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What is the relationship between the arithmetic, geometric and harmonic means of two numbers, the sides of a right angled triangle and the Golden Ratio?

### Classical Means

Use the diagram to investigate the classical Pythagorean means.

# Pythagorean Golden Means

##### Stage: 5 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.