KöMaL 2019 B5026

Let $P$ be an arbitrary point of a given ellipse, different from the endpoints of the major axis. $P$ is connected to the foci $F_1$ and $F_2$. The angle bisector of angle $\angle F_1PF_2$ intersects $F_1F_2$ at $E$. The circle which passes through $P$ and touches $F_1F_2$ at $E$ intersects $PF_1$ at $G$ and $PF_2$ at $H$. Show that the length of $GH$ does not depend on the choice of $P$.