abstract
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Conjugate conics and closed chains of Poncelet polygons

Lorenz Halbeisen and Norbert Hungerbühler

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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.