IGO 2021 E4

In isosceles trapezoid $ABCD$ ( $AB \parallel CD$ ) points $E$ and $F$ lie on the segment $CD$ in such a way that $D, E, F$ and $C$ are in that order and $DE = CF$ . Let $X$ and $Y$ be the reflection of $E$ and $C$ with respect to $AD$ and $AF$ . Prove that circumcircles of triangles $ADF$ and $BXY$ are concentric.

Proposed by Iman Maghsoudi - Iran