Suppose you have a figure which is obtained by rescaling and rotating another one. One can suppose that the rotation and rescaling have the same center (the only fixed point of the transformation). Is there a geometrical construction to find such a center?
For example, how do I find the center of the roto-scaling in the following picture? (the red L-shaped polygon is a rotated enlargement of the blue polygon)
Take two corresponding points in the transformation, e.g. $D$ and $H$. Under the rescaling $D$ goes to $D'$, which is then rotated by $\alpha=90°$ to $H$. If $O$ is the center of both transformations, then $\angle HOD'=\angle HOD=\alpha$, hence $O$ lies on the locus of points subtending an angle $\alpha$ with chord $DH$: in general this locus is formed by two circle arcs (a full circle of diameter $DH$ in this particular case).
Take then three couples of corresponding points ($DH$, $BJ$ and $EG$ in the diagram): the three loci described above will intersect at center $O$.
EDIT.
In practice, the construction can be simplified because the circle passing through the intersection point of a line with its roto-translated image and a couple of corresponding points on those lines, also contains the transformation center. See diagram below for an example.