$ABCD$ is a given parallelogram. What is the locus of the remaining vertices of a parallelogram equal in area to $ABCD$ having $AC$ as its diagonal?
I came up with the answer that the remaining vertices will have a locus as lines parallel to the diagonal $AC$.
Am I correct?
Yep. To illustrate, draw with AC horizontal, origin in the middle. If B and D are located at $(x,y)$ and $(-x,-y)$, the shape will be a parallelogram, and the area will be $|AC|\cdot |y|$.
Thus the loci you are seeking are horizontal lines, which are parallel to AC.