Benedetto Scimemi, Semi-similar complete quadrangles,
Forum Geometricorum, 14 (2014) 87--106.
Abstract. Let A = A_1A_2A_3A_4 and B =B_1B_2B_3B_4 be complete quadrangles and assume that each side A_iA_j is parallel to B_hB_k (i,j,h,k is a permutation of 1,2,3,4 ). Then A and B, in general, are not homothetic; they are linked by another strong geometric relation, which we study in this paper. Our main result states that, modulo similarities, the mapping A_i --> B_i is induced by an involutory affinity (an oblique reflection). A and B may have quite different aspects, but they share a great number of geometric features and turn out to be similar when A belongs to the most popular families of quadrangles: cyclic and trapezoids.
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