Without using a calculator, computer or tables find the exact
values of cos36cos72 and also cos36 - cos72.
Show that the arithmetic mean, geometric mean and harmonic mean of
a and b can be the lengths of the sides of a right-angles triangle
if and only if a = bx^3, where x is the Golden Ratio.
Three triangles ABC, CBD and ABD (where D is a point on AC) are all
isosceles. Find all the angles. Prove that the ratio of AB to BC is
equal to the golden ratio.
One solution is $x = 2$.
Can you discover whether this is the only
solution and justify your claim?