Around the Incircle in a Polygon
Elsewhere I described the problem from C. W. Trigg's Mathematical Quickies. Trigg's incorrectly attributes the problem to Howard Eves, whereas the problem has been posed by A. H. Stone of the Institute for Advanced Studies. H. Eves was the author of the published solution. The proof is accompanied by a note (with no indication of the source) which is repeated by Trigg as well: The proof generalizes to any odd polygon possessing an incircle.
The applet below shows that this is indeed so. (The hollow blue and the solid red points are draggable.) In addition, it demonstrates that for even polygons, the proof also works, except that the second loop around the circle generates exactly the same points such that the process closes in on the first loop regardless of the starting point. This is similar to the behavior of the process for odd polygons that start at one of the points of tangency of the incircle. For the even polygons, the choice of the starting point is inconsequential.
What if applet does not run? |
The even polygons with side lengths a_{1}, ..., a_{n} have the property that the sum of odd numbered sides coincides with the sum of even numbered sides:
(1) | a_{1} + a_{3} + ... = a_{2} + a_{4} + ... |
The proof of which is an immediate consequence of the fact that two tangents from a point to a circle are equal. For a quadrilateral, the condition is equivalent to saying that the sums of the opposite sides are equal. For a quadrilateral, the condition is both necessary and sufficient for having an incircle. For larger n, the condition is only necessary. I.e., there exist n-gons, with even n, for which (1) holds but which do not possess an incircle. For such polygons, Stone's sequence of rotations still closes. On realization of this fact, it also becomes clear that an analogous statement must be true for odd polygons as well. The original note in the American Mathematical Monthly mentions that Joseph Rosenbaum remarked that, concerning odd polygons, the construction may close even if there is no incircle. For, given a polygon, it may be possible to make the sides touch a circle, without altering their lengths, by suitably changing the angles. This process would not affect the positions of the intermediate points on their respective sides.
It is perhaps easier to see this point starting from the end: a triangle is a rigid figure, but not so an n-gon, with
References
- C. W. Trigg, Mathematical Quickies, Dover, 1985, #181 (Am Math Monthly, 50 (June, 1943), 391)
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