Let ` z_1 , z _2 and z_3 ` be three complex numbers such that `z_1 + z_2+ z_3 = z_1z_2 + z_2z_3 + z_1 z_3 = z_1 z_2z_3 = 1`. Then the area of triangle formed by points `A(z_1 ), B(z_2) and C(z_3)` in complex plane is _______.

Explanation

To solve the problem, we need to find the area of the triangle formed by the complex numbers \( z_1, z_2, z_3 \) under the given conditions. Let's go through the solution step by step.

### Step 1: Set up the equations
We are given three conditions:
1. \( z_1 + z_2 + z_3 = 1 \)
2. \( z_1 z_2 + z_2 z_3 + z_1 z_3 = 1 \)
3. \( z_1 z_2 z_3 = 1 \)

### Step 2: Form a cubic equation
Since \( z_1, z_2, z_3 \) are the roots of a cubic polynomial, we can express this polynomial as:
\[ z^3 - (z_1 + z_2 + z_3)z^2 + (z_1 z_2 + z_2 z_3 + z_1 z_3)z - z_1 z_2 z_3 = 0 \]
Substituting the values from the conditions:
\[ z^3 - 1z^2 + 1z - 1 = 0 \]
This simplifies to:
\[ z^3 - z^2 + z - 1 = 0 \]

### Step 3: Factor the cubic equation
We can factor the polynomial:
\[ z^3 - z^2 + z - 1 = (z - 1)(z^2 + 1) = 0 \]
This gives us the roots:
1. \( z - 1 = 0 \) which implies \( z = 1 \)
2. \( z^2 + 1 = 0 \) which implies \( z = i \) and \( z = -i \)

Thus, we have:
\[ z_1 = 1, \quad z_2 = i, \quad z_3 = -i \]

### Step 4: Plot the points in the complex plane
Now we can plot the points:
- \( A(z_1) = A(1, 0) \)
- \( B(z_2) = B(0, 1) \)
- \( C(z_3) = C(0, -1) \)

### Step 5: Calculate the area of the triangle
The area \( A \) of a triangle formed by points \( (x_1, y_1), (x_2, y_2), (x_3, y_3) \) in the coordinate plane can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Substituting the coordinates:
- \( (1, 0) \)
- \( (0, 1) \)
- \( (0, -1) \)

We have:
\[ \text{Area} = \frac{1}{2} \left| 1(1 - (-1)) + 0(-1 - 0) + 0(0 - 1) \right| \]
\[ = \frac{1}{2} \left| 1 \cdot 2 \right| = \frac{1}{2} \cdot 2 = 1 \]

### Final Answer
Thus, the area of the triangle formed by the points \( A(z_1), B(z_2), C(z_3) \) in the complex plane is \( \boxed{1} \).