Classical means

Use the diagram to investigate the classical Pythagorean means.
Age
16 to 18
Challenge level
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Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving
Being curious Being resourceful Being resilient Being collaborative

Problem



For any two numbers $a$ and $b$ three classical Pythagorean means are defined, the arithmetic mean $A$, the geometric mean $G$ and the harmonic mean $H$ such that:

$$\eqalign{

A &= \frac{1}{2}(a+b) \cr

G &= \sqrt {ab}\cr

H &= \frac{2}{\frac{1}{a}+\frac{1}{b}}.}$$

(i) Prove that $H=\frac{G^2}{A}$.

Classical Means


(ii) In this diagram the semicircle has diameter $a+b$. Prove that the lengths $A$, $G$ and $H$ (shown in blue, red and green) are equal to the three means and deduce from the diagram the inequality
$$A> G> H.$$


(iii) Prove from the diagram that the length $Q$ is equal to the quadratic mean (or root mean square) such that $$Q=\sqrt{\frac {a^2+b^2}{2}}.$$