InvertingToEqual

Inverting to equal 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. For every point D on circle (i) and every circle (c) centered at D, the inversion with respect to (c) sends the two circles (a), (b) to two circles (a*) , (b*) having equal radii.

For the existence and some properties of circle (i) see the file InversionInterchanging.html .
The rest is a simple consequence of the c-inversion transformation of the two bundles I(a,b) and its orthogonal II(e,d), where d is the circle orthogonal to both a, b and passing through D.
1) The bundle (I) of (a, b) contains (i) and transforms to the bundle I*(a*,b*) containing line i* as its common radical axis.
2) The bundle II of (d, e) transforms to II*(d*, e*), line d* containing all centers of circles of the bundle I*.
3) Since I, II are orthogonal to each other, I* and II* are too.

Consider the inverses a*, b* of a, b with respect to the inversion on c. By the same inversion (i) maps to line i* and by the preservetion of the circle interchanging two circles by inversion (see the file MidCircleInverted.html ), the inversion on i*, wich is a reflection maps a* to b*. Hence the two are always congruent.

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