The complex numbers ` z_(1) , z_(2) , z_(3)` are the vertices of a triangle . Then the complex numbers z which make the triangle into a parallelogram is

Options

A. `z_(1) + z_(2) - z_(3)`
B. ` z_(2) + z_(3) - z_(1)`
C. `z_(3) + z_(1) - z_(2)`
D. none

Answer

To solve the problem of finding the complex numbers \( z \) that can transform the triangle formed by the vertices \( z_1, z_2, z_3 \) into a parallelogram, we can follow these steps:

### Step 1: Understand the properties of a parallelogram
In a parallelogram, the diagonals bisect each other. This means that the midpoint of one diagonal is the same as the midpoint of the other diagonal.

### Step 2: Identify the midpoints
Let’s denote the vertices of the triangle as:
- \( A = z_1 \)
- \( B = z_2 \)
- \( C = z_3 \)

We want to find a point \( D \) such that the quadrilateral \( ABCD \) forms a parallelogram. The point \( D \) can be represented by the complex number \( z \).

### Step 3: Set up the midpoint equations
The midpoint \( E \) of diagonal \( AC \) is given by:
\[ E = \frac{z_1 + z_3}{2} \]
The midpoint \( F \) of diagonal \( BD \) is given by:
\[ F = \frac{z_2 + z}{2} \]

### Step 4: Set the midpoints equal
Since \( E \) and \( F \) must be equal for \( ABCD \) to be a parallelogram, we can set the two expressions for the midpoints equal to each other:
\[ \frac{z_1 + z_3}{2} = \frac{z_2 + z}{2} \]

### Step 5: Clear the fractions
To eliminate the fractions, multiply both sides by 2:
\[ z_1 + z_3 = z_2 + z \]

### Step 6: Solve for \( z \)
Rearranging the equation to solve for \( z \):
\[ z = z_1 + z_3 - z_2 \]

### Step 7: Consider other configurations
The above calculation gives us one configuration. However, we can also consider the other vertices being opposite to \( z \) in the parallelogram. Thus, we can derive two more expressions for \( z \):
1. If \( z_1 \) is opposite \( z \):
\[ z = z_2 + z_3 - z_1 \]
2. If \( z_2 \) is opposite \( z \):
\[ z = z_1 + z_3 - z_2 \]

### Final Result
Thus, the complex numbers \( z \) that can transform the triangle into a parallelogram are:
1. \( z = z_1 + z_3 - z_2 \)
2. \( z = z_2 + z_3 - z_1 \)
3. \( z = z_1 + z_2 - z_3 \)