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This makes use of the following lemma:
We have a circle with diameter and two chords, GC and DH, related by a fixed point F on the diameter (namely G,F,H are collinear). We need to show that tan(α) is a fixed constant times tan(γ). But, the perpendicular dropped from F to the bottom chord DH splits the chord in the same proportion that F splits the diameter DC, because that dropped perpendicular line is parallel to the chord HC (both being perpendicular to DH). Thus, tan(α)*CF = tan(γ)*DF, so the ratio tan(α)/tan(γ) = DF/CF, which is fixed because F is fixed.