Perfect Proportion

Since

$${2a\over a+b}={2ab\over (a+b)b},$$ (1)

it follows that

$${a\over {a+b\over 2}} = {{2ab\over a+b}\over b},$$ (2)

so

$${a\over A} = {H\over b},$$ (3)

where $A$ and $H$ are the Arithmetic Mean and Harmonic Mean of $a$ and $b$. This relationship was purportedly discovered by Pythagoras.

See also Arithmetic Mean, Harmonic Mean