Problem E

A Square in a Triangle

In \(\triangle ABC\), there is a right angle at \(B\) and the length of \(BC\) is twice the length of \(AB\). In other words, \(BC=2AB\).

Square \(DEFB\) is drawn inside \(\triangle ABC\) so that vertex \(D\) is somewhere on \(AB\) between \(A\) and \(B\), vertex \(E\) is somewhere on \(AC\) between \(A\) and \(C\), vertex \(F\) is somewhere on \(BC\) between \(B\) and \(C\), and the final vertex is at \(B\).

Square \(DEFB\) is called an
*inscribed* square. Determine the ratio of the area of the
inscribed square \(DEFB\) to the area
of \(\triangle ABC\).