radical axis

Let $\backslash vec\; x,\backslash vec\; m\_1,\backslash vec\; m\_2$ be the position vectors of the points $P,M\_1,M\_2$. Then the defining equation of the radical line can be written as:

:$(\backslash vec\; x-\backslash vec\; m\_1)^2-r\_1^2=(\backslash vec\; x-\backslash vec\; m\_2)^2-r\_2^2\; \backslash quad\; \backslash leftrightarrow$

\quad 2\vec x\cdot(\vec m_2-\vec m_1)+\vec m_1^2-\vec m_2^2+r_2^2-r_1^2=0

File:Potenz-gerade-ber-d1d2.svg

From the right equation one gets

• The pointset of the radical axis is indeed a line and is perpendicular to the line through the circle centers.

($\backslash vec\; m\_2-\backslash vec\; m\_1$ is a normal vector to the radical axis !)

Dividing the equation by $2|\backslash vec\; m\_2-\backslash vec\; m\_1|$, one gets the Hessian normal form. Inserting the position vectors of the centers yields the distances of the centers to the radical axis:

:$d\_1\; =\; \backslash frac\{d^2+\{r\_1\}^2-\{r\_2\}^2\}\{2d\}\backslash \; ,\backslash qquad\; d\_2\; =\; \backslash frac\{d^2+\{r\_2\}^2-\{r\_1\}^2\}\{2d\}$,

:with $d\; =\; |M\_1\; M\_2|=|\backslash vec\; m\_2-\backslash vec\; m\_1|$.

($d\_i$ may be negative if $L$ is not between $M\_1,M\_2$.)

If the circles are intersecting at two points, the radical line runs through the common points. If they only touch each other, the radical line is the common tangent line.

File:Potenz-gerade-var.svg

• The radical axis of two intersecting circles is their common secant line.

:The radical axis of two touching circles is their common tangent.

:The radical axis of two non intersecting circles is the common secant of two convenient equipower circles (see below Orthogonal cicles).

File:Potenz-gerade-co.svg

• For a point $P$ outside a circle $c\_i$ and the two tangent points $S\_i,T\_i$ the equation $|PS\_i|^2=|PT\_i|^2=\backslash Pi\_i(P)$ holds and $S\_i,T\_i$ lie on the circle $c\_o$ with center $P$ and radius $\backslash sqrt\{\backslash Pi\_i(P)\}$. Circle $c\_o$ intersects $c\_i$ orthogonal. Hence:

:If $P$ is a point of the radical axis, then the four points $S\_1,T\_1,\; S\_2,T\_2$ lie on circle $c\_o$, which intersects the given circles $c\_1,c\_2$ orthogonally.

• The radical axis consists of all centers of circles, which intersect the given circles orthogonally.

The method described in the previous section for the construction of a pencil of circles, which intersect two given circles orthogonally, can be extended to the construction of two orthogonally intersecting systems of circles:A. Schoenfliess, R. Courant: Einführung in die Analytische Geometrie der Ebene und des Raumes, Springer-Verlag, 1931, p. 113C. Carathéodory: Funktionentheorie, Birkhäuser-Verlag, Basel, 1961, ISBN 978-3-7643-0064-7, p. 46

Let $c\_1,c\_2$ be two apart lying circles (as in the previous section), $M\_1,M\_2,r\_1,r\_2$ their centers and radii and $g\_\{12\}$ their radical axis. Now, all circles will be determined with centers on line

$\backslash overline\{M\_1M\_2\}$, which have together with $c\_1$ line $g\_\{12\}$ as radical axis, too. If $\backslash gamma\_2$ is such a circle, whose center has distance $\backslash delta$ to the center $M\_1$ and radius $\backslash rho\_2$. From the result in the previous section one gets the equation

:$d\_1=\backslash frac\{\backslash delta^2+r\_1^2-\backslash rho\_2^2\}\{2\backslash delta\}\; \backslash quad$, where $d\_1>r\_1$ are fixed.

With $\backslash delta\_2=\backslash delta-d\_1$ the equation can be rewritten as:

:$\backslash delta\_2^2=d\_1^2-r\_1^2+\backslash rho\_2^2$.

