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
Π 1 ( P ) = | P M 1 | 2 − r 1 2 , Π 2 ( P ) = | P M 2 | 2 − r 2 2 . {\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
Π 1 ( P ) = Π 2 ( P ) . {\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 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 M 1 M 2 ¯ . {\displaystyle {\overline {M_{1}M_{2}}}.}
On notations The term radical axis was used by the French mathematician M. Chasles as axe radical. J.V. Poncelet used the term chorde ideale. J. Plücker introduced the term Chordale. J. Steiner called the radical axis line of equal powers (German: Linie der gleichen Potenzen) which led to the term power line (Potenzgerade).
Properties
Geometric shape and its position Let x → , m → 1 , m → 2 {\displaystyle {\vec {x}},{\vec {m}}_{1},{\vec {m}}_{2}} be the position vectors of the points P , M 1 , M 2 {\displaystyle P,M_{1},M_{2}} . Then the defining equation of the radical line can be written as:
( x → − m → 1 ) 2 − r 1 2 = ( x → − m → 2 ) 2 − r 2 2 ↔ 2 x → ⋅ ( m → 2 − m → 1 ) + m → 1 2 − m → 2 2 + r 2 2 − r 1 2 = 0 {\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
… excerpt ends here. Continue reading the full article.
