Power of a point explained

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

\Pi(P)

of a point

P

with respect to a circle

c

with center

O

and radius

r

is defined by

\Pi(P)=|PO|2-r2.

If

P

is outside the circle, then

\Pi(P)>0

,
if

P

is on the circle, then

\Pi(P)=0

and
if

P

is inside the circle, then

\Pi(P)<0

.

Due to the Pythagorean theorem the number

\Pi(P)

has the simple geometric meanings shown in the diagram: For a point

P

outside the circle

\Pi(P)

is the squared tangential distance

|PT|

of point

P

to the circle

c

.

Points with equal power, isolines of

\Pi(P)

, are circles concentric to circle

c

.

Steiner used the power of a point for proofs of several statements on circles, for example:

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

P

outside of the circle

c

there are two tangent points

T1,T2

on circle

c

, which have equal distance to

P

. Hence the circle

o

with center

P

through

T1

passes

T2

, too, and intersects

c

orthogonal:

P

and radius

\sqrt{\Pi(P)}

intersects circle

c

orthogonal.

If the radius

\rho

of the circle centered at

P

is different from

\sqrt{\Pi(P)}

one gets the angle of intersection

\varphi

between the two circles applying the Law of cosines (see the diagram):

\rho2+r2-2\rhor\cos\varphi=|PO|2

→   \cos\varphi=

\rho2+r2-|PO|2=
2\rhor
\rho2-\Pi(P)
2\rhor
(

PS1

and

OS1

are normals to the circle tangents.)

If

P

lies inside the blue circle, then

\Pi(P)<0

and

\varphi

is always different from

90\circ

. If the angle

\varphi

is given, then one gets the radius

\rho

by solving the quadratic equation

\rho2-2\rhor\cos\varphi-\Pi(P)=0

.

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:

P

outside a circle

c

and the intersection points

S1,S2

of a secant line

g

with

c

the following statement is true:

|PS1||PS2|=\Pi(P)

, hence the product is independent of line

g

. If

g

is tangent then

S1=S2

and the statement is the tangent-secant theorem.

P

inside a circle

c

and the intersection points

S1,S2

of a secant line

g

with

c

the following statement is true:

|PS1||PS2|=-\Pi(P)

, hence the product is independent of line

g

.

Radical axis

Let

P

be a point and

c1,c2

two non concentric circles with centers

O1,O2

and radii

r1,r2

. Point

P

has the power

\Pii(P)

with respect to circle

ci

. The set of all points

P

with

\Pi1(P)=\Pi2(P)

is a line called radical axis. It contains possible common points of the circles and is perpendicular to line

\overline{O1O2}

.

Secants theorem, chords theorem: common proof

Both theorems, including the tangent-secant theorem, can be proven uniformly:

Let

P:\vecp

be a point,

c:\vecx2-r2=0

a circle with the origin as its center and

\vecv

an arbitrary unit vector. The parameters

t1,t2

of possible common points of line

g:\vecx=\vecp+t\vecv

(through

P

) and circle

c

can be determined by inserting the parametric equation into the circle's equation:

(\vecp+t\vecv)2-r2=0 t2+2t\vecp\vecv+\vecp2-r2=0.

From Vieta's theorem one finds:

t1 ⋅ t2=\vecp2-r2=\Pi(P)

. (independent of

\vecv

)

\Pi(P)

is the power of

P

with respect to circle

c

.

Because of

|\vecv|=1

one gets the following statement for the points

S1,S2

:

|PS1||PS2|=t1t2=\Pi(P)

, if

P

is outside the circle,

|PS1||PS2|=-t1t2=-\Pi(P)

, if

P

is inside the circle (

t1,t2

have different signs !).

In case of

t1=t2

line

g

is a tangent and

\Pi(P)

the square of the tangential distance of point

P

to circle

c

.

Similarity points, common power of two circles

Similarity points

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

Given two circles

c1:(\vecx-\vecm1)-r

2=0,
1

c2:(\vecx-\vecm2)-r

2=0
2

.

