Consider a circle bundle (I) of circles f.e. generated by two given circles c_{1} and c_{2}. Consider also an arbitrary circle c. Find the envelope of circles (c') resulting by inverting (c) with respect to member circles d of the family (II) orthogonal to (I).

The solution is found by determining the two circle-members c_{3}, c_{4} of family (I) that are tangent to c (see TangentMember.html ). Since inversion with respect to (d) leaves all circles of family (I) invariant c' will contact both c_{3} and c_{4}.
Notice that the centers C' of circles c' describe an ellipse (f) with foci at C_{3}, C_{4} (centers of c_{3}, c_{4}) and major axis 2*a = r_{3}+r_{4}, r_{3}, r_{4} being the raddii of circle c_{3}, c_{4} correspondingly (supply trivial proof).