Power of a point

Definition

Let $C$ be a circle in the Euclidean plane $\mathbb{R}^2$ with the centre $O$ and the radius $r$ , and let $d$ be the Euclidean distance. The power of a point $P \in \mathbb{R}^2$ with respect to $C$ is the real number $\Pi(P) = d(P, O)^2 - r^2.$

Remarks

$\Pi(P) > 0$ if $P$ is outside $C$ , $\Pi(P) = 0$ if $P$ is on $C$ , and $\Pi(P) < 0$ if $P$ is inside $C$ . For $P$ outside $C$ and a tangent from $P$ that touches $C$ at $T$ , the Pythagorean theorem gives $\Pi(P) = d(P, T)^2$ . Let a line through $P$ meet $C$ at $S_1$ and $S_2$ . Then $d(P, S_1)\, d(P, S_2) = |\Pi(P)|$ . This contains the intersecting chords theorem and the intersecting secants theorem. Jakob Steiner introduced the power of a point in 1826.

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Circle concept
Euclidean distance concept
Power of a point definition

Tag	`00PPY3RPNB`
URI	`m2://00PPY3RPNB`
Kind	definition
Writing	W1 stated
MSC	51M04
Defines	Power of a point
Uses	Circle, Euclidean distance

Power of a point

Definition

Remarks

Reading path

Further reading

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