I am looking for (a class of) symmetric polygon(s) that can have any number (odd and even) of points on its boundary, equidistant to their neighboring points, and when all pairs of points are connected by a straight line, the centers of those lines do not overlap.
For example, when the corners of a regular pentagon are connected, the centers of the four lines do not overlap, and the condition holds. However, when the corners of a square are connected, the centers of both lines overlap, and thus the condition does not hold.
I have no background in geometry, please help clarify the question if needed!