# A Chain of Touching Circles in a Polygon

(à la Bui Quang Tuan)

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The applet illustrates an extension of a 6 circles theorem due to Bui Quang Tuan from triangles to convex polygons with more than 3 sides.

Given a convex n-gon AC

_{1}A_{2}...A_{n}(n ≥ 3 and odd), start with inscribing a circle C_{1}into ∠A_{3}A_{1}A_{2}and note point T_{12}of tangency on the side A_{1}A_{2}. Next inscribe circle C_{2}into ∠A_{1}A_{2}A_{3}so that it is tangent to A_{1}A_{2}at T_{12}and note point T_{23}of tangency with side A_{2}A_{3}. Continue inscribing circles C_{3}, C_{4}, C_{5}, and so on, into successive angles. Then_{2n}= C

_{1}.

For regular polygons, the length of the chain may be 1 or 2N.

The situation appears to be analogous to an extension of another 6 circles theorem.

(The applet can emphasize successive pairs of circles counting from 0 to

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