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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