Arithmetic, geometric and harmonic means

Use the diagram to investigate the classical Pythagorean means.

Can you use the diagram to prove the AM-GM inequality?

If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.

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.

Given that u>0 and v>0 find the smallest possible value of 1/u + 1/v given that u + v = 5 by different methods.

A and B are two points on a circle centre O. Tangents at A and B cut at C. CO cuts the circle at D. What is the relationship between areas of ADBO, ABO and ACBO?