Let the complex numbers `z_(1),z_(2)` and `z_(3)` be the vertices of an equailateral triangle. If `z_(0)` is the circumcentre of the triangle , then prove that ` z_(1)^(2) + z_(2)^(2) + z_(3)^(2) = 3z_(0)^(2)`.

Explanation

To prove that \( z_1^2 + z_2^2 + z_3^2 = 3z_0^2 \) where \( z_1, z_2, z_3 \) are the vertices of an equilateral triangle and \( z_0 \) is the circumcenter, we can follow these steps:

### Step 1: Define the vertices and circumcenter
Let \( z_1, z_2, z_3 \) be the complex numbers representing the vertices of the equilateral triangle. The circumcenter \( z_0 \) of the triangle can be expressed as:
\[ z_0 = \frac{z_1 + z_2 + z_3}{3} \]

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