A homothety (similarity)

\sigma

, that maps

c1

onto

c2

stretches (jolts) radius

r1

to

r2

and has its center

Z:\vecz

on the line

\overline{M1M2}

, because

\sigma(M1)=M2

. If center

Z

is between

M1,M2

the scale factor is

s=-\tfrac{r2}{r1}

. In the other case

s=\tfrac{r2}{r1}

. In any case:

\sigma(\vecm1)=\vecz+s(\vecm1-\vecz)=\vecm2

.Inserting

s=\pm\tfrac{r2}{r1}

and solving for

\vecz

yields:

\vecz=

r1\vecm2\mpr2\vecm1
r1\mpr2
.Point E:\vec e=\frac is called the exterior similarity point and I:\vec i=\frac is called the inner similarity point.

In case of

M1=M2

one gets

E=I=Mi

.
In case of

r1=r2

:

E

is the point at infinity of line

\overline{M1M2}

and

I

is the center of

M1,M2

.
In case of

r1=|EM1|

the circles touch each other at point

E

inside (both circles on the same side of the common tangent line).
In case of

r1=|IM1|

the circles touch each other at point

I

outside (both circles on different sides of the common tangent line).

Further more:

E

and the inner ones at

I

.

E,I

lie within both circles.

M1,M2;E,I

are projective harmonic conjugate: Their cross ratio is

(M1,M2;E,I)=-1

.

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

Common power of two circles

Let

c1,c2

be two circles,

E

their outer similarity point and

g

a line through

E

, which meets the two circles at four points

G1,H1,G2,H2

. From the defining property of point

E

one gets
|EG1|=
|EG2|
r1=
r2
|EH1|
|EH2|

|EG1||EH2|=|EH1||EG2|

and from the secant theorem (see above) the two equations

|EG1||EH1|=\Pi1(E),|EG2||EH2|=\Pi2(E).

Combining these three equations yields:\begin\Pi_1(E)\cdot\Pi_2(E) &=|EG_1|\cdot|EH_1|\cdot|EG_2|\cdot|EH_2| \\&=|EG_1|^2\cdot|EH_2|^2= |EG_2|^2\cdot|EH_1|^2 \ .\endHence: |EG_1|\cdot|EH_2|= |EG_2| \cdot |EH_1|=\sqrt (independent of line

g

!).The analog statement for the inner similarity point

I

is true, too.

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

The pairs

G1,H2

and

H1,G2

of points are antihomologous points. The pairs

G1,G2

and

H1,H2

are homologous.[7] [8]

Determination of a circle that is tangent to two circles

For a second secant through

E

:

|EH1||EG2|=|EH'1||EG'2|

From the secant theorem one gets:

The four points

H1,G2,H'1,G'2

lie on a circle.And analogously:

The four points

G1,H2,G'1,H'2

lie on a circle, too.Because the radical lines of three circles meet at the radical (see: article radical line), one gets:

The secants

\overline{H1H'1},\overline{G2G'2}

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

\overline{M1H1},\overline{M2G2}

. The secants

\overline{H1H'1},\overline{G2G'2}

become tangents at the points

H1,G2

. The tangents intercept at the radical line

p

(in the diagram yellow).

Similar considerations generate the second tangent circle, that meets the given circles at the points

G1,H2

(see diagram).

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

g

.
Positions of the centers

If

X

is the center and

\rho

the radius of the circle, that is tangent to the given circles at the points

H1,G2

, then:

\rho=|XM1|-r1=|XM2|-r2

|XM2|-|XM1|=r2-r1.

Hence: the centers lie on a hyperbola with

foci

M1,M2

,

distance of the vertices

2a=r2-r1

,

center

M

is the center of

M1,M2

,

linear eccentricity

c=\tfrac{|M1M2|}{2}

and

b2=e2-a

2=\tfrac{|M
1M
2-(r
2-r
2}{4}
1)
.

Considerations on the outside tangent circles lead to the analog result:

If

X

is the center and

\rho

the radius of the circle, that is tangent to the given circles at the points

G1,H2

, then:

\rho=|XM1|+r1=|XM2|+r2

|XM2|-|XM1|=-(r2-r1).

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.[9] 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[10]

\left|A1A2

2
\right|
2

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

2r1r2\cos\varphi

where φ is the angle of intersection (see section orthogonal circle).

Laguerre's theorem

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 . 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

External links

Notes and References

  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,, 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,, p. 97