Radical axis

This article is about the radical axis used in geometry. For the animation studio, see Radical Axis (studio).

"Radical line" redirects here. For the Kangxi radical, see Radical 2.

  Two circles, centered at M₁, M₂

  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, T₁, T₂ 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 powers 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 respectively, 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 the both 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 ${\overline {M_{1}M_{2}}}.$