KURSCHAK 1952 Problem 3

$ABC$ is a triangle. The point A' lies on the side opposite to $A$ and $BA'/BC = k$ , where $1/2 < k < 1$ . Similarly, $B'$ lies on the side opposite to $B$ with $CB'/CA = k$ , and $C'$ lies on the side opposite to $C$ with $AC'/AB = k$ . Show that the perimeter of $A'B'C'$ is less than $k$ times the perimeter of $ABC$ .