Animated Solution for Mathematics - Complex Numbers: Let the minimum value $v_0$ of $v=|z{|}^2+|z-3{|}^2+|z-6i{|}^2$, $z\in \mathbb{C}$ is attained at $z=z_0$. Then $|2z_0^2-{\bar{z}}_0^3+3{|}^2+v_0^2$ is equal to

When you look at the expression $v=|z{|}^2+|z-3{|}^2+|z-6i{|}^2$, it is easy to see an algebraic mess. However, in the complex plane, the modulus $|z-z_1|$ represents the distance between two points.

Therefore, the expression $v$ is the sum of the squared distances from a variable point $z$ to three fixed vertices: $0$, $3$, and $6i$.

There is an elegant theorem in geometry: the sum of the squares of the distances from any point to the vertices of a triangle is minimized at the centroid. The centroid acts as the "center of mass" of the triangle, representing the point of perfect balance.

Since $z_0$ is the centroid, we find it by calculating the arithmetic mean of the three vertices:

This is our minimizing point. Now, we substitute $z_0=1+2i$ back into the original expression to find the minimum value $v_0$:

Breaking this down, we evaluate the individual components:

Calculating these magnitudes yields:

Thus, we have determined that $v_0=30$.

We now evaluate the final expression: $|2z_0^2-{\bar{z}}_0^3+3{|}^2+v_0^2$. First, we compute the powers of $z_0$:

Next, we find the cube of the conjugate ${\bar{z}}_0=1-2i$:

Now, we assemble the internal expression:

Finally, we calculate the squared magnitude and add $v_0^2$:

The final result is 1000.