KöMaL 2013 C1167

The line segments $AA_1$, $BB_1$, $CC_1$ are the angle bisectors of a triangle ABC . Through the endpoint of the bisector of each angle, draw parallels to the lines forming the angle. Consider the segments of these six lines that lie in the interior of the triangle. Prove that the sum of their lengths cannot be longer than the perimeter of the triangle.