### Gold Again

Without using a calculator, computer or tables find the exact values of cos36cos72 and also cos36 - cos72.

### Golden Triangle

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.

### Golden Powers

You add 1 to the golden ratio to get its square. How do you find higher powers?

# Pythagorean Golden Means

##### Age 16 to 18 Challenge Level:

This question involves the sides of a right-angled triangle, the Golden Ratio, and the arithmetic, geometric and harmonic means of two numbers. Take any two numbers $a$ and $b$, where $0 < b < a$.

The arithmetic mean (AM) is $(a+b)/2$;

the geometric mean (GM) is $\sqrt{ab}$;

the harmonic mean (HM) is $${1 \over {{1 \over 2}\left( {1 \over a} + {1\over b } \right)}};$$

and the arithmetic mean is always the largest.

Show that the AM, GM and HM of $a$ and $b$ can be the lengths of the sides of a right-angled triangle if and only if $$a = b\varphi^3,$$ where $\varphi = {1\over 2}(1+\sqrt{5})$, the Golden Ratio.