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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 ¯. 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]
The radical axis of a pair of circles is defined as the set of points that have equal power h with respect to both circles. For example, for every point P on the radical axis of circles 1 and 2, the powers to each circle are equal: h 1 = h 2. Similarly, for every point on the radical axis of circles 2 and 3, the powers must be equal, h 2 = h 3.
The radical axis of two intersecting circles. The power diagram of the two circles is the partition of the plane into two halfplanes formed by this line. In the case n = 2, the power diagram consists of two halfplanes, separated by a line called the radical axis or chordale of the two circles. Along the radical axis, both circles have equal power.
The set of all points with () = is a line called radical axis. It contains possible common points of the circles and is perpendicular to line O 1 O 2 ¯ {\displaystyle {\overline {O_{1}O_{2}}}} . Secants theorem, chords theorem: common proof
The radical axis is always perpendicular to the line of centers, and if two circles intersect, their radical axis is the line joining their points of intersection. For three circles, three radical axes can be defined, one for each pair of circles ( C 1 / C 2 , C 1 / C 3 , C 2 / C 3 ); remarkably, these three radical axes intersect at a single ...
The line connecting these common intersection points is the radical axis for all three circles. The two isodynamic points are inverses of each other relative to the circumcircle of the triangle. The centers of these three circles fall on a single line (the Lemoine line). This line is perpendicular to the radical axis, which is the line ...
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The radical axis of two circles is the set of points of equal tangents, or more generally, equal power. Circles may be inverted into lines and circles into circles. [clarification needed] If two circles are internally tangent, they remain so if their radii are increased or decreased by the same amount.