power of point


If a secant of the circle is drawn through a point (P), then the productPlanetmathPlanetmath of the line segmentsMathworldPlanetmath on the secant between the point and the perimeterPlanetmathPlanetmath of the circle is on the direction of the secant.  The product is called the power of the point with respect to the circle.

Proof.  Let PA and PB be the segments of a secant and PA and PB the segments of another secant.  Then the trianglesMathworldPlanetmath PAB and PAB are similarMathworldPlanetmathPlanetmath since they have equal angles, namely the central anglesMathworldPlanetmath APB and BPA and the inscribed angles PAB and PAB.  Thus we have the proportion (http://planetmath.org/ProportionEquation)


which implies the asserted equation


Notes.  If the point P is outside a circle, then value of the power of the point with respect to the circle is equal to the square of the limited tangent of the circle from P; this square (http://planetmath.org/SquareOfANumber) may be considered as the limit case of the power of point where the both intersectionMathworldPlanetmath points of the secant with the circle coincide.  Another of the notion power of point is got when the line through P does not intersect the circle; we can think that then the intersecting points are imaginary; also now the product of the “imaginary line segments” is the same.

Denote by p2 the power of the point  P:=(a,b)  with respect to circle


Then, by the Pythagorean theoremMathworldPlanetmathPlanetmath, we obtain

p2=(a-x0)2+(b-y0)2-r2 (1)

if P is outside the circle and

p2=r2-((a-x0)2+(b-y0)2) (2)

if P is inside of the circle.  If in the latter case, we change the definition of the power of point to be the negative value -p2 for a point inside the circle, then in both cases the power of the point (a,b) is equal to

Title power of point
Canonical name PowerOfPoint
Date of creation 2013-03-22 15:07:02
Last modified on 2013-03-22 15:07:02
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 15
Author PrimeFan (13766)
Entry type Theorem
Classification msc 51M99
Synonym power of the point
Synonym power of a point
Related topic InversionOfPlane
Related topic VolumeOfSphericalCapAndSphericalSector