
Parallel lines theorem
Let m and n be two parallel straight lines. Let AB be a line segment that first intersects m and then n, that is perpendicular to both m and n, and whose length is twice the distance between m and n. Then, $\displaystyle \rho_{n} \circ \rho_{m} = \tau_{AB}$; the compositions of the two reflections is a translation.
Once again, I'm clueless as to proving this.

Well, thet composition of two indirect isometries is a direct isometry...
A direct isometry can only be a rotation or translation....show the composition cannot be a rotation