KöMaL 2012 B4415

Let $B$ be an interior point of a line segment $AC$. The Thales circles of the line segments $AB$, $BC$ and $AC$ are $k_1$, $k_2$ and $k_3$, respectively, with radii of $r_1$, $r_2$ and $r_3$. A common exterior tangent of circles $k_1$ and $k_2$ cuts the circle $k_3$ into two circular segments. The radius of the circle inscribed in the smaller circular segment is $r_4$. Prove that $r_1 ^. r_2$=$r_3 ^. r_4$.