How to prove that this geometric constructions for the pythagorean means are right?

Question

According to wikipedia the Pythagorean means (and the quadratic one) of two numbers can be constructed geometrically in this way:

While the arithmetic mean it's obvious, and I think I understood the construction of GM, I cannot clearly see why this construction works for HM and QM, can someone give me an hint on how to prove the correctness of this construction?

Thanks in advance

Alessandro

Answer 1

By similarity of triangles $\frac{A}{G} = \frac{G}{H}$ so $H=\frac{G^2}{A}=\frac{2ab}{a+b}$ is the harmonic mean.

$$Q^2 = A^2+ \left(\frac{a-b}{2}\right)^2 = \frac{a^2+b^2}{2}$$

So:

$$Q=\sqrt{\frac{a^2+b^2}{2}}$$ is the quadratic mean (or root mean square.)