I am trying to write a clean proof (which is visually clear to anybody, so I am not even sure how much explanation I need to write for this proof) of a geometric fact:
I am given a circle (with changing radius) that has an non-tangent intersection with a fixed conic section at time 0 on the real plane. The radius of the circle is a continuous positive function of the position of the center. I am allowed to move this circle continuously such that the number of intersections is always constant (so the circle is not allowed to be tangent to the conic at any point in time while it is moving). I want to prove that any such motion for which the center of the circle at time 0 and at time 1 are equal, cannot permute the intersection points of the circle with the conic. So, for such a motion, if $Z(t)$ is the center of the circle at time $t$, and $Z(0) = Z(1)$ and A,B,C,D are points of intersections of the circle and the conic at time 0 in clockwise ordering at a reference angle then the intersection points of the conic with the circle at center $Z(1)$ will also be A,B,C,D (following the same ordering). I think the proof can be written in two or three lines but I am just really stumbling with words.
I don't mind using some sophisticated algebraic topology (if need be), I just want it to be short and clean. Has anyone a clean proof of this?