Euclidean plane satisfies all the projective geometry postulates, but there are finite projective geometries - so not all Euclidean geometry propositions are provable within projective geometry. Does the same apply to Cartesian and Euclidean geometries?