InversionInterchanging

Inversion interchanging two circles

Given two circles (a), (b) (one of them can be a line) with different radii, there is always a circle (i) inverting the one to the other. If both (a), (b) are circles then the center S of (i) is on a similarity center of the two circles. Circle (i) is orthogonal to every circle simultaneously externally/internally and tangent to (a), (b).

The proofs are easy applications of properties of the inversion transformation. EucliDraw has a ready to use tool constructing the circle (i), for given (a) and (b). [Circle-Tools\Inversion Interchanging 2 cir/lin _ _ ]. Besides (i) this tool constructs also another circle (i*) definining an anti-inversion with the same property: to interchange the circles (a) and (b). These circles are known under various names: Potenzkreis (Steiner), Mid-Circle (Coxeter), Circle of Antisimilitude (Johnson). More details can be found in the files starting with MidCircles.html and the references given there.

http://users.math.uoc.gr/~pamfilos/