You can see now the usefulness of proving Problem 18.2 . Problem 18.2 . If you have trouble getting started with this problem, then take a specific example such as the two congruent figures, F and G , at the start of this chapter and label three non-collinear points, A, B, C . Pick another point X and convince yourself as concretely as possible that any sequence of isometries that takes F onto G must always take X to the same location. Problem 18.3 .
Every isometry is the composition of one, two, or three reflections .
Every composition of two reflections is either a translation, a rotation, or the identity. Every composition of three reflections is either a reflection or a glide reflection. A glide reflection is defined as a reflection through a line l composed with a translation along the same l . Every isometry of the plane or a sphere or a hyperbolic plane is either a reflection, a translation, a rotation, a glide reflection, or the identity . Problem 18.2 allows you to be more concrete when thinking about Problem 18.3 because you only need to look at the effect of the isometries on any three non-collinear points that you pick.