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'Golden Construction' printed from https://nrich.maths.org/
Thank you to Shaun from Nottingham High School and Andrei from
Tudor Vianu National College, Bucharest, Romania for these
solutions.
(1) Drawing the figure, I observe that ratios $AE/AD$ and $BC/BE$
are approximately equal, having a value of 1.6.
(2) From Pythagoras' Theorem I calculate $MC$ (in the right-angled
triangle $MBC$): $$\eqalign{ MC^2 &= BC^2 + MB^2 = 1 + 1/4 \cr
MC &= \sqrt5 /2 }.$$ So $AE=(\sqrt 5 + 1)/2$ and $BE=(\sqrt 5 -
1)/2$. The ratios are: $${AE\over AD}= {\sqrt 5 + 1\over 2}$$ and
$${BC\over BE}= {1\over (\sqrt 5 - 1)/2} = {\sqrt 5 + 1\over 2}.$$
So, $AE/AD = BC/BE.$
(3) From this equality of ratios, I find out that $$BE =
{AD.BC\over AE} = {1\over \phi}$$ But $AE = AB + BE$ so $$\phi = 1
+ {1\over \phi}.$$
(4) Substituting $\phi = 1$ the left hand side of this expression
is less than the right hand side. If we increase the value given to
$\phi$ the left hand side increases and the right hand side
decreases continuously. Substituting $\phi = 2$ the left hand side
is greater than the right hand side so the value of $\phi$ which
satisfies this equation must lie between $1$ and $2$.
The two solutions of the equation can be found at the intersection
of the cyan curve ($y=1 + 1/x$) and magenta curve ($y=x)$. Only the
positive value is considered and it is approximately 1.618.
(5) Now, I solve the equation. It is equivalent to $\phi^2 - \phi
-1 = 0$ so the solutions are $$\phi_1 = {1-\sqrt 5\over 2}$$ and
$$\phi_1 = {1+\sqrt 5 \over 2}.$$ Only the second solution is valid
because $\phi > 0$ .