Explanation:

To solve this problem, we will use complex numbers. Let's assign complex numbers to the vertices of the equilateral triangle as follows:

z1 = a + bi
z2 = c + di
z3 = e + fi

where a, b, c, d, e, f are real numbers.

Step 1: Find the centroid of the equilateral triangle
The centroid, zo, of an equilateral triangle with vertices z1, z2, and z3 can be found by taking the average of the coordinates of the vertices:

zo = (z1 + z2 + z3)/3

Using the assigned complex numbers, we have:

zo = (a + bi + c + di + e + fi)/3

Simplifying, we get:

zo = (a + c + e)/3 + (b + d + f)/3i

Step 2: Calculate z12, z22, z32
To calculate z12, z22, and z32, we square the complex numbers z1, z2, and z3:

z12 = (a + bi)(a + bi) = a^2 + 2abi - b^2
z22 = (c + di)(c + di) = c^2 + 2cdi - d^2
z32 = (e + fi)(e + fi) = e^2 + 2efi - f^2

Step 3: Simplify z12 + z22 + z32
Adding z12, z22, and z32 together, we get:

z12 + z22 + z32 = (a^2 + 2abi - b^2) + (c^2 + 2cdi - d^2) + (e^2 + 2efi - f^2)

Grouping the real and imaginary terms separately, we have:

z12 + z22 + z32 = (a^2 + c^2 + e^2) + (2ab + 2cd + 2ef)i - (b^2 + d^2 + f^2)

Step 4: Simplify the real and imaginary parts
Since we know that the triangle is equilateral, the lengths of its sides are equal. Therefore, the real and imaginary parts of z12 + z22 + z32 should be equal to each other.

Equating the real parts, we have:

a^2 + c^2 + e^2 = b^2 + d^2 + f^2

Equating the imaginary parts, we have:

2ab + 2cd + 2ef = 0

Step 5: Simplify the equations using the centroid
Using the centroid zo = (a + c + e)/3 + (b + d + f)/3i, we can rewrite the equations as:

(a + c + e)^2 = (b + d + f)^2
2(a + c + e)(b + d + f) = 0

Expanding and simplifying the equations, we get:

a^2 + c^2 + e^2 + 2ac + 2ae + 2ce = b^2 + d^2 + f^2 +