In `Delta ABC` with usual notation `r_1/(bc)+r_2/(ca)+r_3/(ab)` is

Options

A. `(1)/(2R) -(1)/(r )`
B. `2R-r`
C. `r-2R`
D. `(1)/(r )-(1)/(2R)`

Answer

4

Explanation

`(r_(1))/(bc)=(4R sin (A//2)cos(B//2)cos(c//2))/(2R sin B.2R sin c)`
`=(sin(A//2))/(4R(sin(B//2)sin(C//2)`
`=(sin^(2)(A//2))/(R )`
`"So that" (r_(1))/(bc)+(r_(2))/(ca)+(r_(3))/(ab)`
`=(1)/(r )[(sin^(2)(A//2)+sin^(2)(B//2)+sin^(2)(C//2)]`
`=(1)/(2r)[3-(1+4 sin(A//2)sin(B//2)sin(C//2))]`
`=(1)/(2r)[2-(r)/(R)]=1/r-1/R`