Given:
The lengths of the sides of the triangle are:
1. Side 1: 3x - 2y
2. Side 2: 4x - (4/3)y
3. Side 3: 3(x - 1) - (3/2)(y - 1)

To prove:
If the triangle is equilateral, then the length of its side is equal.

Proof:
To prove that the triangle is equilateral, we need to show that all three sides are equal in length.

Step 1: Set up equations for the sides of the triangle
We set up equations for the three sides of the triangle:

1. Side 1: 3x - 2y
2. Side 2: 4x - (4/3)y
3. Side 3: 3(x - 1) - (3/2)(y - 1)

Step 2: Equate the lengths of the sides
To prove that the triangle is equilateral, we need to show that all three sides are equal in length. We equate the lengths of the sides:

3x - 2y = 4x - (4/3)y = 3(x - 1) - (3/2)(y - 1)

Step 3: Solve the equations
We solve the equations to find the values of x and y:

3x - 2y = 4x - (4/3)y
3(x - 1) - (3/2)(y - 1) = 4x - (4/3)y

Expanding and simplifying the equations, we get:

3x - 2y = 4x - (4/3)y
3x - 3 - (3/2)y + (3/2) = 4x - (4/3)y

Combining like terms, we get:

-2x + (2/3)y = 3

Step 4: Determine the values of x and y
We solve the equation -2x + (2/3)y = 3 to find the values of x and y. However, since we have only one equation and two variables, we cannot uniquely determine the values of x and y. Therefore, we cannot determine the exact length of the sides of the equilateral triangle.

Conclusion:
Based on the given information and the equations obtained, we cannot determine the exact length of the sides of the equilateral triangle. Further information or constraints are required to find the length of the sides.