KöMaL 2022 C1729

Semicircles $k_1$ and $k_2$, respectively, are drawn outside a square $ABCD$, over the sides $BC$ és $CD$ as diameters. The midpoints of the semicircular arcs are $E$ and $F$, and the midpoints of line segments $DE$ and $AF$ are $P$ and $Q$, respectively. Show that $P$ lies on diagonal $AC$ and $Q$ lies on diagonal $BD$ of the square.