Suppose that three points `z_(1), z_(2), z_(3)` are connected by the relation `a z_(1) + b z_(2) + c z_(3) = 0`, where a + b + c = 0, then the points are

Options

A. vertices of a right triangle
B. vertices of an isosceles triangle
C. vertices of an equilateral triangle
D. collinear

Answer

To solve the problem, we need to analyze the given relation involving the complex numbers \( z_1, z_2, z_3 \) and the coefficients \( a, b, c \).

### Step-by-Step Solution:

1. **Understanding the Relation**:
We are given the relation:
\[ a z_1 + b z_2 + c z_3 = 0 \]
where \( a + b + c = 0 \).

2. **Expressing the Complex Numbers**:
Let's express the complex numbers in terms of their real and imaginary parts:
\[ z_1 = x_1 + i y_1, \quad z_2 = x_2 + i y_2, \quad z_3 = x_3 + i y_3 \]
where \( x_i \) and \( y_i \) are the real and imaginary parts of \( z_i \) respectively.

3. **Substituting into the Relation**:
Substituting the expressions for \( z_1, z_2, z_3 \) into the relation gives:
\[ a(x_1 + i y_1) + b(x_2 + i y_2) + c(x_3 + i y_3) = 0 \]
This can be separated into real and imaginary parts:
\[ (a x_1 + b x_2 + c x_3) + i(a y_1 + b y_2 + c y_3) = 0 \]

4. **Setting Up the Equations**:
From the above equation, we can derive two equations:
- Real part: \( a x_1 + b x_2 + c x_3 = 0 \) (Equation 1)
- Imaginary part: \( a y_1 + b y_2 + c y_3 = 0 \) (Equation 2)

5. **Using the Condition \( a + b + c = 0 \)**:
Since \( a + b + c = 0 \), we can express \( c \) in terms of \( a \) and \( b \):
\[ c = - (a + b) \]
Substituting this into Equations 1 and 2 gives us two linear equations.

6. **Forming the Matrix**:
We can form the following matrix from the coefficients of \( x_1, x_2, x_3 \) and \( y_1, y_2, y_3 \):
\[ \begin{vmatrix} x_1 & x_2 & x_3 \\ y_1 & y_2 & y_3 \\ 1 & 1 & 1 \end{vmatrix} = 0 \]
This determinant being zero indicates that the points \( z_1, z_2, z_3 \) are collinear.

7. **Conclusion**:
Since the determinant is zero, it implies that the points \( z_1, z_2, z_3 \) must lie on the same straight line in the complex plane. Therefore, the final conclusion is:
\[ \text{The points } z_1, z_2, z_3 \text{ are collinear.} \]