id: 06381163
dt: j
an: 2015d.00645
au: Conway, John
ti: A characterization of the equilateral triangles and some consequences.
so: Math. Intell. 36, No. 2, 1-2 (2014).
py: 2014
pu: Springer US, New York, NY
la: EN
cc: G60
ut: equilateral triangles
ci:
li: doi:10.1007/s00283-014-9447-3
ab: 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 $45^\circ$.
rv: Wolfgang Globke (Adelaide)