Three vertices of a parallelogram are (a + b, a - b), (2a + b, 2a - b), (a - b, a + b). Find the fourth vertex.

Let A(a + b, a - b), B(2a + b, 2a - b), C(a - b, a + b) and fourth vertex be D(x, y).

It is given that □ABCD is parallelogram.

We know that diagonals of parallelogram bisect each other.

Let intersection of diagonals be E(x_m, y_m )

By midpoint formula.

x_m = , y_m =

For midpoint E of diagonal AC,

x_m = , y_m =

∴ x_m = a , y_m = a

∴E(x_m, y_m ) ≡ (a, a)

For diagonal BD,

a = , a=

∴ 2a = 2a + b +x , 2a = 2a – b +y

∴ x = -b and y = b

Hence, the fourth vertex is D(-b, b)