ATRACTOR

Obviously, an equilateral triangle satisfies the conditions \(Q_1\) and \(Q_2\), with the particularity that the six ratios in question are all equal to \(1\). Let us see that the reciprocal implication is also valid, that is:

Let \(\mathcal{T}\) be a triangle of sides \(a,\) \(b\) and \(c\) and angles \(\angle A,\) \(\angle B,\) and \(\angle C\), satisfying \(Q_1\) and \(Q_2\). Rescaling it, as explained before, we can assume that \(a,\) \(b,\) \(c\) are rational (or integers). Moreover, we can rewrite \[\angle A=\pi\,\alpha_A, \quad \angle B=\pi\,\alpha_B \quad \text{and} \quad \angle C=\pi\,\alpha_C\] for an appropriate choice of positive reals \(\alpha_A,\) \(\alpha_B,\) \(\alpha_C.\) By assumption \(Q_2\), there are rationals \(s_1,\) \(s_2,\) \(s_3\) such that \[\frac{\angle A}{\angle B}=s_1, \quad \frac{\angle B}{\angle C}=s_2 \quad \text{and} \quad \frac{\angle A}{\angle C}=s_3.\] Therefore \[\alpha_A=s_3 \, \alpha_C, \quad \alpha_B=s_2 \, \alpha_C\] and, as the sum of the angles of a plane triangle must be equal to \(\pi\), we have that \[s_3 \, \alpha_C + s_2 \, \alpha_C + \alpha_C = 1.\] Consequently,\[\alpha_C=\frac{1}{1+s_2+s_3} \in \mathbb{Q}.\] In summary: If the ratios of the amplitudes of the angles of a plane triangle are rational, then the angles are rational numbers multiple of \(\pi.\)