A problem by Sagnik Saha

Level pending

\(\triangle ABC\) is an acute-angled triangle with \(AB=13\) , \(BC=14\) and \(AC=15\). The incircle \(\Gamma\) of \(ABC\) with centre \(I\) touches \(BC\) at \(D\) . Let \(AI\) produced meet \(BC\) at \(E\) . Let \(K\) be the mid-point of \(BC\). \(KI\) meets \(AD\) at \(L\). Find the numerical value of

\[ \dfrac{[R_1] \times [R_2] \times [R_3]}{[ABC]-\frac{4}{3}[LDK]}. \]

Details and assumptions

\(R_1\) is the circumradius of \(\triangle ABC\), \(R_2\) is the circumradius of \(\triangle ALI\) , \(R_3\) is the circumradius of \(\triangle IDE\)

\([PQRS]\) denotes the area of the figure \(PQRS\).

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