`' 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. 0
B. (a+b+c)BC
C. (a+b+c)AC
D. (a+b+c)AB

Answer

A

Explanation

Let the incentre be at the origin and be
`A(p),B(q) and C(r)`. Then
`IA=p,IB=q and IC=r`
Incentre I is `(ap+bq+cr)/(a+b+c)`, where p=BC,q=AC and r=AB incentre is at the origin. Therefore,
`(ap+bq+cr)/(a+b+c)=0`,
or `ab+bq+cr=0`
`implies aIA+bIB+cIC=0`.