A Chain of Touching Circles in a Polygon
(à la Evelyn)
The applet illustrates an extension of a 6 circles theorem from triangles to convex polygons with more than 3 sides.
In a convex n-sided polygon, n ≥ 3 and odd, any chain of circles that touch their neighbors and successive pairs of sides of the polygon counts either at most 2n circles. For even n, the distance between the 1 and the 2n-th circle is independent of the radius of the first circle. For some polygons (e.g., inscriptible
if n = 4)
the chain closes with n circles; for others, it remains always open.
For regular polygons, the length of the chain may be N, N+1, and 2N.
The situation appears to be analogous to the theorem of a chain of circles with fixed centers.
(The applet can emphasize successive pairs of circles counting from 0 to 2n - 1. When that attribute equals 2n, all the circles are displayed in the same color.)
Copyright © 1996-2018 Alexander Bogomolny