Unavoidable golden ratio

It is demonstrated that iterative repeating of some simple geometric construction leads unavoidably in the limit to the golden ratio. The procedure appears to be quickly convergent regardless of a ratio of initial elements sizes. This could explain a widespread occurrence of the golden ratio in various constructions, including puzzling proportions encountered in structures of ancient cultures (e.g., in the Great Pyramid of Giza), as well as in the natural growth and self-organization processes in organic and inorganic matter.