Seven Circles Theorem

Draw an initial circle, and arrange six circles tangent to it such that they touch both the original circle and their two neighbours. Then the three lines joining opposite points of tangency are concurrent in a point.

This theorem was not discovered until 1974 by Evelyn, Money-Coutts, and Tyrrell.

This is the mathematical description but what results has a balance and elegance. In particular, the small seemingly insignificant sections play a vital role in completing the picture.