KöMaL 2022 B5271

$ABC$ is an isosceles right angled triangle with the right angle lying at vertex $C$. $A'$, $B'$ and $C'$ are interior points of sides $AB$, $BC$ and $CA$, respectively, such that triangle $A'B'C'$ is similar to $ABC$. Show that the midpoint of the side $AB$, the midpoint of line segment $A'B'$, and point $C$ are collinear.