Why are the side lengths of the squares inscribed in a triangle so close to each other? (Q2871645)

Given a non-obtuse triangle, it is well known that it is possible to inscribe three different squares, each of them based on each of the sides of the triangle.NEWLINENEWLINEOne can empirically verify that the sides of these squares are very close to each other. This paper explains and quantifies this phenomenon.NEWLINENEWLINEIn particular, if \(x_a\) and \(x_b\) are the sides of the inscribed squares based on two of the sides of a given non-obtuse triangle, it is proved that NEWLINE\[NEWLINE1\geq \frac{x_a}{x_b}\geq\frac{2\sqrt{2}}{3}.NEWLINE\]NEWLINE Incidentally, the authors also prove that the smaller inscribed square is based on the longest side of the triangle.