The points `z_1, z_2, z_3, z_4` in the complex plane form the vertices of a parallelogram iff :

Options

A. `z_1 + z_2 = z_3 + z_4`
B. `z_1 + z_3 = z_2 + z_4`
C. `z_1 + z_3 gt z_2 + z_4`
D. None of these

Answer

To determine the condition under which the points \( z_1, z_2, z_3, z_4 \) in the complex plane form the vertices of a parallelogram, we can follow these steps:

### Step 1: Understand the properties of a parallelogram
A parallelogram has two important properties:
1. Opposite sides are equal in length.
2. The diagonals bisect each other.

### Step 2: Represent the points in the complex plane
Let the points be represented as:
- \( z_1 \) for vertex A
- \( z_2 \) for vertex B
- \( z_3 \) for vertex C
- \( z_4 \) for vertex D

### Step 3: Identify the diagonals
In a parallelogram, the diagonals are:
- Diagonal AC: connects \( z_1 \) and \( z_3 \)
- Diagonal BD: connects \( z_2 \) and \( z_4 \)

### Step 4: Calculate the midpoints of the diagonals
The midpoint of diagonal AC is given by:
\[ \text{Midpoint of AC} = \frac{z_1 + z_3}{2} \]
The midpoint of diagonal BD is given by:
\[ \text{Midpoint of BD} = \frac{z_2 + z_4}{2} \]

### Step 5: Set the midpoints equal to each other
For the points to form a parallelogram, the midpoints of the diagonals must be equal:
\[ \frac{z_1 + z_3}{2} = \frac{z_2 + z_4}{2} \]

### Step 6: Eliminate the fraction
To eliminate the fraction, we can multiply both sides by 2:
\[ z_1 + z_3 = z_2 + z_4 \]

### Conclusion
Thus, the points \( z_1, z_2, z_3, z_4 \) form the vertices of a parallelogram if and only if:
\[ z_1 + z_3 = z_2 + z_4 \]