Six Circles Theorem Illustrated

Given a triangle, start with a circle tangent to two sides. Then draw a new circle tangent to this first circle and a different pair of sides. Continue this process in the same direction. The sixth circle will be tangent to both the fifth and the first circle, thus producing a cyclic chain of touching circles.

In the illustration above, the corners of the triangle can be dragged around. Similarly, the center of one of the circles is marked with a little dot which can also be dragged to change the size of the first circle of the chain. A full-screen version of the illustration is also available.

The paper The Six Circles Theorem revisited by D. Ivanov and S. Tabachnikov looks at some generalizations of the theorem. See also Theorem of the Day for further information.