The points `A(a,b+c),B(b,c+a),C(c,a+b)` are

Options

A. collinear
B. non collinear
C. one is mid point of other two
D. none

Answer

To determine whether the points \( A(a, b+c) \), \( B(b, c+a) \), and \( C(c, a+b) \) are collinear, we can use the area of the triangle formed by these points. If the area is zero, then the points are collinear.

### Step-by-Step Solution:

1. **Identify the Coordinates**:
- Point \( A \) has coordinates \( (a, b+c) \)
- Point \( B \) has coordinates \( (b, c+a) \)
- Point \( C \) has coordinates \( (c, a+b) \)

2. **Use the Area Formula**:
The area \( A \) of the triangle formed by points \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \) is given by the formula:
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

3. **Substitute the Coordinates**:
Substituting the coordinates of points \( A \), \( B \), and \( C \) into the area formula:
\[ \text{Area} = \frac{1}{2} \left| a((c+a) - (a+b)) + b((a+b) - (b+c)) + c((b+c) - (c+a)) \right| \]

4. **Simplify the Expression**:
- For the first term:
\[ a((c+a) - (a+b)) = a(c - b) \]
- For the second term:
\[ b((a+b) - (b+c)) = b(a - c) \]
- For the third term:
\[ c((b+c) - (c+a)) = c(b - a) \]
- Thus, the area becomes:
\[ \text{Area} = \frac{1}{2} \left| a(c-b) + b(a-c) + c(b-a) \right| \]

5. **Factor the Expression**:
Rearranging the terms:
\[ \text{Area} = \frac{1}{2} \left| ac - ab + ba - bc + cb - ca \right| \]
This simplifies to:
\[ \text{Area} = \frac{1}{2} \left| 0 \right| = 0 \]

6. **Conclusion**:
Since the area of the triangle formed by points \( A \), \( B \), and \( C \) is zero, the points are collinear.

### Final Answer:
The points \( A(a, b+c) \), \( B(b, c+a) \), and \( C(c, a+b) \) are collinear.