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A characterization of the equilateral triangles and some consequences. (English)
Math. Intell. 36, No. 2, 1-2 (2014).
Equilateral triangles are characterised as having three sides of equal length. The author generalises this characterisation to what he calls Conway’s Little Theorem: Equilateral triangles are characterised by the assertion that each ratio of two sides and each ratio of two angles are rational. For the proof, the triangle $ABC$ is placed in the complex plane and scaled such that the side lengths $a$, $b$, $c$ are rational and the angles rational multiples of $π$. Expressing the complex number $C$ through $A$, $B$ and the angles and side lengths, Conway finds an equation of the form $c+aω^{kq}=bω^{kp}$, which holds for all $k$ prime to $n$ (modulo $n$) and where $ω$ is a primitive $n$th root of unity. This leads to $ϕ(n)$ triangles with the given lengths, angles and base $AB$. But there are only two such triangles, which then implies that the angles are positive multiples of $60^\circ$. Hence the triangle is equilateral. Conway also gives two consequences of his Little Theorem: Let a {\it rational angle} be an angle that is a rational multiple of $π$. Then, the only rational angle $θ$ in the open interval $(0,90^\circ)$ for which $\cosθ$ is rational is $θ=60^\circ$. The only rational angle $ϕ$ in the open interval $(0,90^\circ)$ for which $\sinϕ$ is rational is $ϕ=30^\circ$. The other consequence is, the only rational angles for which the square of any of the six standard trigonometric functions is rational (or $\infty$) are the multiples of $30^\circ$ and $45^\circ$.