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.
[As a calculator can only give approximate answers, you cannot use a calculator for this question.]
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.
[As a calculator can only give approximate answers, you cannot use a calculator for this question.]
Quadratic equations. Inequality/inequalities. Manipulating algebraic expressions/formulae. Dynamical systems. Number theory. Golden ratio. Mean. Mathematical reasoning & proof. Geometric mean. Harmonic mean.
Published May 2001.
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