Circle of antisimilitude

If α and β are non-intersecting or tangent, a single circle of antisimilitude exists; if α and β intersect at two points, there are two circles of antisimilitude.

[1][2] If the two circles α and β cross each other, another two circles γ and δ are each tangent to both α and β, and in addition γ and δ are tangent to each other, then the point of tangency between γ and δ necessarily lies on one of the two circles of antisimilitude.

If α and β are disjoint and non-concentric, then the locus of points of tangency of γ and δ again forms two circles, but only one of these is the (unique) circle of antisimilitude.

If a circle γ crosses circles α and β at equal angles, then γ is crossed orthogonally by one of the circles of antisimilitude of α and β; if γ crosses α and β in supplementary angles, it is crossed orthogonally by the other circle of antisimilitude, and if γ is orthogonal to both α and β then it is also orthogonal to both circles of antisimilitude.

Altogether, at most eight triple crossing points may be generated in this way, for there are two ways of choosing each of the first two circles and two points where the two chosen circles cross.