Begin with the original diagram and add a few auxiliary lines. The angle CDG is a right angle since this angle subtends a diameter of the circle. Then CDG and CHD are similar right triangles. By our construction, |CD| = 2 and |CG| = 6, and by similar triangles (or trigonometry) |CH| / |CD| = |CD| / |CG| so |CH| = 2/3. Since DH is a perpendicular bisector of the chord CF, we have |CF| = 4/3. Since |CA| = 1, we have |AF| = |CF| - |CA| = 1/3.
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Pages revised 16 Sept 2009