File:Kreise-orth-sys-e.svg

If radius $\backslash rho\_2$ is given, from this equation one finds the distance $\backslash delta\_2$ to the (fixed) radical axis of the new center. In the diagram the color of the new circles is purple. Any green circle (see diagram) has its center on the radical axis and intersects the circles $c\_1,c\_2$ orthogonally and hence all new circles (purple), too. Choosing the (red) radical axis as y-axis and line $\backslash overline\{M\_1M\_2\}$ as x-axis, the two pencils of circles have the equations:

:purple: $\backslash \; \backslash \; \backslash \; (x-\backslash delta\_2)^2+y^2=\backslash delta\_2^2+r\_1^2-d\_1^2$

:green: $\backslash \; x^2+(y-y\_g)^2=y\_g^2+d\_1^2-r\_1^2\backslash \; .$

($\backslash ;\; (0,y\_g)$ is the center of a green circle.)

Properties:

a) Any two green circles intersect on the x-axis at the points $P\_\{1/2\}=\backslash big(\backslash pm\backslash sqrt\{d\_1^2-r\_1^2\},0\backslash big)$, the poles of the orthogonal system of circles. That means, the x-axis is the radical line of the green circles.

b) The purple circles have no points in common. But, if one considers the real plane as part of the complex plane, then any two purple circles intersect on the y-axis (their common radical axis) at the points $Q\_\{1/2\}=\backslash big(0,\backslash pm\; i\; \backslash sqrt\{d\_1^2-r\_1^2\}\backslash big)$.

File:Kreis-sys-orth-pa.svg

File:Kreis-buesch-typen.svg

Special cases:

a) In case of $d\_1=r\_1$ the green circles are touching each other at the origin with the x-axis as common tangent and the purple circles have the y-axis as common tangent. Such a system of circles is called coaxal parabolic circles (see below).

b) Shrinking $c\_1$ to its center $M\_1$, i. e. $r\_1=0$, the equations turn into a more simple form and one gets $M\_1=P\_1$.

Conclusion:

a) For any real $w$ the pencil of circles

:$\backslash ;c(\backslash xi):\backslash ;\; (x-\backslash xi)^2+y^2-\backslash xi^2-w=0\backslash \; :$

:has the property: The y-axis is the radical axis of $c(\backslash xi\_1),c(\backslash xi\_2)$.

:In case of $w>0$ the circles $c(\backslash xi\_1),c(\backslash xi\_2)$ intersect at points $P\_\{1/2\}=(0,\backslash pm\backslash sqrt\; w)$.

:In case of they have no points in common.

:In case of $w=0$ they touch at $(0,0)$ and the y-axis is their common tangent.

b) For any real $w$ the two pencils of circles

:$c\_1(\backslash xi):\backslash ;\; (x-\backslash xi)^2+y^2-\backslash xi^2-w=0\backslash \; ,$

:$c\_2(\backslash eta):\backslash ;\; x^2+(y-\backslash eta)^2-\backslash eta^2\; +\; w=0\; \backslash$

:form a system of orthogonal circles. That means: any two circles $c\_1(\backslash xi),c\_2(\backslash eta)$ intersect orthogonally.

c) From the equations in b), one gets a coordinate free representation:

File:Kreise-orth-sys-p1p2.svg

:For the given points $P\_1,P\_2$, their midpoint $O$ and their line segment bisector $g\_\{12\}$ the two equations

::$|XM|^2=|OM|^2-|OP\_1|^2\backslash \; ,$

::$|XN|^2=|ON|^2+|OP\_1|^2=|NP\_1|^2$

:with $M$ on $\backslash overline\{P\_1P\_2\}$, but not between $P\_1,P\_2$, and $N$ on $g\_\{12\}$

:describe the orthogonal system of circles uniquely determined by $P\_1,P\_2$ which are the poles of the system.

:For $P\_1=P\_2=O$ one has to prescribe the axes $a\_1,a\_2$ of the system, too. The system is parabolic:

::$|XM|^2=|OM|^2\backslash \; ,\; \backslash quad\; |XN|^2=|ON|^2$

:with $M$ on $a\_1$ and $N$ on $a\_2$.

