Prove that (r+r1)tan((B-C)/2)+(r+r2)tan((C-A)/2)+(r+r3)tan((A-B)/2)=0

Explanation

`(r + r_(1)) tan ((B -C)/(2))`
`= [4R sin.(A)/(2) sin.(B)/(2) sin.(C)/(2) + 4R sin.(A)/(2) cos.(B)/(2) cos.(C)/(2)] xx tan ((B- C)/(2))`
`=4R sin.(A)/(2) cos ((B -C)/(2)) tan ((B -C)/(2))`
`=4R sin.(A)/(2) sin ((B-C)/(2))`
`= 2R (sin B - sin C)`...(i)
Similarly,
`(r + r_(2)) tan ((C -A)/(2)) = 2R (sin C - sin A)`...(ii)
`(r + r_(3)) tan ((A - B)/(2)) = 2R (sin A - sin B)`...(iii)
On adding Eqs. (i) (ii) and (iii), we get the result
`rArr (I_(1) I_(2))/(cos.(C)/(2)) = (4R sin.(A)/(2))/(sin.(A)/(2))`
`rArr I_(1) I_(2) = 4 R cos.(C)/(2)`
Similarly, `I_(2) I_(3) = 4 R cos.(A)/(2) and I_(1) I_(3) = 4R cos.(B)/(2)`