Not For Those Who Are Slaves of Trigonometry

Geometry Level 4

In the diagram, triangle \(ABC\) with the incircle touches \(AB\) and \(BC\) at \(F\) and \(E\), respectively.

\(M\) and \(N\) are midpoints of \(AC\) and \(BC\), respectively.

\(EF\) cuts \(MN\) at point \(P\). \(AP\) cuts \(BC\) at point \(D\).

If \(AB=4, AC=5\) and \(BC=6\), length of \(AD\) can be written as \(\dfrac ab\), where \(a\) and \(b\) are coprime positive integers, find \(a+b\).


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