The power of a point is used in many geometrical definitions and proofs.

The th power of a point process, is defined on the product space as follows :

The power of a point arises in the special case that one of the radii is zero.

More generally, Laguerre defined the power of a point with respect to any algebraic curve in a similar way.

As another example, the inscribed angle theorem is the basis for several theorems related to the power of a point with respect to a circle.

The fundamental purpose of this weapon seems to have been to develop a sling shot with the penetrative power of a point.

Book 3 deals with circles and their properties: inscribed angles, tangents, the power of a point, Thales' theorem.

This proof uses the power of a point theorem directly, without the auxiliary triangles obtained by constructing a tangent or a chord.

Specifically, the power of a point P with respect to a circle C of radius r is defined (Figure 1)

The power of a point is also known as the point's circle power or the power of a circle with respect to the point.

In the other two cases, when A is inside the circle, or A is outside the circle, the power of a point theorem has two corollaries.

In elementary plane geometry, the power of a point is a real number h that reflects the relative distance of a given point from a given circle.

From this, he derived a lemma corresponding to the power of a point theorem, which he used to solve the LPP case (a line and two points).

The use of the Pythagorean theorem and the tangent secant theorem can be replaced by a single application of the power of a point theorem.

Whereas Poncelet's proof relies on homothetic centers of circles and the power of a point theorem, Gergonne's method exploits the conjugate relation between lines and their poles in a circle.

Using the intersecting chords theorem (also known as power of a point or secant tangent theorem) it is possible to calculate the radius of a circle given the height and the width of an arc:

In the case when the algebraic curve is a circle this is not quite the same as the power of a point with respect to a circle defined in the rest of this article, but differs from it by a factor of d.