Triplication #2

Let $ABC$ be a non-obtuse triangle such that $\overline{AB}$ $>$ $\overline{AC}$ and $\widehat{B}$ $=$ $45^{\circ}$ . Let $O$ and $I$ denote the circumcenter and incenter of triangle $ABC$ , respectively. Suppose that $\sqrt{2}\cdot \overline{OI}= \overline{AB}-\overline{AC}$ . If the product of all the possible values of $\sin{A}$ can be written as $\sqrt{\frac{1}{\sqrt{F}}-\frac{1}{G}},$ where $F,G$ are all integers. Find $\large \frac{G}{F}$ .