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Classical Means

Stage: 5 Short Challenge Level: Challenge Level:1

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 $H=\frac{G^2}{A}$.

means2.jpg
(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 $$Q=\sqrt{\frac {a^2+b^2}{2}}.$$