Thierry Gensane and Philippe Ryckelynck, On the Maximal Inflation of Two Squares,
Forum Geometricorum, 5 (2005) 23--31.

Abstract:  We consider two non-overlapping congruent squares q_1, q_2 and the homothetic congruent squares q_1^k, q_2^k  obtained from two similitudes centered at the centers of the squares. We study the supremum of the ratios of these similitudes for which  q_1^k, q_2^k are non-overlapping. This yields a function \psi =\psi (q_1,q_2) for which the squares q_1^\psi, q_2^\psi} are non-overlapping although their boundaries intersect. When the squares q_1 and q_2 are not parallel, we give a 8-step construction using straight edge and compass of the intersection q_1^\psi  \cap q_2^\psi and we obtain two formulas for \psi. We also give an angular characterization of a vertex which belongs to q_1^\psi  \cap q_2^\psi.

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