KöMaL 2016 C1390

Triangle $ABC$ is right-angled at vertex $C$. A central similitude of centre $C$ and negative scale factor is applied, to obtain a diminished triangle $A'B'C$. Given that $BC=5$, $B'C=0.2$ and $\tan \alpha =2.4$, prove that the interior angle bisector from $B$ and the exterior angle bisector from $B'$ intersect at the midpoint of $AA'$.