Touching Circles With Given Centers
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A Mathematical Droodle
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Copyright © 1996-2018 Alexander Bogomolny
Touching Circles With Given Centers
The applet illustrates the following theorem [Beardon]:
Given an n-gon. If n is odd, there always exists a unique sequence of circles with centers at the vertices of the polygon in which each circle touches its immediate neighbors. If n is even, no such sequence may exist; if it does it is not unique.
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Proof
Let a_{1}, a_{2}, ..., a_{n} denote the successive side lengths of the polygon. The circles that solve the problem exist iff there exist real numbers r_{1}, r_{2}, ..., r_{n} that could serve as the radii of the pairwise touching circles:
(1) |
r_{1} + r_{2} = a_{1} r_{2} + r_{3} = a_{2} ... r_{n} + r_{1} = a_{n}. |
Assume r_{1} has been found. Then, using the first n-1 equations (1), we can successively determine the other radii:
(2) |
r_{2} = a_{1} - r_{1} r_{3} = a_{2} - r_{2} = a_{2} - a_{1} + r_{1} r_{4} = a_{3} - r_{3} = a_{3} - a_{2} + a_{1} - r_{1} ... r_{n} = a_{n-1} - r_{n-1} = a_{n-1} - a_{n-2} + a_{n-3} - ... ±r_{1}. |
What about the last equation in (1)? Apparently, we should be able to exclude r_{n} as we did with all other equations (2):
(3) | r_{1} = a_{n} - r_{n} = a_{n} - a_{n-1} + a_{n-2} - ... ±r_{1}, |
where ±r_{1} means (-1)^{n-1}, i.e., for n even, r_{1} comes with the sign "+", whereas, for n odd, the sign is "-". In the latter case, we get an equation for r_{1} from:
(4) | r_{1} = (a_{n} - a_{n-1} + a_{n-2} - ... + a_{1}) / 2. |
Which serves as the starting point for (2), such that all the radii are found in a unique way. For n even, (3) becomes
(5) | a_{n} - a_{n-1} + a_{n-2} - ... - a_{1} = 0. |
Unless (5) holds, the problem has no solution. When a solution exists, it is not unique. Indeed, assume that the sequence of radii r_{1}, ..., r_{n}, where n is even, solves (1). Pick any number s. By direct inspection, it is clear that the sequence
(6) | r_{1} + s, r_{2} - s, r_{3} + s, ..., r_{n-1} + s, r_{n} - s |
also satisfies (1). There are no other solutions. (For n even, the applet displays a scroll bar that permits modification of a single parameter. By shifting any of the vertices, one can "find" a geometric solution to the problem. Once one is found, toying with the scroll bar shows that indeed there are many more.)
Note 1
The system (1) may have some solutions negative. I strip the sign minus so that all the circles are still defined. Those whose radii is a negative number turned positive touch their neighbors internally.
Note 2
In an unexpected way exactly same system of equations describes another problem: construct a polygon by the midpoints of its edges. For n odd, the problem has a unique solution. For n even, it either has no solution, or infinitely many solutions.
Note 3
For n = 4, the condition (5), which says that the sums of the opposite side lengths are equal, is equivalent to saying that the four points where the circles touch their neighbors are concyclic.
For a different outlook on the configuration see a separate article.
References
- T. Beardon, The Mid-Edges Theorem, in The Changing Shape of Geometry: Celebrating a Century of Geometry and Geometry Teaching, edited by Chris Pritchard, Cambridge University Press and MAA, 2002, pp. 179-183
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