Let ABCD be a rectangle with sides sizes of $1$ and $CD=AB=\sqrt{2}$. And let a semicircle with center $O$ of radius $1$ be such that its diameter is on the $AB$-produced so $CD$ will be tangent to the circle at point $Q$. $AB$ is always within the diameter of the semicircle. So semicircle cuts AC and BD at points P and R respectively. How to prove that the angle $POR$ always is at least (or maybe exactly) $\pi/2$ ?

PS - I am trying to prove a theorem relating "lattice point covering property" and I reduced the proof to the question I asked above; so $POR$-angle to be $\ge \pi/2$ will finish the proof. Please help!