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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