Sol:
Slope of the diagonal BD = 3x - 7y + 6 = 0 is -(3) / (-7) = 3/7
Diagonals of a square are perpendicular to each other.
So, the slope of AC = -7/3 [Product of the slopes of the perpendicular lines is -1.]
AC passes through the vertex A(3, -2).

Equation of AC = (y - y1) = m(x - x1)
(y + 2) = -7/3(x - 3)
3y + 6 = -7x + 21
7x + 3y - 15 = 0

Point of intersection of the diagonals of the square is the midpoint of the diagonals. Solve the equations of the diagonals to find their midpoint.

Equation of Diagonal AC:
To find the equation of diagonal AC, we need to determine the coordinates of points A and C.

Given that the coordinates of vertex A are (3, -2), we can use this information to find the coordinates of vertex C.

Since a square is a quadrilateral with all sides equal in length and all interior angles equal to 90 degrees, we can determine the coordinates of C by finding the midpoint of the diagonal BD and then using the midpoint formula.

Midpoint of Diagonal BD:
The equation of diagonal BD is 3x - 7y + 6 = 0. To find the midpoint, we need to find the coordinates of points B and D.

To find point B, we can use the fact that a diagonal of a square bisects the opposite vertex. Since A is the opposite vertex of B, the coordinates of B will be the negative of the coordinates of A, which gives us (-3, 2).

To find point D, we can use the fact that the diagonals of a square intersect at right angles and bisect each other. Since B is the midpoint of diagonal AC, the coordinates of D will be the negative of the coordinates of B. Thus, the coordinates of D are (3, -2).

Using the coordinates of B and D, we can find the midpoint of BD using the midpoint formula:

Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
= ((-3 + 3)/2, (2 + (-2))/2)
= (0, 0)

Coordinates of Vertex C:
The coordinates of the midpoint of BD are (0, 0). Since C is the opposite vertex of A, the coordinates of C will be the negative of the coordinates of A, which gives us (-3, 2).

Equation of Diagonal AC:
Now that we have the coordinates of A and C, we can use them to find the equation of diagonal AC.

The equation of a line passing through two points (x1, y1) and (x2, y2) can be found using the point-slope form:

(y - y1) = m(x - x1)

where m is the slope of the line.

Let's find the slope first:

m = (y2 - y1)/(x2 - x1)
= (2 - (-2))/(-3 - 3)
= 4/-6
= -2/3

Using the coordinates of A and the slope, we can write the equation of diagonal AC:

(y - (-2)) = (-2/3)(x - 3)
y + 2 = (-2/3)(x - 3)
3y + 6 = -2x + 6
2x + 3y = 0

Therefore, the equation of diagonal AC is 2x + 3y = 0.

Coordinates of Center of the Square:
The center of a square is the midpoint of its diagonals. Since we have already found the midpoint of diagonal BD to be (0, 0), the coordinates of the center of the square is also (0, 0).

Thus, the coordinates of the center of the square are (0, 0).