Five Circles Theorem
Five circles have centers on a fixed circle with radii such that each intersects its two neighbors on the fixed circle. Then their other points of intersection give rise to the sides of a star polygon with vertices on the five circles. The radii of the first four circles can be picked at random as long as the circles complete the chain.
Click the button to pick circle radii at random.
 D. Wells, The Penguin Dictionary of Curious and Interesting Geometry, London: Penguin, 1991.