Center (of a Circle)
The center of a circle is the fixed interior point that is equidistant from every point on the circle.
Definition
The center of a circle is the point in its exact middle, equidistant from every point on the circle; it is not part of the circle itself, but the reference point every point on the circle is measured from. For the circle $(x-h)^2+(y-k)^2=r^2$, the center is $(h,k)$, and it can be found from any three points on the circle as the intersection of the perpendicular bisectors of two chords, equal to the circumcenter of the triangle those three points form. In inversive geometry, the center is the unique point mapped to infinity by inversion in the circle.
Example
The center of a coin or a bull's-eye target is its exact middle point, and the compass pin marking a drawn circle marks its center. For points $A(0,0)$, $B(4,0)$, $C(0,3)$ on a circle: the perpendicular bisector of $AB$ is $x=2$, of $AC$ is $y=1.5$, giving center $(2, 1.5)$, and the distance from this center to $A$ is $\sqrt{4+2.25}=2.5$, the radius. In inversive geometry, inverting circle $C$ (center $O$, radius $r$) sends $O$ itself to the point at infinity, so all circles through $O$ map to lines.
Key Insight
Change the center and you get a different (shifted) circle; the center defines the circle even though it is not part of it. Finding the center as the intersection of perpendicular bisectors is the same procedure as finding a triangle's circumcenter, showing the deep connection between circle centers and triangle circumcenters, and the center's special role under inversion, being sent to infinity, reveals the deep projective structure of circle geometry, where lines are understood as circles through the "point at infinity," unifying circles and lines into a single family.