[alogo] 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).

[0_0] [0_1] [0_2] [0_3]
[1_0] [1_1] [1_2] [1_3]
[2_0] [2_1] [2_2] [2_3]
[3_0] [3_1] [3_2] [3_3]
[4_0] [4_1] [4_2] [4_3]


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.


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