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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.