If `z_(1), z_(2), z_(3), z_(4)` are complex numbers, show that they are vertices of a parallelogram In the Argand diagram if and only if `z_(1) + z_(3)= z_(2) + z_(4)`

Answer

To show that the complex numbers \( z_1, z_2, z_3, z_4 \) are the vertices of a parallelogram in the Argand diagram if and only if \( z_1 + z_3 = z_2 + z_4 \), we can follow these steps:

### Step 1: Understanding the properties of a parallelogram
In a parallelogram, the diagonals bisect each other. This means that the midpoints of the diagonals are the same.

### Step 2: Define the midpoints of the diagonals
Let’s denote the vertices of the parallelogram as follows:
- Vertex A corresponds to \( z_1 \)
- Vertex B corresponds to \( z_2 \)
- Vertex C corresponds to \( z_3 \)
- Vertex D corresponds to \( z_4 \)

The diagonals are AC and BD. The midpoints of these diagonals can be calculated as follows:
- Midpoint of AC:
\[ M_{AC} = \frac{z_1 + z_3}{2} \]
- Midpoint of BD:
\[ M_{BD} = \frac{z_2 + z_4}{2} \]

### Step 3: Setting the midpoints equal
Since the diagonals bisect each other, we have:
\[ M_{AC} = M_{BD} \]
This leads to the equation:
\[ \frac{z_1 + z_3}{2} = \frac{z_2 + z_4}{2} \]

### Step 4: Simplifying the equation
To eliminate the fraction, we can multiply both sides by 2:
\[ z_1 + z_3 = z_2 + z_4 \]

### Step 5: Proving the converse
Now, we need to show the converse: if \( z_1 + z_3 = z_2 + z_4 \), then the points are vertices of a parallelogram.

Assuming \( z_1 + z_3 = z_2 + z_4 \), we can again calculate the midpoints:
- From \( z_1 + z_3 = z_2 + z_4 \), we can conclude that:
\[ \frac{z_1 + z_3}{2} = \frac{z_2 + z_4}{2} \]
This means that the midpoints of the diagonals are equal, confirming that the diagonals bisect each other.

### Conclusion
Thus, we have shown that the complex numbers \( z_1, z_2, z_3, z_4 \) are the vertices of a parallelogram if and only if \( z_1 + z_3 = z_2 + z_4 \).

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