On subdivisions of polygons

By a subdivision of a polygon, we mean an orthogonal net such that the vertices of the polygon are nodes of the net, and the edges are composed of diagonals and sides of its cells. We study the subdivisions of convex polygons in which all edges have only diagonal directions. Such a polygon has four supporting vertices disposed on the different sides of the described rectangular. From each nonsupporting vertex, toward the interior of the polygon, there emanates a pair of broken lines in the directions of the orthogonal net. After a finite number of reflections in the boundary (the sum of the incidence and reflection angles is equal to $90^\circ$), the broken lines of such a pair can either get stuck at the supporting vertices or meet each other and form a closed orbit. It is proved that in the case of the pentagon, the second variant is not possible.