Six Circles Theorem (Bui Quang Tuan)
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A Mathematical Droodle

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Copyright © 1996-2018 Alexander Bogomolny

The applet is supposed to illustrate the following problem invented by Bui Quang Tuan:

Given Δ A1A2A3, start with inscribing a circle C1 into ∠A3A1A2 and note point T12 of tangency on the side A1A2. Next inscribe circle C2 into ∠A1A2A3 so that it is tangent to A1A2 at T12 and note point T23 of tangency with side A2A3. Continue inscribing circles C3, C4, C5, C6, and so on, into angles A3, A1, A2, A3 and so on, tangent to the previous circle. Then C7 = C1. In addition, the six common points of tangency are concyclic.

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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Copyright © 1996-2018 Alexander Bogomolny


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