The Sum of the Perimeters of Three Subtriangles

Let ABC be a triangle. Let A'A'', B'B'', and C'C'' be tangents to the incircle of ABC and parallel to BC, AC, and AB, respectively. Let be the perimeter of ABC and , and be the perimeters of AA'A'', BB'B'', and CC'C'', respectively. Then . Also, opposite sides of the hexagon A'A''B'B''C'C'' are equal, that is, A'A'' = C'B'', B'B'' = A'C'', and C'C'' = A''B'.