Martin(2), your mirror by midpoint method, with construction line and circle tangent command is very nice.
In your example, there are two circle solutions. There are other cases...




I've never studied inversion/reflection in a circle, nor radical axis, and haven't grasped it yet. Here are a couple of pdf's on the subject:

http://geometer.org/mathcircles/inversion.pdf

http://jwilson.coe.uga.edu/MATH7200/InversionCompanion/inversion/inversionSupplement.pdf

So what would a MoI command to invert some geometry, with some circle do?