ATRACTOR

In the previous argument we made use of several properties of plane triangles that are not valid in non-Euclidean geometries. It is true that with the Euclidean metric, in spaces of constant non-zero curvature \([2]\) it is also true that a triangle is equilateral if and only if it is an equiangle triangle (the proof of this fact is similar to the one that Euclides presented in the planar case because it does not use the parallels postulate). Note, however, that in a sphere there are not similar triangles that are not congruent; and therefore we have to stop right away at the first step of the previous proof because it does not work in the world of positive constant curvature. Furthermore, the sum of the angles of spherical triangles is not constant.

In spite of all this, the characterization of the equilateral triangles we tested in the plane could be valid in a sphere, needing only another proof. But that is not the case. Let us note that in the sphere of radius \(2/\pi\), it is possible to draw a triangle (pythagorean but not equilateral) of sides \(1,\,1,\,2\) and angles \(\pi/2,\,\pi/2,\,\pi\) and another triangle with the same sides and angles \(3\pi/2\), \(3\pi/2\), \(\pi\) (see the next figure).

And, in the same sphere, there are also: an equilateral triangle whose sides are on the equator, dividing the equator in three equal parts of length \(\frac{4}{3}\), which has the three angles equal to \(\pi\); an equilateral pythagorean triangle of side \(1\) and with the three angles equal to \(\pi/2\).

What will be the spherical version of the attribute that we find in the plane equilateral triangles? To explore this question, we suggest to use the interactive module, which enables to draw spherical triangles and test some of their properties.