Power of a point

In elementary plane geometry, the power of a point is a real number that reflects the relative distance of a given point from a given circle. It was introduced by Jakob Steiner in 1826.^[1]

Specifically, the power ${\displaystyle \Pi (P)}$ of a point ${\displaystyle P}$ with respect to a circle ${\displaystyle c}$ with center ${\displaystyle O}$ and radius ${\displaystyle r}$ is defined by

${\displaystyle \Pi (P)=|PO|^{2}-r^{2}.}$

If ${\displaystyle P}$ is outside the circle, then ${\displaystyle \Pi (P)>0}$,
if ${\displaystyle P}$ is on the circle, then ${\displaystyle \Pi (P)=0}$ and
if ${\displaystyle P}$ is inside the circle, then ${\displaystyle \Pi (P)<0}$.

Due to the Pythagorean theorem the number ${\displaystyle \Pi (P)}$ has the simple geometric meanings shown in the diagram: For a point ${\displaystyle P}$ outside the circle ${\displaystyle \Pi (P)}$ is the squared tangential distance ${\displaystyle |PT|}$ of point ${\displaystyle P}$ to the circle ${\displaystyle c}$.

Points with equal power, isolines of ${\displaystyle \Pi (P)}$, are circles concentric to circle ${\displaystyle c}$.

Steiner used the power of a point for proofs of several statements on circles, for example:

• Determination of a circle, that intersects four circles by the same angle.^[2]
• Solving the Problem of Apollonius
• Construction of the Malfatti circles:^[3] For a given triangle determine three circles, which touch each other and two sides of the triangle each.
• Spherical version of Malfatti's problem:^[4] The triangle is a spherical one.

Essential tools for investigations on circles are the radical axis of two circles and the radical center of three circles.

The power diagram of a set of circles divides the plane into regions within which the circle minimizing the power is constant.

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