Begin with the original diagram and add a few auxiliary lines. Angle ADG is a right angle since it is an inscribed angle subtending a diameter. Now GDA and DJA are similar right triangles. By our construction, |AG| = 6 and |AD| = 1, and by similar triangles (or trigonometry) |AG| / |AD| = |AD| / |AJ|. Thus, |AJ| = 1 / |AG| = 1/6. Since DJ is a perpendicular bisector of the chord AF, we have |AF| = 2 |AJ| and so |AF| = 1/3.
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Pages revised 26 Jan 2006