ATRACTOR

At this point, it is appropriate to sumarize what we already know about \(\mathcal{T}\):

1. It is similar to a triangle with rational sides (a consequence of \(Q_1\)).
2. Its angles are rational multiples of \(\pi\) (a consequence of \(Q_2\)).

We can obtain furher information about such a triangle if we combine these two properties \(Q_1\) and \(Q_2\). That can be done using the Cosines law (which generalizes the Pythagorean Theorem) that states that \[ \begin{eqnarray*} \cos(\angle A) &=& \frac{b^2+c^2-a^2}{2bc}\\ \cos(\angle B) &=& \frac{a^2+c^2-b^2}{2ac}\\ \cos(\angle C) &=& \frac{a^2+b^2-c^2}{2ab}. \end{eqnarray*}\] Then \(\cos(\angle A),\) \(\cos(\angle B)\) and \(\cos(\angle C)\) are rational numbers because the sides \(a,\) \(b,\) \(c\) are rational. We notice now that all angles \(\theta \in\,[0,\pi]\) that are rational multiples of \(\pi\) and which have a rational cosine are known: \(\theta\) is equal to \(0\), \(\pi\), \(\pi/2\), \(\pi/3\) or \(2\pi/3\), with cosine equal to \(\pm 1\), \(0\) and \(\pm \frac{1}{2}\), respectively. We will prove this result later on, for now we only point out that with it we can characterize triangle \(\mathcal{T}\): \(\mathcal{T}\) is similar to a triangle with rational sides such that

• Their angles are in \(]0,\pi[\).
• They cannot be bigger or equal than \(\pi/2\), because otherwise one of the other angles would be at most \(\pi/4\) and that is not one of the options.

We conclude that the three angles of triangle \(\mathcal{T}\) are equal to \(\pi/3\), which means that \(\mathcal{T}\) is equilateral.