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$a+b:a=a:b$
i.e. $ \frac{a+b}{a}=\frac{a}{b}=\Phi\ \quad $(phi)
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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.