Consecutive Isosceles Decomposition

Some isosceles triangles can be decomposed into a family of isosceles triangles with bases on the legs of the base triangle. Determine for which apex angle such a decomposition is possible.

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Interestingly, there is a different kind of decomposition of an isosceles triangle into four isosceles triangles that holds for all apex angles that do not exceed 60°. The construction is the same for all such angles.

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