Here a line of fixed length is moved along the edge of an ellipse, tracing out a collection of new shapes.

Here a line of fixed length is moved along the edge of an ellipse, tracing out a collection of new shapes. Consider the area of the shape traced by a point p units from one end of the line and q from the other. Holditch’s theorem, regarded as a milestone in the history of maths, tells us this area is less than the area of the ellipse by at least π×p×q. Curiously, this formula holds not just for an ellipse, but any closed curve.

Reference:
http://en.wikipedia.org/wiki/Holditch's_theorem
Story and animation via  Matthew Henderson