# Triplication #2

Geometry Level 4

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 $$\large \frac{G}{F}$$.

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