It appears I have discovered something for myself:
Take a polygon with an odd number of sides, e.g. a nonagon (9 sides for clarification.) Draw it so a horizontal side is facing down and a point is at the top. Then, draw a polygon with twice the number of sides so that the bottom side is in the same space as the bottom side of the smaller polygon. then draw a circle the size of the larger polygon (circumscribed) and its center should land right on the topmost point of the smaller polygon. When this is done with circles, where one is half the size of the other, and is tangent to the inner surface of the larger one, one end touches the circle, and the other passes through the radius.
But polygons aren't exactly circular, and their heights aren't proportional in the same way circles are. Another thing is that the triangle and hexagon are pretty obvious, as one stacks to form an analog of the other, but a pentagon doesn't stack to make a decagon in the same way.
If anyone else has figured this out, please explain, though I feel the explanation is rather simple. I am a stickler for knowing what is known and what is unknown, so not knowing the possibility of someone else knowing this bugs me.
The phenomenon I've seen:
edit Another thing I've noticed is that when I draw a line along any of the smaller polygon's sides, each line passes through at least 2 points of the larger, or at least comes very close to the latter.
Best Answer
The reason is the angle at $C$ is one half of the central angle at $O$ (see figure below). Hence $C$ is the center of a polygon with the same side $AB$ but doubled number of sides.