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If the incircle of the Delta ABC touches...

If the incircle of the `Delta ABC` touches its sides at `L`, `M` and `N` as shown in the figure and if `x`, `y`, `z` be thecircumradii of the triangles `MIN`, `NIL` and `LIM` respectively, where `I` is the incentre, then the product `xyz` is equal to:
(A) `R r^2`         (B)` r R^2`
(C) `1/2R r^2 `         (D) `1/2r R^2`

Text Solution

Verified by Experts

In the given figure, ANIM is a cyclic quadrilateral

Also, AI is the diameter of circumcircle MNI
`:. AI = 2x`. Then
`cosec. (A)/(2) = (2x)/(r)`
`rArr x = (r)/(2sin.(A)/(2)), y = (r)/(2sin.(B)/(2)), z = (r)/(2sin.(C)/(2))`
`rArr xyz = (r_(3))/(8 sin.(A)/(2) sin.(B)/(2) sin.(C)/(2)) = (r^(3))/(2(r)/(R)) = (r^(2)R)/(2)`
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