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. Specifically, the power Π ( P ) {\displaystyle \Pi (P)} of a point P {\displaystyle P} with respect to a circle c {\displaystyle c} with center O {\displaystyle O} and radius r {\displaystyle r} is defined by
Π ( P ) = | P O | 2 − r 2 . {\displaystyle \Pi (P)=|PO|^{2}-r^{2}.}
If P {\displaystyle P} is outside the circle, then Π ( P ) > 0 {\displaystyle \Pi (P)>0} , if P {\displaystyle P} is on the circle, then Π ( P ) = 0 {\displaystyle \Pi (P)=0} and if P {\displaystyle P} is inside the circle, then Π ( P ) < 0 {\displaystyle \Pi (P)<0} . Due to the Pythagorean theorem the number Π ( P ) {\displaystyle \Pi (P)} has the simple geometric meanings shown in the diagram: For a point P {\displaystyle P} outside the circle, Π ( P ) {\displaystyle \Pi (P)} is the squared tangential distance | P T | {\displaystyle |PT|} of point P {\displaystyle P} to a point T {\displaystyle T} on the circle c {\displaystyle c} . Points with equal power, isolines of Π ( P ) {\displaystyle \Pi (P)} , are circles concentric to the original circle c {\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. Solving the problem of Apollonius Construction of the Malfatti circles: For a given triangle determine three circles, which touch each other and two sides of the triangle each. Spherical version of Malfatti's problem: 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.
Geometric properties Besides the properties mentioned in the lead there are further properties:
Orthogonal circle
For any point P {\displaystyle P} outside of the circle c {\displaystyle c} there are two tangent points T 1 , T 2 {\displaystyle T_{1},T_{2}} on circle c {\displaystyle c} , which have equal distance to P {\displaystyle P} . Hence the circle o {\displaystyle o} with center P {\displaystyle P} through T 1 {\displaystyle T_{1}} passes T 2 {\displaystyle T_{2}} , too, and intersects c {\displaystyle c} orthogonal:
The circle with center P {\displaystyle P} and radius Π ( P ) {\displaystyle {\sqrt {\Pi (P)}}} intersects circle c {\displaystyle c} orthogonal.
If the radius ρ {\displaystyle \rho } of the circle centered at P {\displaystyle P} is different from Π ( P ) {\displaystyle {\sqrt {\Pi (P)}}} one gets the angle of intersection φ {\displaystyle \varphi } between the two circles applying the Law of cosines (see the diagram):
ρ 2 + r 2 − 2 ρ r cos φ = | P O | 2 {\displaystyle \rho ^{2}+r^{2}-2\rho r\cos \varphi =|PO|^{2}}
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