This is something I noticed a few years ago, which, as far as I know, has never been noticed before:
1. Take any regular polygon and circumscribe a circle around it.
2. Inscribe a circle in the same polygon. This defines two concentric circles, derived from the parameters of the polygon.
3. Draw a straight line through the common center of these two concentric circles, such that where this line intersects the outer of the two concentric circles, this defines two points x and y, while, where the line intersects the inner of the two concentric circles, this defines two points a and b.
BATTERSON'S THEOREM: If the line xy is the major-axis of an ellipse, and the line ab is the distance between the foci of this same ellipse, then, the length of the minor-axis of the ellipse defined will always equal the length of the side of the polygon used in the construction.
Now, is that pretty, or what?