The triangle $ABC$ is inscribed in circle $\Gamma$. The points X, Y, Z are the midpoints of the arcs $BC$, $CA$ and $AB$ respectively in $\Gamma$ (those that do not contain the third vertex, in each case). The intersection points of the sides of the triangles $\vartriangle ABC$ and $\vartriangle XY Z$ form the hexagon $DEFGHK$. Prove that the diagonals $DG$, $EH$ and $FK$ are concurrent