Given Information:
Three vertices of a parallelogram are (a,b), (a-b,b), and (a+b,b-a).

To Find:
The fourth vertex of the parallelogram.

Explanation:
To find the fourth vertex of the parallelogram, we need to understand the properties of a parallelogram and the relationship between its vertices.

Properties of a Parallelogram:
1. Opposite sides are parallel: The slope of one side is equal to the slope of the opposite side.
2. Opposite sides are equal in length: The distance between two opposite sides is equal.

Steps to Find the Fourth Vertex:
1. Identify the slope and length of one side of the parallelogram.
2. Use the slope and length to find the equation of the line passing through one of the given vertices.
3. Find the point of intersection of the equations of the lines passing through the other two given vertices.
4. The point of intersection will be the fourth vertex of the parallelogram.

Step 1: Identify the Slope and Length of One Side
Let's consider the two given vertices (a,b) and (a-b,b). The slope of this side can be found using the formula:

slope = (y2 - y1) / (x2 - x1)

Here, (x1, y1) = (a,b) and (x2, y2) = (a-b,b).

slope = (b - b) / (a - (a-b))
= 0 / b
= 0

Since the slope is 0, the side is horizontal.

The length of this side can be found using the distance formula:

length = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Here, (x1, y1) = (a,b) and (x2, y2) = (a-b,b).

length = sqrt((a - (a-b))^2 + (b - b)^2)
= sqrt((a - a + b)^2 + 0^2)
= sqrt(b^2)
= b

Step 2: Find the Equation of the Line Passing Through (a,b)
Since the side is horizontal, the equation of the line passing through (a,b) is y = b.

Step 3: Find the Point of Intersection
Now, let's consider the third given vertex (a+b,b-a). The equation of the line passing through this vertex can be found using the slope-intercept form:

y - y1 = m(x - x1)

Here, (x1, y1) = (a+b,b-a) and m is the slope.

Using the slope calculated in Step 1, the equation becomes:

y - (b-a) = 0(x - (a+b))
y - (b-a) = 0
y = (b-a)

To find the point of intersection, we need to solve the equations y = b and y = (b-a) simultaneously.

b = (b-a)
b + a = b
a = 0

The point of intersection is (0,b).

Step 4: Fourth Vertex
Since opposite sides of a parallelogram are