Center (of a Circle)
GeometryThe 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 the exact middle of the circle. Every point on the circle is the same distance from the center. The center is not on the circle itself - it is inside.
Example
The center of a coin is the exact middle point. The center of a bull's-eye target is the innermost point. In a circle drawn with a compass, the compass pin marks the center.
Key Insight
The center is what defines the circle: change the center and you get a different (shifted) circle. The center is NOT part of the circle itself - it is the reference point that every point on the circle is measured from.
Definition
The center of a circle is the point equidistant from all points on the circle. For the circle $(x-h)^2 + (y-k)^2 = r^2$, the center is $(h, k)$. The center can be found from any three points on the circle by finding the intersection of the perpendicular bisectors of two chords.
Example
Three points on a circle: $A(0,0)$, $B(4,0)$, $C(0,3)$. Perpendicular bisector of $AB$: $x=2$. Perpendicular bisector of $AC$: $y=1.5$. Center $= (2, 1.5)$. Verify: distance from center to $A = \sqrt{4+2.25} = \sqrt{6.25} = 2.5 =$ radius.
Key Insight
Finding the center as the intersection of perpendicular bisectors is the same as finding the circumcenter of the triangle formed by three points on the circle. This shows the deep connection between circle centers and triangle circumcenters.
Definition
The center of a circle is the unique point equidistant from all points on the circle, equal to the circumcenter of any triangle inscribed in the circle. In complex analysis, the center is the point $z_0$ such that $|z - z_0| = r$ for all $z$ on the circle. In inversive geometry, the center is special: inversion in the circle maps the center to the point at infinity.
Example
In inversive geometry, a circle $C$ with center $O$ and radius $r$ inverts the center $O$ to the point at infinity: $I(O) = \infty$. All circles passing through $O$ are mapped to lines (circles through infinity). This explains why inversion maps circles to circles or lines.
Key Insight
The center's role in inversion (mapped to infinity) reveals the deep projective structure of circle geometry. In projective geometry, lines are circles through the "point at infinity." Inversion interchanges ordinary points and this ideal point, unifying circles and lines into a single family.