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Martin Josefsson, Five proofs of an area characterization of rectangles,
Forum Geometricorum, 13 (2013) 17--21.
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Abstract. There are a handful of well known characterizations of rectangles, most of which concerns one or all four of the angles of the quadrilateral. One example is that a parallelogram is a rectangle if and only if it has (at least) one right angle. Here we shall prove that \emph{a convex quadrilateral with consecutive sides a, b, c, d is a rectangle if and only if its area K satisfies K = (1/2)Sqrt((a^2+c^2)(b^2+d^2)). We give five different proofs of this area characterization.

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