# Geometry Problem 295

## Archimedean Twin Circles, Arbelos, Semicircles, Harmonic Mean, Radii, Perpendicular

### Proposition

The figure shows the semicircles of diameters AB (center O), AC (center D, radius a), and BC (center E, radius b). Prove that radii of circles G (radius x) and H (radius y) inscribed in ACF and CBF, respectively, are equal to one-half the harmonic mean of a and b, that is: $$x=y = \dfrac{a \cdot b}{(a+b)}=\dfrac{1}{2} \cdot H(a,b)$$.

The figure included between the circumferences of the three semicircles is "what Archimedes called arbelos."