`' 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

Explanation

To solve the problem, we need to analyze the expression \( a \vec{I A} + b \vec{I B} + c \vec{I C} \) where \( I \) is the incenter of triangle \( ABC \) and \( a, b, c \) are the lengths of the sides opposite to vertices \( A, B, C \) respectively.

### Step-by-Step Solution:

1. **Understanding the Incenter**:
The incenter \( I \) of triangle \( ABC \) is the point where the angle bisectors of the triangle intersect. It is also the center of the circle inscribed in the triangle.

2. **Positioning the Incenter**:
...