Question

If A(x1, y1) , B(x2, y2) , C(x3, y3) are the vertices of a parallelogram, how many possible positions of fourth vertex D exists?

Answer 1

We know that in the parallelogram the diagonals bisect each other:

A = (x₁,y₁)

B = (x₂,y₂)

C = (x₃,y₃)

D = (x,y)

Here the midpoint of the line would be AC = (x₁+x₃)/2

And the midpoint of the line BD = x₁ + x/2

x₁ + x₃/₂ = x+x₂/2

x = x₁+x₃+x₂

y₁ +y₃ = y₂ + y/2

y = y₁ + y₃ + y₂

So the coordinates for D will be (x₁+x₃+x₂,y₁+y₃+y₂)