Straightedge and compass construction:

File:Kreise-os-konstr.svg

A system of orthogonal circles is determined uniquely by its poles $P\_1,P\_2$:

1. The axes (radical axes) are the lines $\backslash overline\{P\_1P\_2\}$ and the Line segment bisector $g\_\{12\}$ of the poles.
2. The circles (green in the diagram) through $P\_1,P\_2$ have their centers on $g\_\{12\}$. They can be drawn easily. For a point $N$ the radius is $\backslash ;r\_N=|NP\_1|\backslash ;$.
3. In order to draw a circle of the second pencil (in diagram blue) with center $M$ on $\backslash overline\{P\_1P\_2\}$, one determines the radius $r\_M$ applying the theorem of Pythagoras: $\backslash ;\; r\_M^2=|OM|^2-|OP\_1|^2\backslash ;$ (see diagram).

In case of $P\_1=P\_2$ the axes have to be chosen additionally. The system is parabolic and can be drawn easily.

Definition and properties:

Let $c\_1,c\_2$ be two circles and $\backslash Pi\_1,\backslash Pi\_2$ their power functions. Then for any $\backslash lambda\backslash ne\; 1$

• $\backslash Pi\_1(x,y)-\backslash lambda\backslash Pi\_2(x,y)=0$

is the equation of a circle $c(\backslash lambda)$ (see below). Such a system of circles is called coaxal circles generated by the circles $c\_1,c\_2$.

(In case of $\backslash lambda=1$ the equation describes the radical axis of $c\_1,c\_2$.) Dan Pedoe: Circles: A Mathematical View, mathematical Association of America, 2020, ISBN 9781470457327, p. 16R. Lachlan: An Elementary Treatise On Modern Pure Geometry, MacMillan&Co, New York,1893, p. 200

The power function of $c(\backslash lambda)$ is

:$\backslash \; \backslash Pi(\backslash lambda,x,y)=\backslash frac\{\backslash Pi\_1(x,y)-\backslash lambda\backslash Pi\_2(x,y)\}\{1-\backslash lambda\}$.

The normed equation (the coefficients of $x^2,y^2$ are $1$) of $c(\backslash lambda)$ is $\backslash \; \backslash Pi(\backslash lambda,x,y)=0$.

A simple calculation shows:

• $c(\backslash lambda),c(\backslash mu),\backslash \; \backslash lambda\backslash ne\backslash mu\backslash \; ,$ have the same radical axis as $c\_1,c\_2$.

Allowing $\backslash lambda$ to move to infinity, one recognizes, that $c\_1,c\_2$ are members of the system of coaxal circles: $c\_1=c(0),\backslash ;\; c\_2=c(\backslash infty)$.

(E): If $c\_1,c\_2$ intersect at two points $P\_1,P\_2$, any circle $c(\backslash lambda)$ contains $P\_1,P\_2$, too, and line $\backslash overline\{P\_1P\_2\}$ is their common radical axis. Such a system is called elliptic.

(P): If $c\_1,c\_2$ are tangent at $P$, any circle is tangent to $c\_1,c\_2$ at point $P$, too. The common tangent is their common radical axis. Such a system is called parabolic.

(H): If $c\_1,c\_2$ have no point in common, then any pair of the system, too. The radical axis of any pair of circles is the radical axis of $c\_1,c\_2$. The system is called hyperbolic.

In detail:

Introducing coordinates such that

:$c\_1:\; (x-d\_1)^2+y^2=r\_1^2$

:$c\_2:\; (x-d\_2)^2+y^2=\; d\_2^2+r\_1^2-d\_1^2$,

then the y-axis is their radical axis (see above).

