Radical axis

In Euclidean geometry, the radical axis of two non-concentric circles is the set of points whose power with respect to the circles are equal. For this reason the radical axis is also called the power line or power bisector of the two circles. In detail:

For two circles c₁, c₂ with centers M₁, M₂ and radii r₁, r₂ the powers of a point P with respect to the circles are

${\displaystyle \Pi _{1}(P)=|PM_{1}|^{2}-r_{1}^{2},\qquad \Pi _{2}(P)=|PM_{2}|^{2}-r_{2}^{2}.}$

Point P belongs to the radical axis, if

${\displaystyle \Pi _{1}(P)=\Pi _{2}(P).}$

If the circles have two points in common, the radical axis is the common secant line of the circles.
If point P is outside the circles, P has equal tangential distance to both the circles.
If the radii are equal, the radical axis is the line segment bisector of M₁, M₂.
In any case the radical axis is a line perpendicular to ${\displaystyle {\overline {M_{1}M_{2}}}.}$

On notations

The notation radical axis was used by the French mathematician M. Chasles as axe radical.^[1]
J.V. Poncelet used chorde ideale.^[2]
J. Plücker introduced the term Chordale.^[3]
J. Steiner called the radical axis line of equal powers (German: Linie der gleichen Potenzen) which led to power line (Potenzgerade).^[4]

Let ${\displaystyle {\vec {x}},{\vec {m}}_{1},{\vec {m}}_{2}}$ be the position vectors of the points ${\displaystyle P,M_{1},M_{2}}$. Then the defining equation of the radical line can be written as:

${\displaystyle ({\vec {x}}-{\vec {m}}_{1})^{2}-r_{1}^{2}=({\vec {x}}-{\vec {m}}_{2})^{2}-r_{2}^{2}\quad \leftrightarrow \quad 2{\vec {x}}\cdot ({\vec {m}}_{2}-{\vec {m}}_{1})+{\vec {m}}_{1}^{2}-{\vec {m}}_{2}^{2}+r_{2}^{2}-r_{1}^{2}=0}$

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.

(${\displaystyle {\vec {m}}_{2}-{\vec {m}}_{1}}$ is a normal vector to the radical axis !)

Dividing the equation by ${\displaystyle 2|{\vec {m}}_{2}-{\vec {m}}_{1}|}$, one gets the Hessian normal form. Inserting the position vectors of the centers yields the distances of the centers to the radical axis:

${\displaystyle d_{1}={\frac {d^{2}+{r_{1}}^{2}-{r_{2}}^{2}}{2d}}\ ,\qquad d_{2}={\frac {d^{2}+{r_{2}}^{2}-{r_{1}}^{2}}{2d}}}$,

with ${\displaystyle d=|M_{1}M_{2}|}$.

(${\displaystyle d_{i}}$ may be negative if ${\displaystyle L}$ is not between ${\displaystyle M_{1},M_{2}}$.)

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.

• 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).

• For a point ${\displaystyle P}$ outside a circle ${\displaystyle c_{i}}$ and the two tangent points ${\displaystyle S_{i},T_{i}}$ the equation ${\displaystyle |PS_{i}|^{2}=|PT_{i}|^{2}=\Pi _{i}(P)}$ holds and ${\displaystyle S_{i},T_{i}}$ lie on the circle ${\displaystyle c_{o}}$ with center ${\displaystyle P}$ and radius ${\displaystyle {\sqrt {\Pi _{i}(P)}}}$. Circle ${\displaystyle c_{o}}$ intersects ${\displaystyle c_{i}}$ orthogonal. Hence:

If ${\displaystyle P}$ is a point of the radical axis, then the four points ${\displaystyle S_{1},T_{1},S_{2},T_{2}}$ lie on circle ${\displaystyle c_{o}}$, which intersects the given circles ${\displaystyle c_{1},c_{2}}$ 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:^[5][6]

Let ${\displaystyle c_{1},c_{2}}$ be two apart lying circles (as in the previous section), ${\displaystyle M_{1},M_{2},r_{1},r_{2}}$ their centers and radii and ${\displaystyle g_{12}}$ their radical axis. Now, all circles will be determined with centers on line ${\displaystyle {\overline {M_{1}M_{2}}}}$, which have together with ${\displaystyle c_{1}}$ line ${\displaystyle g_{12}}$ as radical axis, too. If ${\displaystyle \gamma _{2}}$ is such a circle, whose center has distance ${\displaystyle \delta }$ to the center ${\displaystyle M_{1}}$ and radius ${\displaystyle \rho _{2}}$. From the result in the previous section one gets the equation

${\displaystyle d_{1}={\frac {\delta ^{2}+r_{1}^{2}-\rho _{2}^{2}}{2\delta }}\quad }$, where ${\displaystyle d_{1}>r_{1}}$ are fixed.

