Coaxality

Coaxality is a property of a family of circles in the Euclidean plane in which all members share a common radical axis. The radical axis of two circles is the locus of points having equal power with respect to the two circles; it is a straight line. A set of circles is coaxal if every pair in the set has the same radical axis, equivalently the circles form a pencil with a fixed radical axis.

A classic example is a coaxal system consisting of all circles that pass through two fixed points A and B. Any two such circles intersect at A and B, and their radical axis is the line AB.

More generally, a coaxal family may have no real common points; in that case the common radical axis is still defined, and the family is said to be non-intersecting coaxal. There are two fixed points on the radical axis called the limiting points, which are determined by the family (they may be real or imaginary).

Coaxality is preserved under projective similarity and inversion, making it a useful organizing principle in circle geometry. It also generalizes to three dimensions, where a family of spheres is coaxal if they share a common radical plane.