The power of a point with respect to a circle centered at is a measure of distance from the point to the circle, defined by
which implies that
- points inside the circle have negative power;
- points on the circle have zero power;
- points outside the circle have positive power.
For points outside the circle, the power is equivalent to the square of the length of its tangent to the circle. It is also the square of the radius of an orthogonal circle centered at . Unfortunately, points inside the circle have no particularly good geometric interpretation.
The power of a point is used extensively in computational geometry, including in defining the radical axis (and thus radical center). It is better known for the power of a point theorem, which writes the power of a point in two different ways to show the equality between them. There are two cases to this theorem; the two secants form and the tangent-secant form.
Let be a circle and a point. Let a line through which meets at point and , and let be another line through which meets at points and . Then
There is a special case when lies outside of the circle and one of the lines is a tangent, in which case