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 $${1 \over {{1 \over 2}\left( {1 \over a} + {1\over b } \right)}};$$
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.