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' I ' is the incentre of triangle A B C ...

`' 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`

A

0

B

(a+b+c)BC

C

(a+b+c)AC

D

(a+b+c)AB

Text Solution

Verified by Experts

The correct Answer is:
A
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`.
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