Oliver from Madras College sent a good
solution to this question. From the fact that the angles of a
regular pentagon are $108^{\circ}$ all the angles marked in the
diagram can be found and we leave the reader to prove these
results.
Triangles $CDF$ and $EDB$ are congruent because they have the
same angles and $CD$ and $ED$ are equal in length.
If the sides of the regular pentagon are 1 unit and $FD = x$
units then, because $CDF$ and $EDB$ are congruent isosceles
triangles it follows that $BE = x$ units.
Triangles $BEF$ and $EDB$ are similar, hence $$\frac{x}{1} =
\frac{x + 1}{x}$$ Simplifying this expression gives the equation:
$x^2  x  1 = 0$.
Solving this equation (and taking the positive root as $x$ is
positive)
$x = (1 + \sqrt{5})/2$ which is equal to the golden
ratio.
Interestingly this gives us an exact value for $\cos
36^{\circ}$ because $2 \cos 36^{\circ} = x$, so $\cos 36^{\circ} =
(1 + \sqrt{5})/4$.
