KöMaL 2020 B5099

The angle at vertex $A$ of a rhombus $ABCD$ is $60^\circ$. An ellipse is inscribed in the rhombus, with the axes lying along the diagonals of the rhombus. The points of tangency of the ellipse on sides divide the sides in a ratio $1:3$. On sides $AB$ and $AD$ it is the point closer to $A$, and on sides $BC$ and $CD$ it is the point closer to $C$. Let some point $P$ move along the ellipse. Draw lines through $P$, parallel to the midlines of the rhombus, and consider the intersections with the other midline. Let these point be $Q$ and $R$. Show that the length of the line segment $QR$ is independent of the position of $P$.