In elementary plane
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, the power of a point is a
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
that reflects the relative distance of a given point from a given circle. It was introduced by
Jakob Steiner
Jakob Steiner (18 March 1796 – 1 April 1863) was a Swiss mathematician who worked primarily in geometry.
Life
Steiner was born in the village of Utzenstorf, Canton of Bern. At 18, he became a pupil of Heinrich Pestalozzi and afterwards st ...
in 1826.
Specifically, the power
of a point
with respect to a
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
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
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
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
A contour line (also isoline, isopleth, or isarithm) of a function of two variables is a curve along which the function has a constant value, so that the curve joins points of equal value. It is a plane section of the three-dimensional graph ...
of
, are circles
concentric
In geometry, two or more objects are said to be concentric, coaxal, or coaxial when they share the same center or axis. Circles, regular polygons and regular polyhedra, and spheres may be concentric to one another (sharing the same center point ...
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
In Euclidean plane geometry, Apollonius's problem is to construct circles that are tangent to three given circles in a plane (Figure 1). Apollonius of Perga (c. 262 190 BC) posed and solved this famous problem in his work (', "Tangencies ...
* Construction of the
Malfatti circles
In geometry, the Malfatti circles are three circles inside a given triangle such that each circle is tangent to the other two and to two sides of the triangle. They are named after Gian Francesco Malfatti, who made early studies of the problem o ...
: For a given triangle determine three circles, which touch each other and two sides of the triangle each.
*
Spherical
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ce ...
version of Malfatti's problem: The triangle is a spherical one.
Essential tools for investigations on circles are the
radical axis
In Euclidean geometry, the radical axis of two non-concentric circles is the set of points whose Power of a point, power with respect to the circles are equal. For this reason the radical axis is also called the power line or power bisector of ...
of two circles and the
radical center
The term radical center can refer to:
* Radical centrism, a political movement
* a mathematical construct: also called the power center (geometry)
{{Disambig ...
of three circles.
The
power diagram
In computational geometry, a power diagram, also called a Laguerre–Voronoi diagram, Dirichlet cell complex, radical Voronoi tesselation or a sectional Dirichlet tesselation, is a partition of the Euclidean plane into polygonal cells defined from ...
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
Edmond Nicolas Laguerre (9 April 1834, Bar-le-Duc – 14 August 1886, Bar-le-Duc) was a French mathematician and a member of the Académie des sciences (1885). His main works were in the areas of geometry and complex analysis. He also investigate ...
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
In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states ...
(see the diagram):
:
:
(
and
are
normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
s 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
For the ''
intersecting secants theorem
The intersecting secant theorem or just secant theorem describes the relation of line segments created by two intersecting secants and the associated circle.
For two lines ''AD'' and ''BC'' that intersect each other in ''P'' and some circle in '' ...
'' and
''chord theorem'' the power of a point plays the role of an
invariant
Invariant and invariance may refer to:
Computer science
* Invariant (computer science), an expression whose value doesn't change during program execution
** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...
:
* ''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
The tangent-secant theorem describes the relation of line segments created by a secant and a tangent line with the associated circle.
This result is found as Proposition 36 in Book 3 of Euclid's ''Elements''.
Given a secant ''g'' intersectin ...
''.
* ''
Intersecting chords theorem
The intersecting chords theorem or just the chord theorem is a statement in elementary geometry that describes a relation of the four line segments created by two intersecting chords within a circle.
It states that the products of the lengths o ...
'': 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
In Euclidean geometry, the radical axis of two non-concentric circles is the set of points whose Power of a point, power with respect to the circles are equal. For this reason the radical axis is also called the power line or power bisector of ...
''. 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
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction vecto ...
. 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
In mathematics, Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. They are named after François Viète (more commonly referred to by the Latinised form of his name, "Franciscus Vieta").
Basic formulas
A ...
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
Similarity points
Similarity points are an essential tool for Steiner's investigations on circles.
Given two circles
:
A
homothety
In mathematics, a homothety (or homothecy, or homogeneous dilation) is a transformation of an affine space determined by a point ''S'' called its ''center'' and a nonzero number ''k'' called its ''ratio'', which sends point X to a point X' by the ...
(
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
In projective geometry, the harmonic conjugate point of an ordered triple of points on the real projective line is defined by the following construction:
:Given three collinear points , let be a point not lying on their join and let any line t ...
: Their
cross ratio
In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points ''A'', ''B'', ''C'' and ''D'' on a line, the ...
is
.
Monge's theorem
In geometry, Monge's theorem, named after Gaspard Monge, states that for any three circles in a plane, none of which is completely inside one of the others, the intersection points of each of the three pairs of external tangent lines are collinea ...
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 (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'').
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:
: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
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, cal ...
with
:foci
,
:distance of the vertices
,
:center
is the center of
,
:linear eccentricity
and
:
.
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
In Euclidean plane geometry, Apollonius's problem is to construct circles that are tangent to three given circles in a plane (Figure 1). Apollonius of Perga (c. 262 190 BC) posed and solved this famous problem in his work (', "Tangencies ...
.
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
[Pierre Larochelle, J. Michael McCarthy:''Proceedings of the 2020 USCToMM Symposium on Mechanical Systems and Robotics'', 2020, Springer-Verlag, , p. 97]
:
where ''A''
1 and ''A''
2 are the centers of the two circles and ''r''
1 and ''r''
2 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
Laguerre
Edmond Nicolas Laguerre (9 April 1834, Bar-le-Duc – 14 August 1886, Bar-le-Duc) was a French mathematician and a member of the Académie des sciences (1885). His main works were in the areas of geometry and complex analysis. He also investigate ...
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 ''n''th 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 ''d''
2.
References
* .
* .
*
* .
*
Further reading
*
*
*
External links
Jacob Steiner and the Power of a Pointa
Convergence* {{mathworld, CirclePower, Circle Power
Intersecting Chords Theoremat
cut-the-knot
Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...
Intersecting Chords TheoremWith interactive animation
With interactive animation
Euclidean plane geometry
Analytic geometry