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 $\sqrt{\mathrm{ab}}$; the harmonic mean (HM) is

${\displaystyle \frac{1}{\frac{1}{2}(\frac{1}{a}+\frac{1}{b})};}$

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

${\displaystyle a=b{\phi }^3,}$

where $\phi =\frac{1}{2}(1+\sqrt{5})$, the Golden Ratio. [As a calculator can only give approximate answers, you cannot use a calculator for this question.]