Arthur Holshouser, Stanislav Molchanov, and Harold Reiter,
A special case of Poncelet's problem,
Forum Geometricorum, 16 (2016) 151--170.

Abstract. A special case of Poncelet's Theorem states that if circle C_2 lies inside of circle C_1 and if a convex n-polygon, n >= 3, or an n-star, n>= 5, is inscribed in C_1 and circumscribed about C_2, then there exists a family of such n-polygons and n -stars. Suppose C_2 lies inside of C_1 and R, r, are the radii of C_1, C_2 respectively and rho is the distance between the centers of C_1, C_2. For n >= 3 we give an algorithm that computes the necessary and sufficient conditions on R, r, rho, where R > r + |rho|, r > 0, so that if we start at any arbitrary point P on C_1 and draw successive tangents to C_2 (counterclockwise about the center of C_2) then we will return to P in exactly n-steps and not return to P in fewer than n-steps. This will create the above family of n-polygons and n-stars. The algorithm uses nothing but rational operations. At the end we illustrate this rational algorithm for n=3,4,5,6,7 and we will then see an invariant begin to emerge.

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