Bariz - Problem

There are usually four such squares (or two, or one, or none). Suppose $ x = OB$ and $ y = OD$. Let $ O'$ be the image of $ O$ under a rotation about $ A$ by $ 90$ degrees (you have a freedom to choose the clockwise or the counterclockwise direction, and it doesn't matter). Let $ D_1$ and $ D_2$ be the points in which the circle at $ O'$ with radius $ x$ meets the circle at $ O$ with radius $ y$. Also, let $ B_3$ and $ B_4$ be the points of intersection between the circle at $ O'$ with radius $ y$ and the circle at $ O$ with radius $ x$. The points $ D_1$ and $ D_2$ determine $ \left(B_1,C_1\right)$ and $ \left(B_2,C_2\right)$. Likewise, $ B_3$ and $ B_4$ determine $ \left(C_3,D_3\right)$ and $ \left(C_4,D_4\right)$. Now, we've got the desired $ AB_iC_iD_i$ for $ i = 1,2,3,4$.