Let `S = {z in C : |z - 2| = |z + 2i| = |z - 2i|}` then `sum_(z in S) |z + 1.5|` is equal to _________

Answer

To solve the problem, we need to determine the set \( S \) defined by the conditions \( |z - 2| = |z + 2i| = |z - 2i| \) and then find the sum \( \sum_{z \in S} |z + 1.5| \).

### Step 1: Understand the conditions
The conditions \( |z - 2| = |z + 2i| = |z - 2i| \) imply that the complex number \( z \) is equidistant from the points \( 2 \), \( -2i \), and \( 2i \) in the complex plane.

### Step 2: Interpret the distances geometrically
1. The point \( 2 \) corresponds to the point \( (2, 0) \) on the Argand plane.
2. The point \( -2i \) corresponds to the point \( (0, -2) \).
3. The point \( 2i \) corresponds to the point \( (0, 2) \).

### Step 3: Set up the equations
We can express \( z \) as \( z = x + yi \) where \( x \) is the real part and \( y \) is the imaginary part. The conditions can be rewritten as:
- \( |(x + yi) - 2| = |(x + yi) + 2i| \)
- \( |(x + yi) - 2| = |(x + yi) - 2i| \)

### Step 4: Solve the first equation
The first condition gives:
\[ \sqrt{(x - 2)^2 + y^2} = \sqrt{x^2 + (y + 2)^2} \]
Squaring both sides:
\[ (x - 2)^2 + y^2 = x^2 + (y + 2)^2 \]
Expanding both sides:
\[ x^2 - 4x + 4 + y^2 = x^2 + y^2 + 4y + 4 \]
Simplifying:
\[ -4x + 4 = 4y + 4 \]
\[ -4x = 4y \implies y = -x \]

### Step 5: Solve the second equation
The second condition gives:
\[ \sqrt{(x - 2)^2 + y^2} = \sqrt{x^2 + (y - 2)^2} \]
Squaring both sides:
\[ (x - 2)^2 + y^2 = x^2 + (y - 2)^2 \]
Expanding both sides:
\[ x^2 - 4x + 4 + y^2 = x^2 + y^2 - 4y + 4 \]
Simplifying:
\[ -4x + 4 = -4y + 4 \]
\[ -4x = -4y \implies x = y \]

### Step 6: Solve the system of equations
Now we have two equations:
1. \( y = -x \)
2. \( x = y \)

Substituting \( y = -x \) into \( x = y \):
\[ x = -x \implies 2x = 0 \implies x = 0 \implies y = 0 \]
Thus, the only solution is \( z = 0 \).

### Step 7: Calculate the required sum
Now we need to find \( |z + 1.5| \) where \( z = 0 \):
\[ |0 + 1.5| = |1.5| = 1.5 \]

### Final Answer
The sum \( \sum_{z \in S} |z + 1.5| \) is equal to \( 1.5 \).