JOMO 7, Short 9

Geometry Level 4

Let \(I\) be the incenter of \(\triangle ABC\), and let \(AI,BI\), and \(CI\) intersect \(BC\), \(CA\), and \(AB\) at \(D\), \(E\), and \(F\), respectively. If \(AB=20, BC=14\) and \(AC = \frac{438}{53}\), then \(\dfrac{ID}{IF} =\dfrac{a}{b}\) for positive, co-prime integers \(a\) and \(b\) Find the value of \(a+b\).

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