The question: Five points define a conic, so let points $A,B,C,D,E$ and $F,G,H,I,J$ define two conics $c$ and $d$. Is there a geometric construction that identifies the common points of $c$ and $d$?
(It's easy enough to do it in Geogebra, or algebraically. What I'm asking for here is a ruler and compass construction.)
Background and context Given, say, an ellipse $c$ it is not too difficult to construct tangents from a point $P$ to $c$ using just a straightedge. But this assumes that the ellipse has been drawn as a given. If only the points $A,B,C,D,E$ on $c$ have been given, you can easily construct the polar of $P$ but then have to construct the intersection of the polar and the ellipse. This is less straightforward but the construction is described in several 19th century projective geometry texts (e.g. Cremona or Russell),and requires a reference conic (or compass) in addition to a straightedge.
What I haven't been able to find is a construction for the intersection of two ellipses/conics that are not pre-drawn but given only as two sets of five points each. Hence my question. It came up in following a certain construction for "imaginary" chords that involved finding the common points of two conical loci (Russell, XXVII 6.).
Update: Given two or three points of intersection it is possible to construct the remaining points of intersection.