Angle between squares at which they just touch along the circumference of a circle

Question

Say I have two squares whose centers fall along the circumference of a circle. The circle has radius $x$. The squares have the same height and width $y$. The height of one square is parallel to the height of the other. The angle that defines the separation of the squares on the circle is $z$.

What is the angle $z$ such that the squares just touch? (And how did you work this out?)

Edit

I realize now that the original question is vaguer than it should have been. It is guaranteed in my case that $x$ and $y$ are such that the squares touch only with a separation smaller than 90 degrees. Some answerers have noted that the answer to the question gets complicated when $y$ is large relative to $x$ and the squares touch with separations equal to or larger than 90 degrees.

Answer 1

I suspect that the problem as stated is not well-defined in the sense that the angle asked for does not strictly depend upon the values of $x$ and $y$. Perhaps it should be divided into the different ways the two squares can touch. Here is a diagram to indicate why I have concerns.

Using Geogebra I have only been able to get the squares to intersect at a corner or to share a common side. For these two cases there are two distinct formulas for the angle $\theta$ between their centers.

In the case of intersection at the corners the angle $\frac{\theta}{2}$ has opposite side half the diagonal of the square, or $\frac{\sqrt{2}\,y}{2}$ so

\begin{equation} \theta= 2\arcsin\left(\frac{\sqrt{2}\,y}{2x}\right) \end{equation}

In the case where the two squares share a side the side opposite $\frac{\theta}{2}$ equals $\frac{y}{2}$ so in that case

\begin{equation} \theta= 2\arcsin\left(\frac{y}{2x}\right) \end{equation}

Here is a diagram to show that for the same angle $\theta$ one can have two different values of $y$--one for the corner intersection case and another for the common side intersection case. I could not get the angles precisely equal using Geogebra but close enough to make the point.

Update: The two squares may indeed intersect other than at one or two corners. But the problem is actually not well defined because the solution actually involves a relationship of several interdependent variables:

1. The radius $r$ of the circle
2. The width $W$ of the two squares
3. The distance $D$ from the circle center of the line along which the squares are tangent
4. The angle between the radii connecting the circle center and the centers of the two squares

Rather than a static Geogebra image, one needs a dynamic Geogebra worksheet to understand this. In the link below, the points D and A can be moved. Observe the resulting changes in the positions of the squares, the amount of overlap between the squares and the angle separating their centers.

Dynamic Geogebra Worksheet