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Consider the rhombus as illustrated where $x$ is an unknown
length, $AP = AD = x$ , angle $DAP = 36$ degrees and $P$ is a point
on the diagonal $AC$ such that $PB = 1$ unit.
Without using a calculator, computer or tables find the exact
values of
1. $\cos36^{\circ}\cos72^{\circ}$
2. $\cos36^{\circ}  \cos72^{\circ}.$

3. Draw these two diagrams as accurately as you can and measure the lengths $a$ and $b$. What do you notice? Can you prove it? (In each diagram there are two right angled triangles). 
Show that the arithmetic mean, geometric mean and harmonic mean of a and b can be the lengths of the sides of a rightangles 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.