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]

Geometric meaning

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.[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.

Geometric propertiesEdit

Besides the properties mentioned in the lead there are further properties:

Orthogonal circleEdit

 
Orthogonal circle (green)

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.
 
Angle between two circles

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 (see the diagram):

 
 

(  and   are normals to the circle tangents.)

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 theoremEdit

 
Secant-, chord-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 axisEdit

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 proofEdit

 
Secant-/chord-theorem: 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   (through  ) and circle   can be determined by inserting the parametric equation into the circle's equation:

 

From Vieta's theorem one finds:

 . (independent of   !)

  is the power of   with respect for circle  .

Because of   one gets the following statement for the points  :

 , if   is outside the circle,
 , if   is inside the circle (  have different signs !).

In case of   line   is a tangent and   the square of the tangential distance of point   to circle  .

Similarity points, common power of two circlesEdit

Similarity pointsEdit

Similarity points are an essential tool for Steiner's investigations on circles.[5]

Given two circles

 

A homothety (similarity)  , that maps   onto   stretches (jolts) 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:

 .
 
Similarity points of two circles: various cases

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 (both circles on the same side of the common tangent line).
In case of   the circles touch each other at point   outside (both circles on different sides of the common tangent line).

Further more:

  • If the circles lie disjoint (the discs have no points in common), 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 circlesEdit

 
Similarity points of two circles and their common power

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 (see above) the two equations

 

Combining these three equations yields:

 
Hence:
 
(independent of line   !). The analog statement for the inner similarity point   is true, too.

The invariants   are called by Steiner common power of the two circles (gemeinschaftliche Potenz der beiden Kreise bezüglich ihrer Ähnlichkeitspunkte).[6]

The pairs   and   of points are antihomologous points. The pairs   and   are homologous.[7][8]

Determination of a circle that is tangent to two circlesEdit

 
Common power of two circles: application
 
Circles tangent to two circles

For a second secant through  :

 

From the secant theorem one gets:

The four points   lie on a circle.

And analogously:

The four points   lie on a circle, too.

Because the radical lines of three circles meet at the radical (see: article radical line), one gets:

The secants   meet on the radical axis of the given two circles.

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 secants   become tangents at the points  . The tangents intercept at the radical line   (in the diagram yellow).

Similar considerations generate the second tangent circle, that meets the given circles at the points   (see diagram).

All tangent circles to the given circles can be found by varying line  .

Positions of the centers
 
Circles tangent to two circles

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

foci  ,
distance of the vertices[clarification needed]  ,
center   is the center of   ,
linear eccentricity   and
 [clarification needed].

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 of a point with respect to a sphere

Power with respect to a sphereEdit

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.

Darboux productEdit

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's theoremEdit

Laguerre defined the power of a point P with respect to an algebraic curve of degree n to be the product 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.

ReferencesEdit

  1. ^ Jakob Steiner: Einige geometrische Betrachtungen, 1826, S. 164
  2. ^ Steiner, p. 163
  3. ^ Steiner, p. 178
  4. ^ Steiner, p. 182
  5. ^ Steiner: p. 170,171
  6. ^ Steiner: p. 175
  7. ^ Michel Chasles, C. H. Schnuse: Die Grundlehren der neuern Geometrie, erster Theil, Verlag Leibrock, Braunschweig, 1856, p. 312
  8. ^ William J. M'Clelland: A Treatise on the Geometry of the Circle and Some Extensions to Conic Sections by the Method of Reciprocation,1891, Verlag: Creative Media Partners, LLC, ISBN 978-0-344-90374-8, p. 121,220
  9. ^ K.P. Grothemeyer: Analytische Geometrie, Sammlung Göschen 65/65A, Berlin 1962, S. 54
  10. ^ Pierre Larochelle, J. Michael McCarthy:Proceedings of the 2020 USCToMM Symposium on Mechanical Systems and Robotics, 2020, Springer-Verlag, ISBN 978-3-030-43929-3, p. 97

Further readingEdit

External linksEdit