OFM 2023 S3

Let $ABCD$ be a convex quadrilateral, with $\measuredangle ABC > 90^\circ$ , $\measuredangle CDA > 90^\circ$ and $\measuredangle DAB = \measuredangle BCD$ . Let $E$ , $F$ and $G$ be the reflections of $A$ with respect to the lines $BC$ , $CD$ and $DB$ . Finally, let the line $BD$ meet the line segment $AE$ at a point $K$ , and the line segment $AF$ at a point $L$ . Prove that the circumcircles of the triangles $BEK$ and $DFL$ are tangent to each other at $G$ .