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Power of a point

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From Wikipedia, the free encyclopedia

Relative distance of a point from a circle

In elementary planegeometry, thepower of a point is areal number that reflects the relative distance of a given point from a given circle. It was introduced byJakob Steiner in 1826.^[1]

Specifically, the power $\Pi (P)$ of a point $P {\displaystyle P}$ with respect to acircle $c {\displaystyle c}$ with center $O {\displaystyle O}$ and radius $r {\displaystyle r}$ is defined by

\Pi (P)=|PO|^{2}-r^{2}.

If $P {\displaystyle P}$ isoutside the circle, then $\Pi (P)>0$ ,
if $P {\displaystyle P}$ ison the circle, then $\Pi (P)=0$ and
if $P {\displaystyle P}$ isinside the circle, then $\Pi (P)<0$ .

Due to thePythagorean theorem the number $\Pi (P)$ has the simple geometric meanings shown in the diagram: For a point $P {\displaystyle P}$ outside the circle $\Pi (P)$ is the squared tangential distance $|PT|$ of point $P {\displaystyle P}$ to the circle $c {\displaystyle c}$ .

Points with equal power,isolines of $\Pi (P)$ , are circlesconcentric to circle $c {\displaystyle c}$ .

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 theproblem of Apollonius
Construction of theMalfatti 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 theradical axis of two circles and theradical center of three circles.

Thepower diagram of a set of circles divides the plane into regions within which the circle minimizing the power is constant.

More generally, French mathematicianEdmond Laguerre defined the power of a point with respect to anyalgebraic curve in a similar way.

Geometric properties

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

Orthogonal circle

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For any point $P {\displaystyle P}$ outside of the circle $c {\displaystyle c}$ there are two tangent points $T_{1},T_{2}$ on circle $c {\displaystyle c}$ , which have equal distance to $P {\displaystyle P}$ . Hence the circle $o {\displaystyle o}$ with center $P {\displaystyle P}$ through $T_{1}$ passes $T_{2}$ , too, and intersects $c {\displaystyle c}$ orthogonal:

The circle with center $P {\displaystyle P}$ and radius ${\sqrt {\Pi (P)}}$ intersects circle $c {\displaystyle c}$ orthogonal.

If the radius $\rho$ of the circle centered at $P {\displaystyle P}$ is different from ${\sqrt {\Pi (P)}}$ one gets the angle of intersection $\varphi$ between the two circles applying theLaw of cosines (see the diagram):

\rho ^{2}+r^{2}-2\rho r\cos \varphi =|PO|^{2}

\rightarrow \ \cos \varphi ={\frac {\rho ^{2}+r^{2}-|PO|^{2}}{2\rho r}}={\frac {\rho ^{2}-\Pi (P)}{2\rho r}}

( $PS_{1}$ and $OS_{1}$ arenormals to the circle tangents.)

If $P {\displaystyle P}$ 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

\rho ^{2}-2\rho r\cos \varphi -\Pi (P)=0

Intersecting secants theorem, intersecting chords theorem

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For theintersecting secants theorem andchord theorem the power of a point plays the role of aninvariant:

Intersecting secants theorem: For a point $P {\displaystyle P}$ outside a circle $c {\displaystyle c}$ and the intersection points $S_{1},S_{2}$ of a secant line $g {\displaystyle g}$ with $c {\displaystyle c}$ the following statement is true: $|PS_{1}|\cdot |PS_{2}|=\Pi (P)$ , hence the product is independent of line $g {\displaystyle g}$ . If $g {\displaystyle g}$ is tangent then $S_{1}=S_{2}$ and the statement is thetangent-secant theorem.
Intersecting chords theorem: For a point $P {\displaystyle P}$ inside a circle $c {\displaystyle c}$ and the intersection points $S_{1},S_{2}$ of a secant line $g {\displaystyle g}$ with $c {\displaystyle c}$ the following statement is true: $|PS_{1}|\cdot |PS_{2}|=-\Pi (P)$ , hence the product is independent of line $g {\displaystyle g}$ .

Radical axis

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Let $P {\displaystyle P}$ be a point and $c_{1},c_{2}$ two non concentric circles with centers $O_{1},O_{2}$ and radii $r_{1},r_{2}$ . Point $P {\displaystyle P}$ has the power $\Pi _{i}(P)$ with respect to circle $c_{i}$ . The set of all points $P {\displaystyle P}$ with $\Pi _{1}(P)=\Pi _{2}(P)$ is a line calledradical axis. It contains possible common points of the circles and is perpendicular to line ${\overline {O_{1}O_{2}}}$ .

