Comparing two geometric problems Resize


Reduction to the case of concentric circles

It would be convenient to find a geometric transformation that sends circles into circles and preserves the tangency property (it sends rings of tangent circles into rings of tangent circles) but does not preserve (necessarily) the property "to be concentric". Well, such a transformation exists: an inversion4 whose center does not lie in any of the circles.

Figure 11

Figure 12

Figure 13

Figure 14

Click on the images to see the animation

As an example, the rings of concentric circles represented in figures 11-12 and 13-14 are sent by inversions in rings of non-concentric circles represented in the corresponding figures. In one case the rings close; in the other they do not. But what is unexpected is that, in both examples of rings of non-concentric circles, as in those of concentric circles, whether the rings closes or not does not depend on the position of the first circle in the ring. We can add: that behaviour - whether the ring closes or not does not depend on the first chosen circle - is not only that of concentric circles but also that of non-concentric circles coming from an inversion of a pair of concentric circles! The interesting question we now ask is: given any pair of disjoint circles, can it be obtained as an inversion of a pair of concentric circles? If the answer is positive, we will have concluded that that behaviour, for any pair of disjoint circles, is the same for all pairs of concentric circles: there will be pairs for which no ring closes (the general situation) and exceptional pairs5 for which all rings close6! At this point, we suggest that the reader with some experience working with inversions try directly to prove that any pair of disjoint circles comes from a pair of concentric inversions. This is the same as proving that for any pair of disjoint circles, there is an inversion that sends it into a pair of concentric circles. Should the reader not follow this suggestion he/she can find below a non-constructive proof (illustrated in figure 15) of such an inversion.

Using this result, we conclude that the answer for any pair of circles is identical to the one already described for concentric circles. All rings close with the same number of circles or none closes.

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4Recall that, in the plane, an inversion with respect to a circle with center \(C\) and radius \(r\) sends any point \(P\) different from \(C\) into a point \(P'\) such that \(\overrightarrow{OP}=k\cdot\overrightarrow{OP'}\), where \(k = \frac{1}{r^{2}}\). And analogously, in the three dimensional space, with respect to a sphere with center \(C\) and radius \(r\).
5Every such a pair has in its neighborhood in which the ring does not close for any other pair.
6It is to a problem having this kind of answer that we usually call "porism".

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