Distance Between Complex Numbers

To find the required value, we use the properties of roots of unity, complex rotation, and conjugates step-by-step:

1. Determine the vertices of the equilateral triangle:
The vertices $$z_1, z_2, z_3$$ are inscribed in the circle $$|z| = 2$$ and form an equilateral triangle. Since $$z_1 = 2$$ lies on the positive real axis, the other two vertices are obtained by rotating $$z_1$$ counter-clockwise about the origin by angles of $$\frac{2\pi}{3}$$ and $$\frac{4\pi}{3}$$ respectively.
Using the complex cube root of unity $$\omega = e^{i\frac{2\pi}{3}}$$, we can write:
$$z_1 = 2$$
$$z_2 = 2\omega$$
$$z_3 = 2\omega^2$$

2. Calculate the conjugate of $$z_3$$:
We find the conjugate of $$z_3 = 2\omega^2$$. Since $$\bar{\omega} = \omega^2$$ and $$\overline{\omega^2} = \omega$$, the conjugate simplifies to:
$$\bar{z_3} = \overline{2\omega^2} = 2\omega$$

3. Evaluate the transformed complex number $$z_4$$:
Substitute the values of $$z_1$$, $$z_2$$, and $$\bar{z_3}$$ into the given expression:
$$z_4 = \frac{z_2 \cdot \bar{z_3}}{z_1}$$
$$z_4 = \frac{(2\omega) \cdot (2\omega)}{2}$$
$$z_4 = \frac{4\omega^2}{2} = 2\omega^2$$

Notice that $$z_4$$ is exactly equal to the vertex $$z_3$$.

4. Compute the final absolute squared value:
We need to find the value of $$|z_4 - z_1|^2$$, which is equivalent to finding the squared distance between the vertices $$z_3$$ and $$z_1$$ of our equilateral triangle.
$$|z_4 - z_1|^2 = |2\omega^2 - 2|^2 = 4|\omega^2 - 1|^2$$

We substitute the algebraic value of $$\omega^2 = -\frac{1}{2} - i\frac{\sqrt{3}}{2}$$ into the magnitude expression:
$$|\omega^2 - 1| = \left| -\frac{1}{2} - i\frac{\sqrt{3}}{2} - 1 \right| = \left| -\frac{3}{2} - i\frac{\sqrt{3}}{2} \right|$$
$$|\omega^2 - 1|^2 = \left(-\frac{3}{2}\right)^2 + \left(-\frac{\sqrt{3}}{2}\right)^2 = \frac{9}{4} + \frac{3}{4} = \frac{12}{4} = 3$$

Now, multiply this by the scalar factor of 4:
$$|z_4 - z_1|^2 = 4 \times 3 = 12$$

Geometrically, the side length $$s$$ of an equilateral triangle inscribed in a circle of radius $$R = 2$$ is given by $$s = R\sqrt{3} = 2\sqrt{3}$$. The square of the side length (the distance between $$z_3$$ and $$z_1$$) is $$s^2 = (2\sqrt{3})^2 = 12$$.