Secants theorem, chords theorem: common proof

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

Let $P:{\vec {p}}$ be a point, $c:{\vec {x}}^{2}-r^{2}=0$ a circle with the origin as its center and ${\vec {v}}$ an arbitraryunit vector. The parameters $t_{1},t_{2}$ of possible common points of line $g:{\vec {x}}={\vec {p}}+t{\vec {v}}$ (through $P {\displaystyle P}$ ) and circle $c {\displaystyle c}$ can be determined by inserting theparametric equation into the circle's equation:

({\vec {p}}+t{\vec {v}})^{2}-r^{2}=0\quad \rightarrow \quad t^{2}+2t\;{\vec {p}}\cdot {\vec {v}}+{\vec {p}}^{2}-r^{2}=0\ .

FromVieta's theorem one finds:

t_{1}\cdot t_{2}={\vec {p}}^{2}-r^{2}=\Pi (P)

. (independent of

{\vec {v}}

)

$\Pi (P)$ is the power of $P {\displaystyle P}$ with respect to circle $c {\displaystyle c}$ .

Because of $|{\vec {v}}|=1$ one gets the following statement for the points $S_{1},S_{2}$ :

|PS_{1}|\cdot |PS_{2}|=t_{1}t_{2}=\Pi (P)\

, if

P {\displaystyle P}

is outside the circle,

|PS_{1}|\cdot |PS_{2}|=-t_{1}t_{2}=-\Pi (P)\

, if

P {\displaystyle P}

is inside the circle (

t_{1},t_{2}

have different signs !).

In case of $t_{1}=t_{2}$ line $g {\displaystyle g}$ is a tangent and $\Pi (P)$ the square of the tangential distance of point $P {\displaystyle P}$ to circle $c {\displaystyle c}$ .

Similarity points, common power of two circles

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Similarity points

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

Given two circles

\ c_{1}:({\vec {x}}-{\vec {m}}_{1})^{2}-r_{1}^{2}=0,\quad c_{2}:({\vec {x}}-{\vec {m}}_{2})^{2}-r_{2}^{2}=0\ .

\sigma ({\vec {m}}_{1})={\vec {z}}+s({\vec {m}}_{1}-{\vec {z}})={\vec {m}}_{2}

Inserting $s=\pm {\tfrac {r_{2}}{r_{1}}}$ and solving for ${\vec {z}}$ yields:

{\vec {z}}={\frac {r_{1}{\vec {m}}_{2}\mp r_{2}{\vec {m}}_{1}}{r_{1}\mp r_{2}}}

Similarity points of two circles: various cases

Point $E:{\vec {e}}={\frac {r_{1}{\vec {m}}_{2}-r_{2}{\vec {m}}_{1}}{r_{1}-r_{2}}}$ is called theexterior similarity point and $I:{\vec {i}}={\frac {r_{1}{\vec {m}}_{2}+r_{2}{\vec {m}}_{1}}{r_{1}+r_{2}}}$ is called theinner similarity point.

In case of $M_{1}=M_{2}$ one gets $E=I=M_{i}$ .
In case of $r_{1}=r_{2}$ : $E {\displaystyle E}$ is thepoint at infinity of line ${\overline {M_{1}M_{2}}}$ and $I {\displaystyle I}$ is the center of $M_{1},M_{2}$ .
In case of $r_{1}=|EM_{1}|$ the circles touch each other at point $E {\displaystyle E}$ inside (both circles on the same side of the common tangent line).
In case of $r_{1}=|IM_{1}|$ the circles touch each other at point $I {\displaystyle I}$ outside (both circles on different sides of the common tangent line).

Further more:

If the circles liedisjoint (the discs have no points in common), the outside common tangents meet at $E {\displaystyle E}$ and the inner ones at $I {\displaystyle I}$ .
If one circle is containedwithin the other, the points $E, I {\displaystyle E,I}$ liewithin both circles.
The pairs $M_{1},M_{2};E,I$ areprojective harmonic conjugate: Theircross ratio is $(M_{1},M_{2};E,I)=-1$ .

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

Common power of two circles

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Similarity points of two circles and their common power

Let $c_{1},c_{2}$ be two circles, $E {\displaystyle E}$ their outer similarity point and $g {\displaystyle g}$ a line through $E {\displaystyle E}$ , which meets the two circles at four points $G_{1},H_{1},G_{2},H_{2}$ . From the defining property of point $E {\displaystyle E}$ one gets

{\frac {|EG_{1}|}{|EG_{2}|}}={\frac {r_{1}}{r_{2}}}={\frac {|EH_{1}|}{|EH_{2}|}}\

\rightarrow \ |EG_{1}|\cdot |EH_{2}|=|EH_{1}|\cdot |EG_{2}|\

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

|EG_{1}|\cdot |EH_{1}|=\Pi _{1}(E),\quad |EG_{2}|\cdot |EH_{2}|=\Pi _{2}(E).

