# Touching Circles With Given Centers II

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A Mathematical Droodle

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Copyright © 1996-2018 Alexander Bogomolny

### Touching Circles With Given Centers II

The applet illustrates an extension of a theorem of touching circles with the centers at the vertices of a polygon. The extension has been suggested by Bui Quang Tuan.

For two points, X and Y, X(Y) denotes the circle centered at X and passing Y. Let A_{0}A_{2}...A_{n-1} be an n-sided polygon. We are going to construct a sequence of points on the side lines of the polygon and circles with centers at its vertices. Let B_{0} be any point on A_{n-1}A_{0}. Form A_{0}(B_{0}) and find its intersection B_{1} with A_{0}A_{1}. Form A_{1}(B_{1}) and find its intersection B_{2} with A_{1}A_{2}. Continue in this manner until you find the intersection of A_{n-1}(B_{n-1}) with A_{n-1}A_{0}. This may or may not coincide B_{0}. Just in case we denote it C_{0} and go on constructing the sequence of circles A_{k}(C_{k}) and points C_{k} as before, until we determine the intersection C_{n-2}(C_{n-2}) with A_{n-2}A_{n-1}.

Let D be the intersection of A_{n-1}(C_{n-1}) with A_{n-1}A_{0}. Then,

- D = B
_{0}and the construction leads to no new point or circles. - B
_{0}C_{0}= B_{1}C_{1}= ... = B_{n-1}C_{n-1}, the common quantity being dependent on B_{0}. - Midpoints M
_{i}of segments B_{i}C_{i}are fixed and do not depend on B_{0}. - For B
_{0}= M_{0}, B_{i}= C_{i}, for all I, and naturally A_{i}(B_{i}) = A_{i}(C_{i}) also.

If n is even,

- D ≠ B
_{0}1 and the chain of circles may be continued indefinitely. - B
_{0}C_{0}= B_{1}C_{1}= ... = B_{n-1}C_{n-1}= t = constant independent of B_{0}. - A pair of circles A
_{i}(B_{i})A_{i}(C_{i}) at vertex A_{i}can be thought of as a single circle drawn with a pen of thickness t. - Thickness t depends on the polygon but not on the position of B
_{0}. For polygons witht = 0, the construction gives a series of n-1 circle, each touching its immediate neighbors (cyclically). For such polygons, the chain of circles could be constructed starting with any B_{0}.

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The reader is referred to several relevant articles:

- Touching Circles With Given Centers II
- Six Circles Theorem (Bui Quang Tuan)
- Two Triples of Concurrent Circles
- About a Line and a Triangle
- Around the Incircle

|Activities| |Contact| |Front page| |Contents| |Geometry|

Copyright © 1996-2018 Alexander Bogomolny

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