# If (x1, y1), (x2, y2), (x3, y3) and (x4, y4) point s are joined in order to form a parallelogram, then prove that x1 + x3 = x2 + x4 and y1 + y3 = y2 + y4.

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

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