# Use the right theorem

Geometry Level 4

In $$\Delta ABC$$ as shown above, straight lines are drawn from each of the vertices to their respective opposite sides and meet at point $$K$$. Furthermore, $$\overline{AB} = 5$$, and $$\overline{AC} = 12$$.

If $$\large \frac {AD}{DB} = 2\frac{AE}{EC}$$, the limiting length of $$\overline {BF}$$ can be expressed in the form $$\frac{a}{b}$$ where $$a$$ and $$b$$ are coprime positive integers. Determine $$a+b$$.

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