Baltic Way 1994 · Problem 11

Let $N S$ and $E W$ be two perpendicular diameters of a circle $\mathcal{C}$. A line $l$ touches $\mathcal{C}$ at point $S$. Let $A$ and $B$ be two points on $\mathcal{C}$, symmetric with respect to the diameter $E W$. Denote the intersection points of $l$ with the lines $N A$ and $N B$ by $A^{\prime}$ and $B^{\prime}$, respectively. Show that $\left|S A^{\prime}\right| \cdot\left|S B^{\prime}\right|=|S N|^{2}$.