Woah what a ratio!!

Level pending

\(\triangle ABC\) is an acute-angled triangle with \(AB=13\) , \(BC=14\) and \(AC=15\). the circle \(\Gamma\) with centre \(I\) touches \(BC\) at \(D\) . Let \(AI\) produced meet \(BC\) at \(E\) . Find the numerical value of \( \dfrac{[ABC]}{[ AID]-[IDE]}\)

\([PQRS]\) means area of the figure \(PQRS\)

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