Radical axis information

Two circles, centered at $M 1, M 2$

  Radical axis, with sample point $P$

  Tangential distances from both circles to $P$
The tangent lines must be equal in length for any point on the radical axis: $|PT_{1}|=|PT_{2}|.$ If $P, T 1, T 2$ lie on a common tangent, then $P$ is the midpoint of ${\overline {T_{1}T_{2}}}.$

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 $c 1, c 2$ with centers $M 1, M 2$ and radii $r 1, r 2$ the powers of a point $P$ with respect to the circles are

\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

\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 $M 1, M 2$ .
In any case the radical axis is a line perpendicular to ${\overline {M_{1}M_{2}}}.$

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]

^ Michel Chasles, C. H. Schnuse: Die Grundlehren der neuern Geometrie, erster Theil, Verlag Leibrock, Braunschweig, 1856, p. 312
^ Ph. Fischer: Lehrbuch der analytische Geometrie, Darmstadt 1851, Verlag Ernst Kern, p. 67
^ H. Schwarz: Die Elemente der analytischen Geometrie der Ebene, Verlag H. W. Schmidt, Halle, 1858, p. 218
^ Jakob Steiner: Einige geometrische Betrachtungen. In: Journal für die reine und angewandte Mathematik, Band 1, 1826, p. 165

[1] Michel Chasles, C. H. Schnuse: Die Grundlehren der neuern Geometrie, erster Theil, Verlag Leibrock, Braunschweig, 1856, p. 312

[2] Ph. Fischer: Lehrbuch der analytische Geometrie, Darmstadt 1851, Verlag Ernst Kern, p. 67

[3] H. Schwarz: Die Elemente der analytischen Geometrie der Ebene, Verlag H. W. Schmidt, Halle, 1858, p. 218

[4] Jakob Steiner: Einige geometrische Betrachtungen. In: Journal für die reine und angewandte Mathematik, Band 1, 1826, p. 165

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