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your main statement in the section "Trisection" is:
"The proportions of the adjacent angles are constantly retained through shifting the vertex along the axis of symmetry."
But this statement in general is not true. For example let the desired point D be the intersection of segments SR and PK. Then angle(RDP) = 90°, and according to your main statement angle(RDG) should be 3*angle(RDP), or 270°, and this is obviously not true.
As for section "pi = 3.1547..." see Kochanski's Approximation: