Show that the arithmetic mean, geometric mean and harmonic mean of
a and b can be the lengths of the sides of a right-angles triangle
if and only if a = bx^3, where x is the Golden Ratio.
What is the relationship between the arithmetic, geometric and
harmonic means of two numbers, the sides of a right angled triangle
and the Golden Ratio?
Use the diagram to investigate the classical Pythagorean means.
Area fomulas and a little trig will help.
Here is the diagram if you need some clues to get started: