John Steinberger asks in
quick question: what is the set of points equidistant from a given
point off a parabola to the parabola? Parabolas are themselves defined
as the set of points equidistant from a given point to a line, and I
would like to know what happens when we apply this definition
recursively, replacing the line by the parabola, then the parabola by
whatever we get next... also it would be nice to know the set of
points equadistant between a parabola and a line (it might look like a
parabola but I know it can't be, because a symmetry argument would
prove there are two focii) or between two parabolas.

I have a suspicion these curves might be the intersection of a plane
and the body of revolution of a conic curve...
The interactive construction below shows this set of points as the orange curve: You must enable Java for an interactive construction.

Created with Cinderella by Ulrich Kortenkamp