Hello, padfoot!
I don't know what you're asking, but I'll take a guess.
You're given a circle with radius r and a square is inscribed on it.
The center of the circle is at the origin.
Given coordinates (x1,y1) of one vertex, locate the other vertices. Code:
Y

* * *
*  * P
*    +    o(x1,y1)
*  * *
  r* 
*   * θ  *
  *     +     *   X
*  O  *
  
*   *
*    +    *
*  *
* * *

The coordinates of the one vertex is: .
Then has coordinates: .
The four vertices are: .