With ${\displaystyle \delta _{2}=\delta -d_{1}}$ the equation can be rewritten as:

${\displaystyle \delta _{2}^{2}=d_{1}^{2}-r_{1}^{2}+\rho _{2}^{2}}$.

If radius ${\displaystyle \rho _{2}}$ is given, from this equation one finds the distance ${\displaystyle \delta _{2}}$ 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 ${\displaystyle c_{1},c_{2}}$ orthogonally and hence all new circles (purple), too. Choosing the (red) radical axis as y-axis and line ${\displaystyle {\overline {M_{1}M_{2}}}}$ as x-axis, the two pencils of circles have the equations:

purple: ${\displaystyle \ \ \ (x-\delta _{2})^{2}+y^{2}=\delta _{2}^{2}+r_{1}^{2}-d_{1}^{2}}$

green: ${\displaystyle \ x^{2}+(y-y_{g})^{2}=y_{g}^{2}+d_{1}^{2}-r_{1}^{2}\ .}$

(${\displaystyle \;(0,y_{g})}$ is the center of a green circle.)

Properties:
a) Any two green circles intersect on the x-axis at the points ${\displaystyle P_{1/2}={\big (}\pm {\sqrt {d_{1}^{2}-r_{1}^{2}}},0{\big )}}$, 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 ${\displaystyle Q_{1/2}={\big (}0,\pm i{\sqrt {d_{1}^{2}-r_{1}^{2}}}{\big )}}$.

Special cases:
a) In case of ${\displaystyle d_{1}=r_{1}}$ 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 ${\displaystyle c_{1}}$ to its center ${\displaystyle M_{1}}$, i. e. ${\displaystyle r_{1}=0}$, the equations turn into a more simple form and one gets ${\displaystyle M_{1}=P_{1}}$.

Conclusion:
a) For any real ${\displaystyle w}$ the pencil of circles

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

has the property: The y-axis is the radical axis of ${\displaystyle c(\xi _{1}),c(\xi _{2})}$.

In case of ${\displaystyle w>0}$ the circles ${\displaystyle c(\xi _{1}),c(\xi _{2})}$ intersect at points ${\displaystyle P_{1/2}=(0,\pm {\sqrt {w}})}$.

In case of ${\displaystyle w<0}$ they have no points in common.

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

b) For any real ${\displaystyle w}$ the two pencils of circles

${\displaystyle c_{1}(\xi ):\;(x-\xi )^{2}+y^{2}-\xi ^{2}-w=0\ ,}$

${\displaystyle c_{2}(\eta ):\;x^{2}+(y-\eta )^{2}-\eta ^{2}+w=0\ }$

form a system of orthogonal circles. That means: any two circles ${\displaystyle c_{1}(\xi ),c_{2}(\eta )}$ intersect orthogonally.

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

For the given points ${\displaystyle P_{1},P_{2}}$, their midpoint ${\displaystyle O}$ and their line segment bisector ${\displaystyle g_{12}}$ the two equations

${\displaystyle |XM|^{2}=|OM|^{2}-|OP_{1}|^{2}\ ,}$

${\displaystyle |XN|^{2}=|ON|^{2}+|OP_{1}|^{2}=|NP_{1}|^{2}}$

with ${\displaystyle M}$ on ${\displaystyle {\overline {P_{1}P_{2}}}}$, but not between ${\displaystyle P_{1},P_{2}}$, and ${\displaystyle N}$ on ${\displaystyle g_{12}}$

describe the orthogonal system of circles uniquely determined by ${\displaystyle P_{1},P_{2}}$ which are the poles of the system.

For ${\displaystyle P_{1}=P_{2}=O}$ one has to prescribe the axes ${\displaystyle a_{1},a_{2}}$ of the system, too. The system is parabolic:

${\displaystyle |XM|^{2}=|OM|^{2}\ ,\quad |XN|^{2}=|ON|^{2}}$

with ${\displaystyle M}$ on ${\displaystyle a_{1}}$ and ${\displaystyle N}$ on ${\displaystyle a_{2}}$.

