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'Gold Yet Again' printed from https://nrich.maths.org/
Nick Lord chose this problem for the 10th Anniversary Edition of
the NRICH website and had this to say about it:
"The equilateral case is genuinely surprising; the square case
echoes back through the centuries to Euclid's original construction
of the golden section, as does the familiar 'diagonal:side' ratio
in the pentagon. I say 'familiar' in that annoying way that
teachers often do, forgetting that everyone has to encounter
beautiful results like this for the first time and that they will
struggle to find them in today's school textbooks. It is fantastic
to have such gems collated in such a convenient form in one place -
gems which I can customise for my own teaching purposes. For
example, adjacent to 'Gold Yet Again' on the menu is
'Pentabuild' so, at the click of a button, I can animate the
construction of a regular pentagon and we can hunt for the
tell-tale evidence of the golden ratio which we now know is needed
to make it work. By the end of all this, my students' mathematics
has been enriched, my mathematics has been enriched and I log-off
smiling that I have struck 'Gold Again' on the NRICH website!
"
This problem gives three examples of the occurrence of the golden
ratio.
For the pentagon, the hint suggests you use the similar triangles
which have angles $72^\circ, 72^\circ$ and $36^\circ$, one with
sides $x$, $x$ and $1$ and one with sides $1$, $1$ and $x-1$.
Equating the ratios of corresponding sides of these triangles gives
a quadratic equation.
Another method of proving the same result is given in
Pent