"Let $W$ be a circle with center $O$ and radius $r$. Let $S$ be a point outside the circle. Let $l_1$ and $l_2$ be two tangent lines to the circle $W$ passing through the point $S$ . Let $T_1$ and $T_2$ be the points of tangency of these lines with the circle. Show that the point of intersection of the segments $OS$ and $T_1$$T_2$ is the image of $S$ under inversion in $W$"
I really do not know how to get started with this question, apart from knowing that $OS$*$OS' = R^2$ where $R$ is the radius of the new circle. Help?
Let $X$ be an intersection point of $T_1T_2$ and $OS$. Since circle around $XT_1S$ touches $OT_1$ we have, by the PoP of $O$ with respect to that new circle:$$ r^2 =OT_1^2 = OC \cdot OS$$ and we are done.