ATRACTOR

We will see how the property that the three sides of a plane triangle are equal results from a unique and harmonious combination of sides and angles.

A triangle is said to be equilateral if all three ratios between the lengths of its sides are equal to \(1\). For which plane triangles are these quotients a rational number? Given a triangle in the plane with sides of lengths \(a,\) \(b,\) \(c\) such that there are natural numbers \(p_{1},\) \(p_{2},\) \(p_{3},\) \(q_{1},\) \(q_{2},\) \(q_{3},\) which satisfy the equalities \[\frac{a}{b}=\frac{p_1}{q_1}, \quad \frac{b}{c}=\frac{p_2}{q_2} \quad \text{ and } \quad \frac{a}{c}=\frac{p_3}{q_3}\] we can rescale the triangle and get a similar one but with rational sides. This is done by using the homothety \[H_1:(x,y) \in \mathbb{R}^2 \mapsto \left(\frac{x}{c}, \frac{y}{c}\right)\] which transforms the initial triangle in one with sides \[a^\prime =\frac{p_1\,p_2}{q_1\,q_2}, \quad b^\prime=\frac{p_2}{q_2}, \quad \text{ and } \quad c^\prime=1.\]
If we apply another homothety
\[H_2:(x,y) \in \mathbb{R}^2 \mapsto (q_1\,q_2\,x,\,\, q_1\,q_2\,\,y)\] we obtain a triangle with integer sides. This conclusion makes us suspect that without additional information on the angles of the triangle we can hardly get more information about it. Therefore we need some assumption about the angles.

(*) Difficulty level: University