ASPECT RATIO AND SLOPE OF ALGEBRAIC RECTANGLES INSCRIBED IN LINES OVER FIELDS
BRUCE OLBERDING AND ELAINE A. WALKER
Abstract. Let k be a field. By an algebraic rectangle in k2 we mean four points in k2 subject to certain conditions that in the case where k is the field of real numbers yield four vertices of a rectangle. We study algebraic rectangles inscribed in lines in k2 by parametrizing these rectangles in two ways, one involving slope and the other aspect ratio. This produces two paths, one that finds rectangles with specified slope and the other rectangles with specified aspect ratio. We describe the geometry of these paths and its dependence on the choice of four lines. (Read the pdf on the arXiv.)

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