Straightedge and compass construction:

A system of orthogonal circles is determined uniquely by its poles ${\displaystyle P_{1},P_{2}}$:

1. The axes (radical axes) are the lines ${\displaystyle {\overline {P_{1}P_{2}}}}$ and the Line segment bisector ${\displaystyle g_{12}}$ of the poles.
2. The circles (green in the diagram) through ${\displaystyle P_{1},P_{2}}$ have their centers on ${\displaystyle g_{12}}$. They can be drawn easily. For a point ${\displaystyle N}$ the radius is ${\displaystyle \;r_{N}=|NP_{1}|\;}$.
3. In order to draw a circle of the second pencil (in diagram blue) with center ${\displaystyle M}$ on ${\displaystyle {\overline {P_{1}P_{2}}}}$, one determines the radius ${\displaystyle r_{M}}$ applying the theorem of Pythagoras: ${\displaystyle \;r_{M}^{2}=|OM|^{2}-|OP_{1}|^{2}\;}$ (see diagram).

In case of ${\displaystyle P_{1}=P_{2}}$ the axes have to be chosen additionally. The system is parabolic and can be drawn easily.

Definition and properties:

Let ${\displaystyle c_{1},c_{2}}$ be two circles and ${\displaystyle \Pi _{1},\Pi _{2}}$ their power functions. Then for any ${\displaystyle \lambda \neq 1}$

• ${\displaystyle \Pi _{1}(x,y)-\lambda \Pi _{2}(x,y)=0}$

is the equation of a circle ${\displaystyle c(\lambda )}$ (see below). Such a system of circles is called coaxal circles generated by the circles ${\displaystyle c_{1},c_{2}}$. (In case of ${\displaystyle \lambda =1}$ the equation describes the radical axis of ${\displaystyle c_{1},c_{2}}$.) ^[7][8]

The power function of ${\displaystyle c(\lambda )}$ is

${\displaystyle \ \Pi (\lambda ,x,y)={\frac {\Pi _{1}(x,y)-\lambda \Pi _{2}(x,y)}{1-\lambda }}}$.

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

A simple calculation shows:

• ${\displaystyle c(\lambda ),c(\mu ),\ \lambda \neq \mu \ ,}$ have the same radical axis as ${\displaystyle c_{1},c_{2}}$.

Allowing ${\displaystyle \lambda }$ to move to infinity, one recognizes, that ${\displaystyle c_{1},c_{2}}$ are members of the system of coaxal circles: ${\displaystyle c_{1}=c(0),\;c_{2}=c(\infty )}$.

(E): If ${\displaystyle c_{1},c_{2}}$ intersect at two points ${\displaystyle P_{1},P_{2}}$, any circle ${\displaystyle c(\lambda )}$ contains ${\displaystyle P_{1},P_{2}}$, too, and line ${\displaystyle {\overline {P_{1}P_{2}}}}$ is their common radical axis. Such a system is called elliptic.
(P): If ${\displaystyle c_{1},c_{2}}$ are tangent at ${\displaystyle P}$, any circle is tangent to ${\displaystyle c_{1},c_{2}}$ at point ${\displaystyle P}$, too. The common tangent is their common radical axis. Such a system is called parabolic.
(H): If ${\displaystyle c_{1},c_{2}}$ 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 ${\displaystyle c_{1},c_{2}}$. The system is called hyperbolic.

In detail:

Introducing coordinates such that

${\displaystyle c_{1}:(x-d_{1})^{2}+y^{2}=r_{1}^{2}}$

${\displaystyle c_{2}:(x-d_{2})^{2}+y^{2}=d_{2}^{2}+r_{1}^{2}-d_{1}^{2}}$,

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

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

${\displaystyle c(\lambda ):\ x^{2}+y^{2}-2{\tfrac {d_{1}-\lambda d_{2}}{1-\lambda }}\;x+d_{1}^{2}-r_{1}^{2}=0\ .}$

Completing the square and the substitution ${\displaystyle \delta _{2}={\tfrac {d_{1}-\lambda d_{2}}{1-\lambda }}}$ (x-coordinate of the center) yields the centered form of the equation

${\displaystyle c(\lambda ):\ (x-\delta _{2})^{2}+y^{2}=\delta _{2}^{2}+r_{1}^{2}-d_{1}^{2}}$.

In case of ${\displaystyle r_{1}>d_{1}}$ the circles ${\displaystyle c_{1},c_{2},c(\lambda )}$ have the two points

${\displaystyle P_{1}={\big (}0,{\sqrt {r_{1}^{2}-d_{1}^{2}}}{\big )},\quad P_{2}={\big (}0,-{\sqrt {r_{1}^{2}-d_{1}^{2}}}{\big )}}$

in common and the system of coaxal circles is elliptic.

