If (x₁

A(x₁, y₁), B(x₂, y₂), C(x₃, y₃) and D(x₄, y₄)

We know that a quadrilateral is a parallelogram if the co-ordinates of mid-point s of its both the diagonals are same.

Therefore, we’ll find the mid-point s of diagonal AC and BD.

Let the co-ordinates of mid-point of AC be (x₀, y₀).

And, we know mid-point formula, i.e. the coordinates of mid-point of line joining (x₁, y₁) and (x₂, y₂) is

Similarly, let the co-ordinates of mid-point of AC be (x₅, y₅).

And, since it is a mid-point –

Now, since ABCD is a parallelogram –

⇒ (x₀, y₀) = (x₅, y₅)

⇒ x₀ = x₅ and y₀ = y₅

⇒x₁ + x₃ = x₂ + x₄ and y₁ + y₃ = y₂ + y₄