# Where's The Centre?

Geometry Level 5

$$ABC$$ is an equilateral triangle with side length $$6$$

Point $$E$$ is on the external angle bisector from $$B$$ such that $$BE=9$$.

The perpendicular dropped from $$E$$ to $$BC$$ intersects $$BC$$ at $$G$$ and $$AC$$ at $$F$$.

Point $$I$$ is on line $$BF$$ extended through $$F$$ such that $$\angle FIG=30^{\circ}$$.

The circumcenter of $$\triangle BIG$$ is $$O$$

Find $$8 \times AO$$.

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