KöMaL 2023 B5292

Given a circle $k$ and points $P$ and $Q$ in its interior, construct (write down the steps of the construction - basic construction steps like bisecting an angle, reflecting over a line, etc. do not need to be described in detail - and explain why the method is correct; it is not required to perform the construction on paper) a circle through points $P$ and $Q$ that intersects the circle $k$ at two diametrically opposite points. Depending on the position of points $P$ and $Q$, how many solutions will the problem have?