Calculating the power function $\backslash Pi(\backslash lambda,x,y)$ gives the normed circle equation:

:$c(\backslash lambda):\; \backslash \; x^2+y^2-2\backslash tfrac\{d\_1-\backslash lambda\; d\_2\}\{1-\backslash lambda\}\backslash ;\; x\; +d\_1^2-r\_1^2=0\backslash \; .$

Completing the square and the substitution

$\backslash delta\_2=\backslash tfrac\{d\_1-\backslash lambda\; d\_2\}\{1-\backslash lambda\}$ (x-coordinate of the center) yields the centered form of the equation

:$c(\backslash lambda):\; \backslash \; (x-\backslash delta\_2)^2+y^2=\backslash delta\_2^2+r\_1^2-d\_1^2$.

In case of $r\_1>d\_1$ the circles $c\_1,c\_2,c(\backslash lambda)$ have the two points

:$P\_1=\backslash big(0,\backslash sqrt\{r\_1^2-d\_1^2\}\backslash big),\backslash quad\; P\_2=\backslash big(0,-\backslash sqrt\{r\_1^2-d\_1^2\}\backslash big)$

in common and the system of coaxal circles is elliptic.

In case of $r\_1=d\_1$ the circles $c\_1,c\_2,c(\backslash lambda)$ have point $P\_0=(0,0)$ in common and the system is parabolic.

In case of hyperbolic.

Alternative equations:

1) In the defining equation of a coaxal system of circles there can be used multiples of the power functions, too.

2) The equation of one of the circles can be replaced by the equation of the desired radical axis. The radical axis can be seen as a circle with an infinitely large radius. For example:

:$(x-x\_1)^2+y^2-r^2\_1\backslash \; -\; \backslash \; \backslash lambda\backslash ;\; 2(x-x\_2)\backslash \; =0\backslash \; \backslash Leftrightarrow$

:$(x-(x\_1+\backslash lambda))^2+y^2\; =(x\_1+\backslash lambda)^2+r\_1^2-x\_1^2-2\backslash lambda\; x\_2$,

describes all circles, which have with the first circle the line $x=x\_2$ as radical axis.

3) In order to express the equal status of the two circles, the following form is often used:

:$\backslash mu\backslash Pi\_1(x,y)+\backslash nu\backslash Pi\_2(x,y)=0\backslash ;\; .$

But in this case the representation of a circle by the parameters $\backslash mu,\backslash nu$ is not unique.

Applications:

a) Circle inversions and Möbius transformations preserve angles and generalized circles. Hence orthogonal systems of circles play an essential role with investigations on these mappings.Carathéodory: Funktionentheorie, p. 47.R. Sauer: Ingenieur-Mathematik: Zweiter Band: Differentialgleichungen und Funktionentheorie, Springer-Verlag, 1962, ISBN 978-3-642-53232-0, p. 105

b) In electromagnetism coaxal circles appear as field lines.Clemens Schaefer: Elektrodynamik und Optik, Verlag: De Gruyter, 1950, ISBN 978-3-11-230936-0, p. 358.

[[File:Potenz-gerade-3k.svg|thumb|upright=1.4|Radical center of three circles

The green circle intersects the three circles orthogonally.]]

• For three circles $c\_1,c\_2,c\_3$, no two of which are concentric, there are three radical axes $g\_\{12\},g\_\{23\},g\_\{31\}$. If the circle centers do not lie on a line, the radical axes intersect in a common point $R$, the radical center of the three circles. The orthogonal circle centered around $R$ of two circles is orthogonal to the third circle, too (radical circle).

:Proof: the radical axis $g\_\{ik\}$ contains all points which have equal tangential distance to the circles $c\_i,c\_k$. The intersection point $R$ of $g\_\{12\}$ and $g\_\{23\}$ has the same tangential distance to all three circles. Hence $R$ is a point of the radical axis $g\_\{31\}$, too.

:This property allows one to construct the radical axis of two non intersecting circles $c\_1,c\_2$ with centers $M\_1,M\_2$: Draw a third circle $c\_3$ with center not collinear to the given centers that intersects $c\_1,c\_2$. The radical axes $g\_\{13\},g\_\{23\}$ can be drawn. Their intersection point is the radical center $R$ of the three circles and lies on $g\_\{12\}$. The line through $R$ which is perpendicular to $\backslash overline\{M\_1M\_2\}$ is the radical axis $g\_\{12\}$.

