Conjugate conics and closed chains of Poncelet polygons

If a point x moves along a conic G then each polar of x with respect to a second conic A is tangent to one particular conic H, the conjugate of G with respect to A. In particular, if P is a Poncelet polygon, inscribed in G and circumscribed about A, then, the polygon P' whose vertices are the contact points of P on A is tangent to the conjugate conic H of G with respect to A. Hence P' is itself a Poncelet polygon for the pair A and H. P' is called dual to P. This process can be iterated. Astonishingly, there are very particular configurations, where this process closes after a finite number of steps, i.e., the n-th dual of P is again P. We identify such configurations of closed chains of Poncelet polygons and investigate their geometric properties.

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