If  $z_1,z_2,z_3$ are three points lying on the circle $\left|z\right|=\sqrt{3}$ then maximum value

of  ${|z_1-z_2|}^2+{|z_2-z_3|}^2+{|z_3-z_1|}^2$ is equal to _______

We are given three points z₁, z₂, z₃ that lie on the circle |z| = 3, and we are tasked with finding the maximum value of the expression:

|z₁ − z₂|² + |z₂ − z₃|² + |z₃ − z₁|²

Step 1: General Setup

Let the points z₁, z₂, z₃ be represented as complex numbers in polar form since they lie on the circle |z| = 3. For any point on the circle, we have:

|z_i| = 3, i = 1, 2, 3

Thus, each of z₁, z₂, z₃ can be written as:

z₁ = 3e^iθ₁, z₂ = 3e^iθ₂, z₃ = 3e^iθ₃

Step 2: Expanding the Expression

Now, let's expand the given expression. For any two complex numbers z_i and z_j, we know that:

|z_i − z_j|² = (z_i − z_j)(̅z_i − ̅z_j) = |z_i|² + |z_j|² − 2Re(z_i̅z_j)

Since |z_i| = |z_j| = 3, this simplifies to:

|z_i − z_j|² = 9 + 9 − 2Re(z_i̅z_j) = 18 − 2Re(z_i̅z_j)

We now apply this formula to each pair:

|z₁ − z₂|² = 18 − 2Re(z₁̅z₂),|z₂ − z₃|² = 18 − 2Re(z₂̅z₃),|z₃ − z₁|² = 18 − 2Re(z₃̅z₁)

Thus, the total expression becomes:

|z₁ − z₂|² + |z₂ − z₃|² + |z₃ − z₁|² = 3 × 18 − 2 (Re(z₁̅z₂) + Re(z₂̅z₃) + Re(z₃̅z₁))

Simplifying:

= 54 − 2 (Re(z₁̅z₂) + Re(z₂̅z₃) + Re(z₃̅z₁))

Step 3: Maximizing the Expression

Now, we need to maximize the real part terms. Since z₁, z₂, z₃ lie on the circle |z| = 3, the maximum value of the sum of the real parts will occur when the points are positioned to maximize the distances between them.

This happens when the points are spaced as far apart as possible, i.e., when they are placed at the vertices of an equilateral triangle inscribed in the circle. For such a configuration, the angle between any two points is 120°.

Step 4: Calculate the Maximum Value

For an equilateral triangle inscribed in the circle, the distance between any two points is the chord length corresponding to an angle of 120°. The chord length d for two points on a circle of radius r separated by an angle θ is given by:

d = 2r sin(θ/2)

For r = 3 and θ = 120°, we have:

d = 2 × 3 × sin(60°) = 6 × √3/2 = 3√3

Thus, the squared distance between any two points is:

|z₁ − z₂|² = (3√3)² = 27