Extensions of a theorem of Van Aubel

Abstract

A theorem of Van Aubel, which first appeared in concerns the figure obtained by erecting squares on the sides of a quadrilateral, and considering various line segments such as the joins of the centres of opposite squares. In parts of the theorem are extended to the case of (i) four similar rhombi, and (ii) four similar rectangles erected on the sides of a quadrilateral. Further, these two results are described as in some sense ‘dual’ to each other. Here we show that they are in fact both special cases of a further extension of the original theorem (see Proposition 6, and the remarks that follow it), and we also find several extra squares in the original Van Aubel figure (Proposition 11).