Here we have the full case of 4 sides. It may be helpful to have seen first the
previous cases of prime numbers,
3 sides and
5 sides.

There are a red circle and inscribed square s, a blue circle and inscribed square
a (with reversed
orientation), red and blue radius control points, green midpoints
r between
corresponding red and blue ones, and green lines making up the
parallelogram that results from the construction. The radius
controls and the points s0 and a0 are all draggable (as in the
previous cases). This time the circles do not start on top of each
other. In addition, now there is a yellow circle and its radius
control point, a yellow digon with the double point d13
on its left, and a final figure
of black t-points
and lines resulting from midpoints between
r0
and d02r1
and d13,
r2
and d02,
and
r3
and d13.

Initially all the r
and t points are nearly
in a straight line. The diagram is cluttered. Moving
s0 round a bit, say thirty degrees up,
makes the figures clearer. Then for reducing the blue radius makes
the final black figure into a kite shape, and making it smaller yet
gives a reentrant quadrilateral (self-crossing). By starting from the
kite and reducing the size of the yellow contribution you can make
the quadrilateral convex again.

This illustrates fully the theory of the Discrete Fourier Transform
of order 4, i.e., harmonic analysis on the cyclic group of order
4---a.k.a. the basic Geometric Fourier Transform that takes
arbitrary quadrilaterals to their square, anti-square and degenerate
double digon harmonic components---. It lets us see that any
quadrilateral can be constructed in the way the mechanism here
illustrates, provided its center of gravity is at the origin. There
is clearly a lot of opportunity for play. Enjoy!