Given two line segments, Construct a rhombus whose diagonals have lengths equal to the lengths of the two given segments.
I can get to finding perpendicular bisectors of each line segment, but have no idea how to move one so that it lies over the other one. Any hints would be appreciated.
You said you know how to construct mid point and perpendicular line. Now we begin.
The question is to move a line segment to somewhere with one end at a given point.
See picture, $AB$ is the given line segment, $O$ is the point we want to move. Construct $OF \perp AB$, and construct $\ell \perp OF$ s.t. $\ell$ pass thru $O$. Connect $OB$ and bisect it at $C$. Connect $AC$ and extend the line to $\ell$ so they intersect at $OD$, then $OD=AB$.
This allows us to "move a line segment around", so for convenience, we can assume that we are allowed to maintain a certain angle of the compass, so that we can record the length of a line segment.
(Remard: from the picture, we supposed that $O$ is not on line $AB$, but the case that $O$ is on line $AB$ is same. In this case, $C$ is on line $AB$.)