KöMaL 2014 B4639

Let $P$ be an exterior point of the ellipse $\mathcal E$ with foci $F_1$ and $F_2$. It does not lie on the line of the major axis. Let $M_1$ be the intersection of the line segment $PF_1$ with $\mathcal E$ , let $M_2$ be the intersection of the line segment $PF_2$ with $\mathcal E$ , and let $R$ be the intersection of the lines $M_1F_2$ and $M_2F_1$. Prove that the quadrilateral $PM_1RM_2$ has an inscribed circle.