If you've ever brushed the sides of a tub with a wide upright broom you'd have noticed that only the ends touch the tub wall. Consider the tub sides elliptical in shape so the broom forms a chord that is deeper at major axis ends of the ellipse than the minor.
I) Is there a condition where the handle (midpoint of brush chord) traces a circle?
Obvious limiting cases are: (i) a zero chord width broom path matching the ellipse itself when circular, and then (ii) a tub Diameter width(ed) broom chord, whose mid-point is fixed to a Center of a then also circular tub.
II) If it ever does traverse a circle between those limits, then is it (i) a unique configuration (of major, minor axes and chord lengths) or is it (ii) a many solution (functional or manifold) in R^2?
For extra credit, III) If we admit Complex fields can we always find a circular path for the now imaginary broom?
Suppose that the midpoints did trace a circle.
By symmetry, the chords bisected by the major and minor axes of the ellipse have to be tangent to the circle. But if $M, E$ are the midpoint and an endpoint of a chord bisected by the major axis, and $m, e$ are the same for the minor axis, and $C$ is the center of the circle, then $CM = Cm$ because $M, m$ are both points on the circle, and $ME = me$ since the chords are all the same length. Further $\triangle CME$ and $\triangle Cme$ are both right triangles. Thus by Pythagorus, we must have $CE = Ce$. But that is only true when the ellipse is a circle.
I.e., the only time the midpoints of the chords trace out a circle is when the original ellipse is a circle.