### Six Circles Theorem (Bui Quang Tuan)

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A Mathematical Droodle

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Copyright © 1996-2018 Alexander BogomolnyThe applet is supposed to illustrate the following problem invented by Bui Quang Tuan:

Given Δ AC In addition, the six common points of tangency are concyclic.

_{1}A_{2}A_{3}, 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}, C_{6}, and so on, into angles A_{3}, A_{1}, A_{2}, A_{3}and so on, tangent to the previous circle. Then_{7}= C

_{1}.

The problem is a reformulation of a H. Eves' tricky quickie, adorned with Bui Quang Tuan's insight that Eves' points on the sides of a triangle can serve as points of tangency common to successive circles in a chain of six. The points of tangency are concyclic on a circle with center at the incenter of the triangle.

In addition, the centers of the circles form a paraxegon.

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