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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.
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.
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.
Gergonne was able to identify two other points for each of the three lines. One of the two points has already been identified: the radical center G lies on all three lines (Figure 6). To locate a second point on the lines L 1, L 2 and L 3, Gergonne noted a reciprocal relationship between those lines and the radical axis R of the solution ...
Radical axis Let be a point and , ... Johnson RA (1960), Advanced Euclidean Geometry: An elementary treatise on the geometry of the triangle and the circle ...
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 ...
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 hyperbolic pencil defined by points C, D (the blue circles) has its radical axis on the perpendicular bisector of line CD, and all its circle centers on line CD. Inversive geometry, orthogonal intersection, and coordinate systems