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.
has the simple geometric meanings shown in the diagram: For a point
Points with equal power, isolines of
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.
between the two circles applying the Law of cosines (see the diagram): (
lies inside the blue circle, then
by solving the quadratic equation For the intersecting secants theorem and chord theorem the power of a point plays the role of an invariant: Let
is a line called radical axis.
It contains possible common points of the circles and is perpendicular to line
the square of the tangential distance of point
Similarity points are an essential tool for Steiner's investigations on circles.
is called the exterior similarity point and
inside (both circles on the same side of the common tangent line).
outside (both circles on different sides of the common tangent line).
Further more: Monge's theorem states: The outer similarity points of three disjoint circles lie on a line.
are called by Steiner common power of the two circles (gemeinschaftliche Potenz der beiden Kreise bezüglich ihrer Ähnlichkeitspunkte).
: From the secant theorem one gets: And analogously: Because the radical lines of three circles meet at the radical (see: article radical line), one gets: Moving the lower secant (see diagram) towards the upper one, the red circle becomes a circle, that is tangent to both given circles.
The center of the tangent circle is the intercept of the lines
The tangents intercept at the radical line
, then: Hence: the centers lie on a hyperbola with Considerations on the outside tangent circles lead to the analog result: If
, then: The centers lie on the same hyperbola, but on the right branch.
The idea of the power of a point with respect to a circle can be extended to a sphere .
[9] The secants and chords theorems are true for a sphere, too, and can be proven literally as in the circle case.
The power of a point is a special case of the Darboux product between two circles, which is given by[10] where A1 and A2 are the centers of the two circles and r1 and r2 are their radii.
The power of a point arises in the special case that one of the radii is zero.
If the two circles are orthogonal, the Darboux product vanishes.
If the two circles intersect, then their Darboux product is where φ is the angle of intersection (see section orthogonal circle).
Laguerre defined the power of a point P with respect to an algebraic curve of degree n to be the sum of the distances from the point to the intersections of a circle through the point with the curve, divided by the nth power of the diameter d. Laguerre showed that this number is independent of the diameter (Laguerre 1905).
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 d2.
