If the triangle fromed by complex numbers `z_(1), z_(2)` and `z_(3)` is equilateral then prove that `(z_(2) + z_(3) -2z_(1))/(z_(3) - z_(2))` is purely imaginary number

Explanation

To prove that \(\frac{z_2 + z_3 - 2z_1}{z_3 - z_2}\) is a purely imaginary number, we will follow these steps:

### Step 1: Define the Complex Numbers
Let \(z_1\), \(z_2\), and \(z_3\) be the vertices of the equilateral triangle in the complex plane. We can express these complex numbers as:
- \(z_1 = x_1 + iy_1\)
- \(z_2 = x_2 + iy_2\)
- \(z_3 = x_3 + iy_3\)

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