Combining these three equations yields: ${\begin{aligned}\Pi _{1}(E)\cdot \Pi _{2}(E)&=|EG_{1}|\cdot |EH_{1}|\cdot |EG_{2}|\cdot |EH_{2}|\\&=|EG_{1}|^{2}\cdot |EH_{2}|^{2}=|EG_{2}|^{2}\cdot |EH_{1}|^{2}\ .\end{aligned}}$ Hence: $|EG_{1}|\cdot |EH_{2}|=|EG_{2}|\cdot |EH_{1}|={\sqrt {\Pi _{1}(E)\cdot \Pi _{2}(E)}}$ (independent of line $g {\displaystyle g}$ !).The analog statement for the inner similarity point $I {\displaystyle I}$ is true, too.

The invariants ${\textstyle {\sqrt {\Pi _{1}(E)\cdot \Pi _{2}(E)}},\ {\sqrt {\Pi _{1}(I)\cdot \Pi _{2}(I)}}}$ are called by Steinercommon power of the two circles (gemeinschaftliche Potenz der beiden Kreise bezüglich ihrer Ähnlichkeitspunkte).^[6]

The pairs $G_{1},H_{2}$ and $H_{1},G_{2}$ of points areantihomologous points. The pairs $G_{1},G_{2}$ and $H_{1},H_{2}$ arehomologous.^[7]^[8]

Determination of a circle that is tangent to two circles

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Common power of two circles: application

For a second secant through $E {\displaystyle E}$ :

|EH_{1}|\cdot |EG_{2}|=|EH'_{1}|\cdot |EG'_{2}|

From the secant theorem one gets:

The four points

H_{1},G_{2},H'_{1},G'_{2}

lie on a circle.

And analogously:

The four points

G_{1},H_{2},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 {H_{1}H'_{1}}},\;{\overline {G_{2}G'_{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 ${\overline {M_{1}H_{1}}},{\overline {M_{2}G_{2}}}$ . The secants ${\overline {H_{1}H'_{1}}},{\overline {G_{2}G'_{2}}}$ become tangents at the points $H_{1},G_{2}$ . The tangents intercept at the radical line $p {\displaystyle p}$ (in the diagram yellow).

Similar considerations generate the second tangent circle, that meets the given circles at the points $G_{1},H_{2}$ (see diagram).

Alltangent circles to the given circles can be found by varying line $g {\displaystyle g}$ .

Positions of the centers

If $X {\displaystyle X}$ is the center and $\rho$ the radius of the circle, that is tangent to the given circles at the points $H_{1},G_{2}$ , then:

\rho =|XM_{1}|-r_{1}=|XM_{2}|-r_{2}

\rightarrow \ |XM_{2}|-|XM_{1}|=r_{2}-r_{1}.

Hence: the centers lie on ahyperbola with

foci

M_{1},M_{2}

distance of the vertices^{[clarification needed]}

2a=r_{2}-r_{1}

center

M {\displaystyle M}

is the center of

M_{1},M_{2}

linear eccentricity

c={\tfrac {|M_{1}M_{2}|}{2}}

and

\ b^{2}=e^{2}-a^{2}={\tfrac {|M_{1}M_{2}|^{2}-(r_{2}-r_{1})^{2}}{4}}

^{[clarification needed]}.

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

If $X {\displaystyle X}$ is the center and $\rho$ the radius of the circle, that is tangent to the given circles at the points $G_{1},H_{2}$ , then:

\rho =|XM_{1}|+r_{1}=|XM_{2}|+r_{2}

\rightarrow \ |XM_{2}|-|XM_{1}|=-(r_{2}-r_{1}).

The centers lie on the same hyperbola, but on the right branch.

Power with respect to a sphere

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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

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The power of a point is a special case of the Darboux product between two circles, which is given by^[10]

\left|A_{1}A_{2}\right|^{2}-r_{1}^{2}-r_{2}^{2}\,

whereA₁ andA₂ are the centers of the two circles andr₁ andr₂ 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

2r_{1}r_{2}\cos \varphi \,

whereφ is the angle of intersection (see sectionorthogonal circle).

Laguerre's theorem

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Laguerre defined the power of a pointP with respect to an algebraic curve of degreen to be the sum of the distances from the point to the intersections of a circle through the point with the curve, divided by thenth power of the diameterd. 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 ofd².

References

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^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