In geometry, the limiting points of two disjoint circles ''A'' and ''B'' in the

Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...

are points ''p'' that may be defined by any of the following equivalent properties: *The pencil of circles defined by ''A'' and ''B'' contains a degenerate (radius zero) circle centered at ''p''. *Every circle or line that is perpendicular to both ''A'' and ''B'' passes through ''p''. *An inversion centered at ''p'' transforms ''A'' and ''B'' into

concentric In geometry, two or more objects are said to be concentric, coaxal, or coaxial when they share the same center or axis. Circles, regular polygons and regular polyhedra, and spheres may be concentric to one another (sharing the same center point ...

circles. The midpoint of the two limiting points is the point where the

radical axis 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. ...

of ''A'' and ''B'' crosses the line through their centers. This intersection point has equal power distance to all the circles in the pencil containing ''A'' and ''B''. The limiting points themselves can be found at this distance on either side of the intersection point, on the line through the two circle centers. From this fact it is straightforward to construct the limiting points algebraically or by

compass and straightedge In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...

. An explicit formula expressing the limiting points as the solution to a quadratic equation in the coordinates of the circle centers and their radii is given by Weisstein. Inverting one of the two limiting points through ''A'' or ''B'' produces the other limiting point. An inversion centered at one limiting point maps the other limiting point to the common center of the concentric circles..

References

Circles Inversive geometry {{elementary-geometry-stub