There is this sample question from my book that I dont know how to go about. please help out
Use the three reflections theorem to show that the only transformations of the Euclidean plane are translations, rotations, reflections and glide transformations.
The three isometry theorem states that any isometry of the plane can be written as the product of at most 3 reflections
Depending on what you book has covered, you should know
A. Composition of 2 isometries is an isometry.
B. Composition of reflections: $T_m T_n = (T_n T_m)^{-1}$
C. Classifying fixed point sets of isometry (must be linear subspace).
D. Classifying orientation preserving of isometry.
E. Classifying translations, reflections, rotations, glide reflections uniquely by existence of fixed point and preservation of isometries.
Show the following:
1. The isometry written as the product of 1 reflection is a reflection.
2. The isometry written as the product of 2 parallel reflections is a translation.
3. The isometry written as the product of 2 intersecting reflections is a rotation.
4. The isometry written as the product of 3 all parallel reflections is a reflection.
5. The isometry written as the product of 3 not all parallel reflections is a glide reflection.