(Logic behind) Geometric construction for imaginary roots of quadratic.

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In the question on this site Geometric derivation of the quadratic equation

there are geometric constructions for real roots of a quadratic. As a follow-up I found the construction below for imaginary roots, which I understand from the paper referenced within the diagram, has some connection to inversion in the Euclidean plane.

While I understand how to work through the construction and demonstrate its validity, I would appreciate anyone's insights into how this connects with inversive geometry (or, if that is a mistaken reference, what the intuitive raison d'etre for this might be!).

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