In case of ${\displaystyle r_{1}=d_{1}}$ the circles ${\displaystyle c_{1},c_{2},c(\lambda )}$ have point ${\displaystyle P_{0}=(0,0)}$ in common and the system is parabolic.

In case of ${\displaystyle r_{1}<d_{1}}$ the circles ${\displaystyle c_{1},c_{2},c(\lambda )}$ have no point in common and the system is 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:

${\displaystyle (x-x_{1})^{2}+y^{2}-r_{1}^{2}\ -\ \lambda \;2(x-x_{2})\ =0\ \Leftrightarrow }$

${\displaystyle (x-(x_{1}+\lambda ))^{2}+y^{2}=(x_{1}+\lambda )^{2}+r_{1}^{2}-x_{1}^{2}-2\lambda x_{2}}$,

describes all circles, which have with the first circle the line ${\displaystyle x=x_{2}}$ as radical axis.
3) In order to express the equal status of the two circles, the following form is often used:

${\displaystyle \mu \Pi _{1}(x,y)+\nu \Pi _{2}(x,y)=0\;.}$

But in this case the representation of a circle by the parameters ${\displaystyle \mu ,\nu }$ 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.^[9][10]
b) In electromagnetism coaxal circles appear as field lines.^[11]

• For three circles ${\displaystyle c_{1},c_{2},c_{3}}$, no two of which are concentric, there are three radical axes ${\displaystyle g_{12},g_{23},g_{31}}$. If the circle centers do not lie on a line, the radical axes intersect in a common point ${\displaystyle R}$, the radical center of the three circles. The orthogonal circle centered around ${\displaystyle R}$ of two circles is orthogonal to the third circle, too (radical circle).

Proof: the radical axis ${\displaystyle g_{ik}}$ contains all points which have equal tangential distance to the circles ${\displaystyle c_{i},c_{k}}$. The intersection point ${\displaystyle R}$ of ${\displaystyle g_{12}}$ and ${\displaystyle g_{23}}$ has the same tangential distance to all three circles. Hence ${\displaystyle R}$ is a point of the radical axis ${\displaystyle g_{31}}$, too.

This property allows one to construct the radical axis of two non intersecting circles ${\displaystyle c_{1},c_{2}}$ with centers ${\displaystyle M_{1},M_{2}}$: Draw a third circle ${\displaystyle c_{3}}$ with center not collinear to the given centers that intersects ${\displaystyle c_{1},c_{2}}$. The radical axes ${\displaystyle g_{13},g_{23}}$ can be drawn. Their intersection point is the radical center ${\displaystyle R}$ of the three circles and lies on ${\displaystyle g_{12}}$. The line through ${\displaystyle R}$ which is perpendicular to ${\displaystyle {\overline {M_{1}M_{2}}}}$ is the radical axis ${\displaystyle g_{12}}$.

Additional construction method:

All points which have the same power to a given circle ${\displaystyle c}$ lie on a circle concentric to ${\displaystyle c}$. 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 ${\displaystyle c_{1},c_{2}}$, there can be drawn two equipower circles ${\displaystyle c'_{1},c'_{2}}$, which have the same power with respect to ${\displaystyle c_{1},c_{2}}$ (see diagram). In detail: ${\displaystyle \Pi _{1}(P_{1})=\Pi _{2}(P_{2})}$. If the power is large enough, the circles ${\displaystyle c'_{1},c'_{2}}$ have two points in common, which lie on the radical axis ${\displaystyle g_{12}}$.

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 ${\displaystyle y}$-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

${\displaystyle \det {\begin{bmatrix}g&k&p\\e&i&m\\f&j&n\end{bmatrix}}:\det {\begin{bmatrix}g&k&p\\f&j&n\\d&h&l\end{bmatrix}}:\det {\begin{bmatrix}g&k&p\\d&h&l\\e&i&m\end{bmatrix}}.}$

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.^[12] 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.

• R. A. Johnson (1960). Advanced Euclidean Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle (reprint of 1929 edition by Houghton Mifflin ed.). New York: Dover Publications. pp. 31–43. ISBN 978-0-486-46237-0.

• C. Stanley Ogilvy (1990). Excursions in Geometry. Dover. pp. 17–23. ISBN 0-486-26530-7.
• H. S. M. Coxeter, S. L. Greitzer (1967). Geometry Revisited. Washington, D.C.: Mathematical Association of America. pp. 31–36, 160–161. ISBN 978-0-88385-619-2.
• Clark Kimberling, "Triangle Centers and Central Triangles," Congressus Numerantium 129 (1998) i–xxv, 1–295.