Let $k_1$ and $k_2$ be internally tangent circles with common point $X$. Let $P$ be a point lying neither on one of the two circles nor on the line through the two centers. Let $N_1$ be the point on $k_1$ closest to $P$ and $F_1$ the point on $k_1$ that is farthest from $P$. Analogously, let $N_2$ be the point on $k_2$ closest to $P$ and $F_2$ the point on $k_2$ that is farthest from $P$. Prove that $\angle N_1 X N_2 = \angle F_1 X F_2$. (Robert Geretschläger)