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.
Specifically, the power of a point with respect to a circle with center and radius is defined by
If is outside the circle, then,
if is on the circle, then and
if is inside the circle, then.
Due to the Pythagorean theorem the number has the simple geometric meanings shown in the diagram: For a point outside the circle is the squared tangential distance of point to the circle.
Points with equal power, isolines of, are circles concentric to circle.
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 outside of the circle there are two tangent points on circle, which have equal distance to. Hence the circle with center through passes, too, and intersects orthogonal:

The circle with center and radius intersects circle orthogonal.

If the radius of the circle centered at is different from one gets the angle of intersection between the two circles applying the Law of cosines :
If lies inside the blue circle, then and is always different from.
If the angle is given, then one gets the radius by solving the quadratic equation

Intersecting secants theorem, intersecting chords theorem

For the intersecting secants theorem and chord theorem the power of a point plays the role of an invariant:

Intersecting secants theorem: For a point outside a circle and the intersection points of a secant line with the following statement is true:, hence the product is independent of line. If is tangent then and the statement is the tangent-secant theorem.
Intersecting chords theorem: For a point inside a circle and the intersection points of a secant line with the following statement is true:, hence the product is independent of line.
Radical axis

Let be a point and two non concentric circles with
centers and radii. Point has the power with respect to circle. The set of all points with is a line called radical axis. It contains possible common points of the circles and is perpendicular to line.

Secants theorem, chords theorem: common proof

Both theorems, including the tangent-secant theorem, can be proven uniformly:
Let be a point, a circle with the origin as its center and an arbitrary unit vector. The parameters of possible common points of line and circle can be determined by inserting the parametric equation into the circle's equation:
From Vieta's theorem one finds:
is the power of with respect to circle.
Because of one gets the following statement for the points :
In case of line is a tangent and the square of the tangential distance of point to circle.

Similarity points, common power of two circles

Similarity points

Similarity points are an essential tool for Steiner's investigations on circles.
Given two circles
A homothety , that maps onto stretches radius to and has its center on the line, because. If center is between the scale factor is. In the other case. In any case:
Inserting and solving for yields:
Point is called the exterior similarity point and is called the inner similarity point.
In case of one gets.
In case of : is the point at infinity of line and is the center of.
In case of the circles touch each other at point inside.
In case of the circles touch each other at point outside.
Further more:

If the circles lie disjoint, the outside common tangents meet at and the inner ones at.
If one circle is contained within the other, the points lie within both circles.
The pairs are projective harmonic conjugate: Their cross ratio is.

Monge's theorem states: The outer similarity points of three disjoint circles lie on a line.

Common power of two circles

Let be two circles, their outer similarity point and a line through, which meets the two circles at four points. From the defining property of point one gets
and from the secant theorem the two equations
Combining these three equations yields:
Hence: .
The analog statement for the inner similarity point is true, too.
The invariants are called by Steiner common power of the two circles.
The pairs and of points are antihomologous points. The pairs and are homologous.

Determination of a circle that is tangent to two circles

For a second secant through :
From the secant theorem one gets:
And analogously:
Because the radical lines of three circles meet at the radical, one gets:
Moving the lower secant 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 secants become tangents at the points. The tangents intercept at the radical line .
Similar considerations generate the second tangent circle, that meets the given circles at the points .
All tangent circles to the given circles can be found by varying line.
;Positions of the centers
If is the center and the radius of the circle, that is tangent to the given circles at the points, then:
Hence: the centers lie on a hyperbola with
Considerations on the outside tangent circles lead to the analog result:
If is the center and the radius of the circle, that is tangent to the given circles at the points, then:
The centers lie on the same hyperbola, but on the right branch.
See also Problem of Apollonius.

Power with respect to a sphere

The idea of the power of a point with respect to a circle can be extended to a sphere
. The secants and chords theorems are true for a sphere, too, and can be proven literally as in the circle case.

Darboux product

The power of a point is a special case of the Darboux product between two circles, which is given by
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.

Laguerre's theorem

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. 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.