Four of their proofs work for non-isosceles right triangles, while the fifth proof works for all right triangles. The first ...

Angle OGH (\(y\)) = angle OHG because triangle GOH is also isosceles. Lengths OH and OG are also both radii. Angle GOH = \(180^\circ - 2y\) (because angles in a triangle add up to 180°).