If `|vec (AO) +vec (OB)| =|vec(BO) + vec(OC)|` , then `A, B, C` form

Options

A. equilateral triangle
B. collinear
C. non-collinear
D. none of these

Explanation

To solve the problem, we start with the given equation:

\[ |\vec{AO} + \vec{OB}| = |\vec{BO} + \vec{OC}| \]

### Step 1: Rewrite the vectors
We can express the vectors in terms of the position vectors of points A, B, C, and O. Let:
- \(\vec{A}\) be the position vector of point A,
- \(\vec{B}\) be the position vector of point B,
- \(\vec{C}\) be the position vector of point C,
- \(\vec{O}\) be the position vector of point O.

Then we can rewrite the vectors as:
- \(\vec{AO} = \vec{O} - \vec{A}\)
- \(\vec{OB} = \vec{B} - \vec{O}\)
- \(\vec{BO} = \vec{O} - \vec{B}\)
- \(\vec{OC} = \vec{C} - \vec{O}\)

### Step 2: Substitute into the equation
Now substituting these into the equation gives us:

\[ |(\vec{O} - \vec{A}) + (\vec{B} - \vec{O})| = |(\vec{O} - \vec{B}) + (\vec{C} - \vec{O})| \]

This simplifies to:

\[ |-\vec{A} + \vec{B}| = |-\vec{B} + \vec{C}| \]

Which can be rewritten as:

\[ |\vec{B} - \vec{A}| = |\vec{C} - \vec{B}| \]

### Step 3: Interpret the equation
The equation \( |\vec{B} - \vec{A}| = |\vec{C} - \vec{B}| \) implies that the distance between points A and B is equal to the distance between points B and C.

### Step 4: Conclusion about the points
Since the distances are equal, points A, B, and C must be positioned such that they are collinear. This means they lie on a straight line.

### Final Answer
Thus, the points A, B, and C form a collinear arrangement.