Power of a point – Wikipedia
Relative distance of a point from a circle
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]
Specifically, the power
of a point
with respect to a circle
with center
and radius
is defined by
If
is outside the circle, then
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 properties[edit]
Besides the properties mentioned in the lead there are further properties:
Orthogonal circle[edit]
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 (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 theorem[edit]
For the intersecting secants theorem and chord theorem the power of a point plays the role of an invariant:
Radical axis[edit]
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[edit]
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 circles[edit]
Similarity points[edit]
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:
- .
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 circles[edit]
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 circles[edit]
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
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 with respect to a sphere[edit]
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 product[edit]
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 theorem[edit]
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.
References[edit]
- ^ Jakob Steiner: Einige geometrische Betrachtungen, 1826, S. 164
- ^ Steiner, p. 163
- ^ Steiner, p. 178
- ^ Steiner, p. 182
- ^ Steiner: p. 170,171
- ^ Steiner: p. 175
- ^ Michel Chasles, C. H. Schnuse: Die Grundlehren der neuern Geometrie, erster Theil, Verlag Leibrock, Braunschweig, 1856, p. 312
- ^ 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
- ^ K.P. Grothemeyer: Analytische Geometrie, Sammlung Göschen 65/65A, Berlin 1962, S. 54
- ^ 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
- Coxeter, H. S. M. (1969), Introduction to Geometry (2nd ed.), New York: Wiley.
- Darboux, Gaston (1872), “Sur les relations entre les groupes de points, de cercles et de sphéres dans le plan et dans l’espace”, Annales Scientifiques de l’École Normale Supérieure, 1: 323–392, doi:10.24033/asens.87.
- Laguerre, Edmond (1905), Oeuvres de Laguerre: Géométrie (in French), Gauthier-Villars et fils, p. 20
- Steiner, Jakob (1826). “Einige geometrischen Betrachtungen” [Some geometric considerations]. Crelle’s Journal (in German). 1: 161–184. doi:10.1515/crll.1826.1.161. S2CID 122065577. Figures 8–26.
- Berger, Marcel (1987), Geometry I, Springer, ISBN 978-3-540-11658-5
Further reading[edit]
- Ogilvy C. S. (1990), Excursions in Geometry, Dover Publications, pp. 6–23, ISBN 0-486-26530-7
- Coxeter H. S. M., Greitzer S. L. (1967), Geometry Revisited, Washington: MAA, pp. 27–31, 159–160, ISBN 978-0-88385-619-2
- Johnson RA (1960), Advanced Euclidean Geometry: An elementary treatise on the geometry of the triangle and the circle (reprint of 1929 edition by Houghton Mifflin ed.), New York: Dover Publications, pp. 28–34, ISBN 978-0-486-46237-0
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