# Touching Circles With Given Centers II What Is This About? A Mathematical Droodle

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Explanation ### 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 A0A2...An-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 B0 be any point on An-1A0. Form A0(B0) and find its intersection B1 with A0A1. Form A1(B1) and find its intersection B2 with A1A2. Continue in this manner until you find the intersection of An-1(Bn-1) with An-1A0. This may or may not coincide B0. Just in case we denote it C0 and go on constructing the sequence of circles Ak(Ck) and points Ck as before, until we determine the intersection Cn-1 of An-2(Cn-2) with An-2An-1.

Let D be the intersection of An-1(Cn-1) with An-1A0. Then,

If n is odd,
1. D = B0 and the construction leads to no new point or circles.
2. B0C0 = B1C1 = ... = Bn-1Cn-1, the common quantity being dependent on B0.
3. Midpoints Mi of segments BiCi are fixed and do not depend on B0.
4. For B0 = M0, Bi = Ci, for all I, and naturally Ai(Bi) = Ai(Ci) also.

If n is even,

1. D ≠ B0 1 and the chain of circles may be continued indefinitely.
2. B0C0 = B1C1 = ... = Bn-1Cn-1 = t = constant independent of B0.
3. A pair of circles Ai(Bi)Ai(Ci) at vertex Ai can be thought of as a single circle drawn with a pen of thickness t.
4. Thickness t depends on the polygon but not on the position of B0. For polygons with t = 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 B0.

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The reader is referred to several relevant articles: 