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)
P
c
O
r
\Pi(P)=|PO|2-r2.
P
\Pi(P)>0
P
\Pi(P)=0
P
\Pi(P)<0
Due to the Pythagorean theorem the number
\Pi(P)
P
\Pi(P)
|PT|
P
c
Points with equal power, isolines of
\Pi(P)
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.
Besides the properties mentioned in the lead there are further properties:
For any point
P
c
T1,T2
c
P
o
P
T1
T2
c
P
\sqrt{\Pi(P)}
c
If the radius
\rho
P
\sqrt{\Pi(P)}
\varphi
\rho2+r2-2\rhor\cos\varphi=|PO|2
→ \cos\varphi=
\rho2+r2-|PO|2 | = | |
2\rhor |
\rho2-\Pi(P) | |
2\rhor |
PS1
OS1
If
P
\Pi(P)<0
\varphi
90\circ
\varphi
\rho
\rho2-2\rhor\cos\varphi-\Pi(P)=0
For the intersecting secants theorem and chord theorem the power of a point plays the role of an invariant:
P
c
S1,S2
g
c
|PS1| ⋅ |PS2|=\Pi(P)
g
g
S1=S2
P
c
S1,S2
g
c
|PS1| ⋅ |PS2|=-\Pi(P)
g
Let
P
c1,c2
O1,O2
r1,r2
P
\Pii(P)
ci
P
\Pi1(P)=\Pi2(P)
\overline{O1O2}
Both theorems, including the tangent-secant theorem, can be proven uniformly:
Let
P:\vecp
c:\vecx2-r2=0
\vecv
t1,t2
g:\vecx=\vecp+t\vecv
P
c
(\vecp+t\vecv)2-r2=0 → t2+2t \vecp ⋅ \vecv+\vecp2-r2=0 .
t1 ⋅ t2=\vecp2-r2=\Pi(P)
\vecv
\Pi(P)
P
c
Because of
|\vecv|=1
S1,S2
|PS1| ⋅ |PS2|=t1t2=\Pi(P)
P
|PS1| ⋅ |PS2|=-t1t2=-\Pi(P)
P
t1,t2
In case of
t1=t2
g
\Pi(P)
P
c
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 |
.
\sigma
c1
c2
r1
r2
Z:\vecz
\overline{M1M2}
\sigma(M1)=M2
Z
M1,M2
s=-\tfrac{r2}{r1}
s=\tfrac{r2}{r1}
\sigma(\vecm1)=\vecz+s(\vecm1-\vecz)=\vecm2
s=\pm\tfrac{r2}{r1}
\vecz
\vecz=
r1\vecm2\mpr2\vecm1 | |
r1\mpr2 |
In case of
M1=M2
E=I=Mi
r1=r2
E
\overline{M1M2}
I
M1,M2
r1=|EM1|
E
r1=|IM1|
I
Further more:
E
I
E,I
M1,M2;E,I
(M1,M2;E,I)=-1
Monge's theorem states: The outer similarity points of three disjoint circles lie on a line.
Let
c1,c2
E
g
E
G1,H1,G2,H2
E
|EG1| | = | |
|EG2| |
r1 | = | |
r2 |
|EH1| | |
|EH2| |
→ |EG1| ⋅ |EH2|=|EH1| ⋅ |EG2|
|EG1| ⋅ |EH1|=\Pi1(E), |EG2| ⋅ |EH2|=\Pi2(E).
g
I
The invariants are called by Steiner common power of the two circles (gemeinschaftliche Potenz der beiden Kreise bezüglich ihrer Ähnlichkeitspunkte).[6]
The pairs
G1,H2
H1,G2
G1,G2
H1,H2
For a second secant through
E
|EH1| ⋅ |EG2|=|EH'1| ⋅ |EG'2|
The four points
H1,G2,H'1,G'2
The four points
G1,H2,G'1,H'2
The secants
\overline{H1H'1}, \overline{G2G'2}
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}
\overline{H1H'1},\overline{G2G'2}
H1,G2
p
Similar considerations generate the second tangent circle, that meets the given circles at the points
G1,H2
All tangent circles to the given circles can be found by varying line
g
If
X
\rho
H1,G2
\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
M1,M2
linear eccentricity
c=\tfrac{|M1M2|}{2}
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
\rho
G1,H2
\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.
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.
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 |
If the two circles are orthogonal, the Darboux product vanishes.
If the two circles intersect, then their Darboux product is
2r1r2\cos\varphi
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.