Poncelet's Closure Theorem

If an $n$-sided Poncelet Transverse constructed for two given Conic Sections is closed for one point of origin, it is closed for any position of the point of origin. Specifically, given one Ellipse inside another, if there exists one Circuminscribed (simultaneously inscribed in the outer and circumscribed on the inner) $n$-gon, then any point on the boundary of the outer Ellipse is the Vertex of some Circuminscribed $n$-gon.

References

Dörrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, p. 193, 1965.