`' I '` is the incentre of triangle `A B C` whose corresponding sides are `a , b ,c ,` rspectively. `a vec I A+b vec I B+c vec I C` is always equal to a. ` vec0` b. `(a+b+c) vec B C` c. `( vec a+ vec b+ vec c) vec A C` d. `(a+b+c) vec A B`

Options

A. `vec0`
B. `(a+ b + c) vec(BC)`
C. `(veca+ vecb+vecc) vec(AC)`
D. `(a+b+c) vec(AB)`

Answer

A

Explanation

Let the incentre be at the origin and be
`A(vecp), B( vecq) and C(vecr) `. Then
`" "vec(IA) = vecp , vec(IB) = vecq and vec(IC) = vecr`.
Incentre I is `(a vecp + bvecq + cvecr)/( a+ b+c)` , where `p= BC, q =AC and r =AB`
Incentre is the origin. Therefore,
`( avecp + b vecq + c vecr)/( a + b+ c) = vec0, or a vecp + b vecq + c vecr= vec0`
`rArr a vec(IA) + b vec(IB) + c vec(IC) = vec0`