Can one describe conic sections using synthetic geometry? I was curious to know whether synthetic geometry is more powerful than analytical geometry. Take as an example, the conic sections. Can one describe conic sections without the co-ordinate system, without the equations of conic sections? How difficult would it be?
I havent read it but I think that this book might be what you are looking for : 
https://archive.org/stream/conicsectionstre00besarich#page/n12/mode/1up

In the introduction we have 


  The object of the following pages is to discuss the general forms and
  characetistics of these (conic sections) curves and to determine their
  most important properties by help of the methods and relations
  developed in the first six books , and in the eleventh book of
  Euclid's Elements and it will be found that for this purpose a
  knowledge of Euclid's Geometry is all that is necessary.


Also, I am also looking for books of this type so if you have found some yourself please post them as an answer or as a comment.
Yes, synthetic geometry works well for plan sections in conics in finding the focus points, the axes, the directors line positions. Danut Dragoi, PhD