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.
$O$ is the centre of a circle with $A$ and $B$ two points NOT on a diameter. The tangents to $A$ and $B$ intersect at $C$. $CO$ cuts the circle at $D$ and a tangent through $D$ cuts $AC$ and $BC$ at $E$ and $F$.
What is the relationship between area of $ADBO$ and the areas of $ABO$ and $ACBO$?