Power of a point

For a point and a circle, its power is the product of the directed distances from the point to the two intersections of any secant. It equals the point’s squared distance from the center minus the radius squared.

Intersecting secants theorem: When two secants meet outside a circle, the products of each secant’s external and whole lengths are equal. Each secant product equals the external point’s power, so the theorem expresses its invariance.
Secant line: A line that intersects a circle at two distinct points. Its two intersection points supply the distances whose product defines power.
Radical center: The common intersection point of the three pairwise radical axes of three circles, when they are not parallel. Equal-power loci for three circles meet at a point with equal powers to all three.
Power of a point theorem: A theorem stating that the product of the two directed secant lengths from a point to a circle is independent of the secant. The theorem asserts the invariance; power names the value that remains invariant.
Tangent-secant theorem: For a tangent and a secant from the same external point, the tangent length squared equals the secant’s external length times its whole length. The tangent length squared is another expression for the same external point’s power.
Directed distance: A distance measured along a line with a chosen positive direction, allowing lengths to carry signs. Directed distances give power a consistent sign for points inside and outside a circle.
Coaxal system: A family of circles sharing the same radical axis. Their shared axis consists of points whose powers agree across the family.
Signed distance: A distance assigned a positive or negative value according to which side of a reference set contains a point. Power’s sign distinguishes interior points from exterior points and points on the circle.
Point-circle distance: The shortest distance from a point to a circle is the absolute difference between its center distance and radius. Together with the center distance and radius, it locates the point relative to the circle.
Euclidean distance: The straight-line distance between two points in Euclidean space. The familiar unsigned version of distance underlies the secant-length definition.

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