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: .$\displaystyle P(x_1,y_1)$
$\displaystyle \text{Let }r\, = \,\text{distance }OP.$
$\displaystyle \text{Let }\theta \,=\,\angle POX,\,\text{ where }\tan\theta \,=\,\frac{y_1}{x_1}$
Then $\displaystyle P$ has coordinates: .$\displaystyle \left(r\cos\theta,\:r\sin\theta\right) $
The four vertices are: .$\displaystyle \bigg[r\cos\!\left(\theta + \frac{\pi}{2}n\right),\: r\sin\!\left(\theta + \frac{\pi}{2}n\right)\bigg]\;\text{ for }n = 0,1,2,3$