Arthur Holshouser, Stanislav Molchanov, and Harold Reiter,
Applying Poncelet's theorem to the pentagon and the pentagonal star,
Forum Geometricorum, 16 (2016) 141--149.
Abstract. A special case of Poncelet's Theorem states that if all points on circle C_2 lie inside of circle C_1 and if a convex n-polygon, n >= 3, or an n-star, n >= 5, is inscribed in circle C_1 and circumscribed about circle C_2, then there exists a family of such n-polygons and n-stars. Suppose all points on C_2 lie inside of C_1, 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, in a companion paper 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 Q on C_1 and draw successive tangents to C_2 (counterclockwise about the center of C_2) then we will return to Q in exactly n steps and not return to Q in fewer than n steps. This will create the above family of n-polygons and n-stars. However, when n>= 5, this companion paper relies on computers to find these conditions. In some ways, this is a sign of defeat. In this paper, we illustrate for n=5 a technique that can compute these exact same necessary and sufficient conditions on R,r,rho without using a computer.
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