Radical axis

The radical axis of two circles is the line containing points with equal powers relative to both circles. It is perpendicular to the line joining their centers.

Power of a point: For a circle, the power of a point is the squared distance to the center minus the squared radius. Subtracting the two circles’ powers gives the linear equation of their radical axis.
Intersecting circles: Intersecting circles are circles that share two distinct points. Their two shared points define the radical axis as a common chord.
Line: A line is a straight one-dimensional set extending without bound in both directions. The equal-power locus is linear even though it is defined using circular distances.
Radical center: The radical center of three circles is the common intersection of their three pairwise radical axes. It extends the equal-power construction from a pair of circles to a triple.
Circle: A circle is the set of points in a plane at a fixed distance from a center. The centers and radii of the two circles determine their power functions.
Tangent circles: Tangent circles meet at exactly one point. Their single common point lies on the radical axis, which is perpendicular to the center line.
Euclidean distance: Euclidean distance is the straight-line distance between two points in Euclidean space. Circle power is computed from the distance between a point and the circle’s center.
Radical axis theorem: The radical axis theorem states that the pairwise radical axes of three circles are concurrent or parallel. It explains why three pairwise equal-power lines meet at one radical center when possible.
Perpendicular lines: Two lines are perpendicular when they meet at a right angle. The radical axis is perpendicular to the line joining the circle centers.
Concentric circles: Concentric circles have the same center and different radii. Their equal-power locus is empty, rather than a finite radical-axis line.

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