Exercise 825 If t12 and g are three distinct parallel lines

Exercise 8.25. If t1,(2, and (g are three distinct parallel lines, prove that F3 o F2o Fi is a reflection (a glide reflection with translation of length 0)

Solution

Let F = F₃ F₂ F₁.

First, suppose that F₂ F₁ is a rotation with center O. Let l₃ be a line fixed by F₃ and let l\'₂ be a line through O parallel to l₃. We can find a line l\'₁ through O such that F₂ F₁ = F\'₂ F\'₁ (where F\'_i is a reflection with respect to l\'_i ).

Then F = F₃ (F\'₂ F\'₁) = (F₃ F\'₂) F\'₁ is a composition of a reflection and a translation (as l\'₂ is parallel to l₃) by some vector v. Now, moving the vector v along the plane (i.e. applying all possible translations to it) we may place it so that the midpoint of v belongs to l\'₁. Let l be the line parallel to l₁ and passing through the head of the vector v. Then l is preserved by F (not point-wise). Moreover, F restricted to l does not change its orientation. So, it is a glide reflection.

Finally, suppose that F₂ F₁ is a translation. Then F is a composition of a translation and a reflection and may be treated exactly as above (the only difference is that the line fixed by F now passes through the tail of v).