Explanation:

Given three consecutive vertices (x1, y1), (x2, y2), and (x3, y3) of a parallelogram, we need to find the fourth vertex.

Properties of a parallelogram:
- Opposite sides are parallel.
- Opposite sides are congruent.
- Opposite angles are congruent.
- Consecutive angles are supplementary.

Method:
- Find the vector between (x1, y1) and (x2, y2). Let's call it vector v1.
- Find the vector between (x2, y2) and (x3, y3). Let's call it vector v2.
- The fourth vertex will be (x3 + v1x + v2x, y3 + v1y + v2y).

Explanation of the method:
- The vector between (x1, y1) and (x2, y2) gives us the direction and magnitude of one of the sides of the parallelogram.
- The vector between (x2, y2) and (x3, y3) gives us the direction and magnitude of the adjacent side of the parallelogram.
- Adding these two vectors gives us the diagonal of the parallelogram.
- Adding this diagonal to the third vertex (x3, y3) gives us the fourth vertex.

Example:
Let's take an example to understand this method better.

We are given the vertices (1, 2), (3, 4), and (5, 6) of a parallelogram.

- The vector between (1, 2) and (3, 4) is (3 - 1, 4 - 2) = (2, 2). Let's call it vector v1.
- The vector between (3, 4) and (5, 6) is (5 - 3, 6 - 4) = (2, 2). Let's call it vector v2.
- The fourth vertex will be (5 + 2 + 2, 6 + 2 + 2) = (9, 10).

Therefore, the fourth vertex of the parallelogram with vertices (1, 2), (3, 4), and (5, 6) is (9, 10).