Pencils of Orthogonal Circles

A fundamental figure in our study of orthogonal circles has been this figure:

There are many stories and starting points that go with the same figure. For example:

Euclidean story.  Start with two points A and B and construct the rest.

• Construct some circles through A and B such as d1 and d2 with center on the perpendicular bisector of AB.  Also, construct two Apollonian circles c1 and c2.  The Apollonian circles can be constructed handily using angle bisectors of a triangle. The circles c1 and c2 are orthogonal to each of d1 and d2.

Inversion story.  Start with the two circles c1 and c2 and construct the rest.

• Inversion question. Construct a point A which is the center of a circle that will invert the circles c1 and c2 to concentric circles. Construct A and also B by first constructing two circles d1 and d2 orthogonal to c1 and c2.  Then d1 and d2 will have to intersect in points A and B so that the inversion of A is B in both circle c1 and c2.  Then invert the figure in any circle m with center A. The circles d1 and d2 will invert to lines through B' and the circles c1 and c2 will invert to circles orthogonal to these intersecting lines d1' and d2'. Thus the two lines are diameters and the center of the circles c1' and c2' is B', the intersection of d1' and d2'.
  

DWEG story. Start with B = O and circle c1. Construct all the rest.

• In the DWEG interpretation, c1 should be a DWEG circle, but if we have chosen B as the point O, then the DWEG diameters of c1 are Euclidean circles through B that are orthogonal to c1 such as d1 and d2.  This means that A in the figure is the DWEG center of c1.  So to construct A we can either construct d1 and d2 as circles through B orthogonal to c1 and intersect them, or we can just invert B in c1 to get A and then construct any diameters we want as circles through A and B.
• Special case:  If the special point O is B again, a Euclidean line not through B is also a DWEG circle.  It will be the perpendicular bisector of AB, and A will be the DWEG center.  (Why?  Check what the DWEG diameters must be.)