KöMaL 2026 A942

Let $ABCD$ be a cyclic quadrilateral without right angles. Let $E$ denote the midpoint of diagonal $AC$, and let $F$ denote the midpoint of diagonal $BD$. Suppose that the circles $AEF$, $BEF$, $CEF$, $DEF$ do not degenerate into lines, and they intersect the circumcircle of $ABCD$ for the second time at $A'$, $B'$, $C'$, $D'$, respectively. Let $E'$ be the midpoint of segment $A'C'$, and let $F'$ be the midpoint of segment $B'D'$. Prove that points $E$, $F$, $E'$, $F'$ lie on the same circle.