A problem by Sagnik Saha

Level pending

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

[ R 1 ] × [ R 2 ] × [ R 3 ] [ A B C ] 4 3 [ L D K ] . \dfrac{[R_1] \times [R_2] \times [R_3]}{[ABC]-\frac{4}{3}[LDK]}.

Details and assumptions

R 1 R_1 is the circumradius of A B C \triangle ABC , R 2 R_2 is the circumradius of A L I \triangle ALI , R 3 R_3 is the circumradius of I D E \triangle IDE

[ P Q R S ] [PQRS] denotes the area of the figure P Q R S PQRS .


The answer is 5.

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