D is adjacent to either A and B, B and C, or A and C.
If D is adjacent to A and B, then it is opposite C. You can see that in this case D = C + (A-C) + (B-C) = A+B-C by drawing a picture.
Permuting the roles of A, B and C gives all possibilities.
If and are the vertices of a parallelogram, find all the possible coordinates for the point D
I managed to find one point using the fact that the vector AB is parallel to the vector CD and hence finding the vector of D. However, I cannot find the remaining two points.