before we begin: Angles that are congruent due to easily seen
relationships have been marked in the diagram, but the explanations
have been omitted to save space (and to avoid annoying the reader).
As Barney travels along the parts of his path, PF, IJ, and MN, he covers the same distance as the perimeter of the original triangle, or AB + BC + AC, since PF, IJ, and MN are opposite sides of rectangles from the three sides of the triangle. So, we need to add in the distances he covers as he goes around the yellow portions of this diagram.
Se will start with the right triangles below BC.