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Take a triangle. For each circle tangent to two triangle sides, draw a circle tangent to it and to two sides of the triangle. Iterating the construction you get a circle tangent to the first circle after six steps. This Demonstration is a sort of visual proof. Choose the triangle with the locators and choose the size of the circle you start with. Choosing solution 1 or 2 selects the corner for the second circle. Increase the number of circles. Finally the seventh circle will appear again as the first one.
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