$(A,C,D,E)$ are four fixed points on a fixed outer (red) ellipse and $B$ is a variable point lying between $A$ and $C$. Alternate vertices are joined to form a pentagonal star. (drawn on Geogebra).
A variable green inner ellipse circumscribing pentagon $(abcde) $ is formed by connecting five internal points of intersection. The green ellipse size and orientation, eccentricity etc. geometric properties are functions of position of $B$ on arc $AC$ as shown. If $B$ does not lie between $(A,C)$ then the green ellipse is not formed at all.
How are equations of red and green ellipses related with respect to the moving point $ B ? $ Do they have any common geometrical property? Can one property be obtained from the other?
Thanks in advance.