To find the coordinates of the fourth vertex of a parallelogram, we need to understand the properties of a parallelogram. A parallelogram is a quadrilateral with opposite sides that are parallel and equal in length.

Given that the three consecutive vertices of the parallelogram are (-2, -1), (1, 0), and (4, 3), we can use this information to find the coordinates of the fourth vertex.

Let's denote the consecutive vertices as A (-2, -1), B (1, 0), and C (4, 3). To find the fourth vertex, let's use the properties of a parallelogram:

1. Opposite sides are parallel:
The line passing through points A and B, AB, will be parallel to the line passing through points C and D, CD (the fourth vertex). Similarly, the line passing through points B and C, BC, will be parallel to the line passing through points D and A, DA.

2. Opposite sides are equal in length:
The length of AB is equal to the length of CD. Similarly, the length of BC is equal to the length of DA.

Using these properties, we can determine the coordinates of the fourth vertex:

1. Find the equation of the line AB passing through points A and B:
The slope of AB can be calculated as:
slope_AB = (y2 - y1) / (x2 - x1) = (0 - (-1)) / (1 - (-2)) = 1/3

Using the point-slope form, the equation of the line AB is:
y - y1 = slope_AB * (x - x1)
y - (-1) = 1/3 * (x - (-2))
y + 1 = 1/3 * (x + 2)

2. Find the equation of the line parallel to AB passing through point C:
Since AB and CD are parallel, they have the same slope.
The equation of the line CD can be written as:
y - y1 = slope_AB * (x - x1)
y - 3 = 1/3 * (x - 4)
y - 3 = 1/3x - 4/3

3. Find the intersection point of the lines AB and CD:
Equate the two equations:
1/3x + 1 = 1/3x - 4/3
1 = -4/3

Since the equation is inconsistent, the lines AB and CD will not intersect. However, they are parallel.

4. Find the equation of the line BC passing through points B and C:
The slope of BC can be calculated as:
slope_BC = (y2 - y1) / (x2 - x1) = (3 - 0) / (4 - 1) = 1

Using the point-slope form, the equation of the line BC is:
y - y1 = slope_BC * (x - x1)
y - 0 = 1 * (x - 1)
y = x - 1

5. Find the equation of the line parallel to BC passing through point A:
Since BC and DA are parallel, they have the same slope.
The equation of the