# 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."

### Geometric Art using iPad or Mobile Apps

Geometric art is a form of art based on the use and application of geometric figures. A geometric figure is any set or combination of points, lines, surfaces and solids.

A mobile app or mobile application software is a computer program designed to run on smartphones and tablet computers.