Comparing two geometric problems (*) Resize


Introduction

We recall a famous geometry problem proposed by Jakob Steiner and compare it with another one that is usually called Hexlet. As we will see, the solution to both problems uses, in the proof, analogous tools and ideas. However, even after knowing the answer to the first problem the answer to the second one is, in general, a surprise.

The Steiner Porism has to do with the following: having fixed a pair of disjoint circles1 in the plane, is it always possible to build a finite sequence of \(n\geq3\) circles, all tangent to the two initial ones and such that each new one is tangent to the previous one and the last one not only tangent to the previous one but also the first one2?

Figure 1

Figure 2

Figure 3

The first figure shows a choice of the inicial pair of circles for which this is possible. From the second figure we cannot conclude anything right away: starting from the light blue circle in that figure, the ring does not close, but a priori we could believe that, for a different position of the first circle in the ring, the ring would close (see figure 3).

Figure 4

Figure 5

Note that both circles of the initial pair can be exterior – this is the case of figure 4 and 5: in the first one, those in the ring are also exterior, in the second one, one of those in the ring contains all the others. But there is a particular case, the one where the initial circles are concentric (see figures 6, 7, 8, 9), in which we can clearly conclude that if a ring closes (resp. does not close) any other ring also closes (resp. does not close).

Figure 6

Figure 7

Figure 8

Figure 9

1Represented in brown and light green in figures 1 to 3.
2In this text, we suppose that only the tagency points are in more than one circle of the sequence. Here you can have access to an applet where you can build sequences without this condition: they can close after more than one turn.
Translated for Atractor by a CMUC team, from its original version in Portuguese. Atractor is grateful for this cooperation.

This text is a slightly modified version of the following article (in Portuguese) published by Atractor in Gazeta de Matemática

(*) Difficulty level: University