This article is about the center of a geometric object. For the center of a group, see Center (group theory).
For example, all the points of a circle are equidistant from the center, however the center is not a point on the circle because the circle is only the rim and has empty interior. In contrast, the center of a disk is a point of the disk.
The center of a square is not the same distance from every point on the square--the corners are further away from the center than the midpoints of the sides. The center of a square is the point where the two diagonals intersect.
Figures like squares, triangles, ellipses, etc. are why we must say "on average" in the definition of "center". While not all points are the same distance away from the center of the figure, the distances will average to be the same value and cancel out.