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Problem 295: Archimedean Twin Circles, Arbelos, Semicircles, Harmonic Mean, Radii, Perpendicular. Level: High School, College, SAT Prep.

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 and H inscribed in ACF and CBF, respectively, are equal to one-half the harmonic mean of a and b. The figure included between the circumferences of the three semicircles is "what Archimedes called arbelos."

Archimedes Twins, Harmonic Mean

Geometry problem solving is one of the most challenging skills for students to learn. When a problem requires auxiliary construction, the difficulty of the problem increases drastically, perhaps because deciding which construction to make is an ill-structured problem. By “construction,” we mean adding geometric figures (points, lines, planes) to a problem figure that wasn’t mentioned as "given."

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