# Find the Distance

Geometry Level 4

Let $$\Delta ABC$$ be the right triangle with side lengths $$AB = 15$$, $$BC = 8$$, and $$AC = 17$$. Let $$D$$ be the point on the segment $$\overline{AC}$$ between $$A$$ and $$C$$ whose distance to $$A$$ is 1, and let $$E$$ be the point on the segment $$\overline{AB}$$ between $$A$$ and $$B$$ whose distance to $$A$$ is 3. Let $$F$$ be the point of intersection of the lines $$\overleftrightarrow{DE}$$ and $$\overleftrightarrow{BC}$$. The distance $$BF$$ can be written as $$\frac{a}{b}$$, where $$a$$ and $$b$$ are coprime positive integers. What is the value of $$a+b$$?

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