Quote: for which the line $AD$ lies on the line $P$, What is line $P$?

I have made a construction here https://www.geogebra.org/calculator/xc8g2twq. To better understand my intuition. I started with https://www.geogebra.org/calculator/mxfumqe6, and tried to find a condition to make the rectangle ABCD a square. I used the fact that $AB=BC$, which means that $PQ\sin\theta=RS\cos\theta$, where $\theta=\angle APQ$. From there we can construct $\theta$ by constructing a right triangle with legs of length $PQ$ and $RS$.

Draw the circles $(PS)$ and $(RQ)$, name $X, Y$ their lower arcs' midpoints, draw the line $XY$ and it will intersect again the 2 circes at $B,D$ respectively, then draw the circle $(BD)$ and its diameter perpendicular onto $BD$, which will cut thiws circle at $A,C$ respectively, done. (See above post drawing for easy reference). Best regards, sunken rock