Since Ancient Greece that there have been attempts to use conic sections to make geometrical constructions. Actually, it seems that the aim of Apollonius of Perga, while writing his treatise on conics, was to develop the properties of these curves necessary for their application to the solution of geometrical problems.
What is known about which problems can be solved using conics? Of course, this assumes that the expression “using conics” becames defined as precisely as “using compass and straightedge”. Has that study ever been made? I would like to have references about this. In fact, I am not so much interested in what can be done using conics. What I would like to know is what can't be done, in a way similar to the proof by Pierre Wantzel of the fact that the problems of doubling the cube and of trisecting an angle cannot be solved using compass and straightedge only.
You might find Gibbins And Smolinsky's Geometric Constructions With Ellipses helpful.
In the final section it states that a regular polygon with $n$ sides is constructible with conics if and only if $\phi(n) = 2^s3^t$. (Here $\phi$ is the totient function, and the statement nicely parallels the requirement $\phi(n) = 2^s$ for ruler and compass constructions.
The proof is given in Videla's On Points Constructible from Conics (full reference in the G+S paper).
So, to your question in the comments, a 13-gon is constructible with conics because $\phi(13)=12$.