Six Circles Theorem

change size of first circle

0.5

which solution?

1

2

solution 2: second circle in the right corner

how many circles?

1

2

3

4

5

6

7

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.