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 72o, 72o and 36o, 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