Additional construction method:

File:Potenz-gerade-konstr-e.svg

All points which have the same power to a given circle $c$ lie on a circle concentric to $c$. Let us call it an equipower circle. This property can be used for an additional construction method of the radical axis of two circles:

For two non intersecting circles $c\_1,c\_2$, there can be drawn two equipower circles $c\text{'}\_1,c\text{'}\_2$, which have the same power with respect to $c\_1,c\_2$ (see diagram). In detail: $\backslash Pi\_1(P\_1)=\backslash Pi\_2(P\_2)$. If the power is large enough, the circles $c\text{'}\_1,c\text{'}\_2$ have two points in common, which lie on the radical axis $g\_\{12\}$.

In general, any two disjoint, non-concentric circles can be aligned with the circles of a system of bipolar coordinates. In that case, the radical axis is simply the $y$-axis of this system of coordinates. Every circle on the axis that passes through the two foci of the coordinate system intersects the two circles orthogonally. A maximal collection of circles, all having centers on a given line and all pairs having the same radical axis, is known as a pencil of coaxal circles.

If the circles are represented in trilinear coordinates in the usual way, then their radical center is conveniently given as a certain determinant. Specifically, let X = x : y : z denote a variable point in the plane of a triangle ABC with sidelengths a = |BC|, b = |CA|, c = |AB|, and represent the circles as follows:

:(dx + ey + fz)(ax + by + cz) + g(ayz + bzx + cxy) = 0

:(hx + iy + jz)(ax + by + cz) + k(ayz + bzx + cxy) = 0

:(lx + my + nz)(ax + by + cz) + p(ayz + bzx + cxy) = 0

Then the radical center is the point

:

e&i&m\\f&j&n\end{bmatrix} : \det \begin{bmatrix}g&k&p\\

f&j&n\\d&h&l\end{bmatrix} : \det \begin{bmatrix}g&k&p\\

d&h&l\\e&i&m\end{bmatrix}.

The radical plane of two nonconcentric spheres in three dimensions is defined similarly: it is the locus of points from which tangents to the two spheres have the same length.See [https://www.merriam-webster.com/dictionary/radical%20plane Merriam–Webster online dictionary]. The fact that this locus is a plane follows by rotation in the third dimension from the fact that the radical axis is a straight line.

The same definition can be applied to hyperspheres in Euclidean space of any dimension, giving the radical hyperplane of two nonconcentric hyperspheres.

{{reflist|1}}

• {{cite book | author = R. A. Johnson | year = 1960 | title = Advanced Euclidean Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle | url = https://archive.org/details/advancedeuclidea00john_668 | url-access = limited | edition = reprint of 1929 edition by Houghton Mifflin | publisher = Dover Publications | location = New York | isbn = 978-0-486-46237-0 | pages = [https://archive.org/details/advancedeuclidea00john_668/page/n44 31]–43 }}

• {{Cite book | author = C. Stanley Ogilvy | year = 1990 | title = Excursions in Geometry | publisher = Dover | isbn = 0-486-26530-7 | pages = [https://archive.org/details/excursionsingeom0000ogil/page/17 17–23] | url = https://archive.org/details/excursionsingeom0000ogil/page/17 }}
• {{cite book | title = Geometry Revisited | url = https://archive.org/details/geometryrevisite00coxe | url-access = limited | author = H. S. M. Coxeter, S. L. Greitzer| year = 1967 | publisher = Mathematical Association of America | location = Washington, D.C. | isbn = 978-0-88385-619-2 | pages = [https://archive.org/details/geometryrevisite00coxe/page/n42 31]–36, 160–161}}
• Clark Kimberling, "Triangle Centers and Central Triangles," Congressus Numerantium 129 (1998) i–xxv, 1–295.

{{commons category|Radical axes}}

• {{mathworld|RadicalLine|Radical line}}
• {{mathworld|ChordalTheorem|Chordal theorem}}
• [http://www.cut-the-knot.org/Curriculum/Geometry/IncircleInSegment1.shtml Animation] at Cut-the-knot

Category:Circles

Category:Elementary geometry

Category:Analytic geometry