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

__ Öab

; the harmonic mean (HM) is

1 1 2 æ ç è 1 a + 1 b ö ÷ ø ;

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-angles triangle if and only if

a = bj3,

where j = ¹/₂(1+Ö5) , the Golden Ratio. [As a calculator can only give approximate answers, you cannot use a calculator for this question.]