#### noobonastick

If A(-1,3,4), B(4,6,3), C(-2,1,1) and D are the verticies of a parallelogram, find all the possible coordinates for the point D.

So I let D be (x,y,z), then I worked out vectors AB,AC,AD,BC,BD,CD and after this I don't know what to do. (am i even doing it right?) Can someone help me with this question?

#### alexmahone

Diagonals of a parallelogram bisect each other.

So, $$\displaystyle (\frac{-1-2}{2}, \frac{3+1}{2}, \frac{4+1}{2})=(\frac{4+x}{2}, \frac{6+y}{2}, \frac{3+z}{2})$$

#### slovakiamaths

the ans by alex is very much correct

#### noobonastick

This is what i have done so far. If vector AB is parallel to CD then x=4,y=5,z=0.

If BC is parallel to AD then x=-6,y=-1,z=2

Diagonals of a parallelogram bisect each other (BC and AC)

$$\displaystyle (\frac{-1-1}{2},\frac{3+2}{2},\frac{4+1}{2}) = (\frac{x+4}{2},\frac{y+6}{2},\frac{z+3}{2})$$

and i get the solution x=-6,y=-1,z=2, which is the same as my 2nd solution set.

The problem is I need the third solution set. How do I do this?

bump?