ATRACTOR

If a triangle is right angled and has integer sides, like the one we just built, then it is called a pythagorean triangle. The next figure shows one: note that, besides the sides being integers, the three angles have rational cosines.

But there are other possibilities. See, for instance, the scalene triangle with sides of lengths \(5,\) \(6\) and \(7\): in this case, the angles have rational cosines since they are \(\arccos(1/5)\), \(\arccos(5/7)\) and \(\arccos(19/35)\).

Now notice the triangle in the next figure: the ratios between the amplitudes of the angles are rational, but neither all ratios of the sides nor all cosines of the angles are.

If, however, we join conditions

then we are reduced to equilateral triangles. The reader will find in the forthcoming sections a proof of this statement.