Conic sections on the plane have the same properties to objects of synthetic geometry as the proofs in, say, Apollonius of Perga's synthetic geometry book on conics. This means algebraically-defined curves derived from a cone (parabola, hyperbola, circle, ellipsis) in the plane will have the same relations as conic sections in a cone to circles, squares, triangles, and 3D synthetic geometrical objects, i.e., geometrical objects defined without algebra.
However, are there other plane curves which can be treated in synthetic geometrical terms, i.e., without the use of algebra, Euclid-style? For instance, do some polynomial or rational curves have properties that can be applied or derived from determinate 2D or 3D synthetic geometrical objects? Are there books written about the subject? I know that Newton's predecessors seemed to have some stuff on the matter, but I'm pretty lost in terms of references to study the matter further, and in particular, I wonder if there has been anything written on this after Newton.
Thank you!