5. Theorems

1. At this point invertible isometries include translations, reflections, and compositions thereof.

2. Our definitions are for arcs which are less than a half circle. With this result we can extend our discussion to arcs that are more than a half circle because any arc which is contained in a full circle can be broken up into two or more arcs which take up less than a half circle.

3. It is possible to prove this analytically. After translating the fixed point of the rotation to the origin, it will suffice to obtain a formula for the image of a point in the plane under a rotation which fixes the origin.

If the x axis is rotated to a line whose slope is m, then a point (x, y) will be rotated to the point

It is fairly reasonable to show that this map preserves distances. However, to show that this is actually the desired rotation would take us too far afield when it is possible to prove that a rotation is a composition of two reflections, which is not only interesting in its own right, but will allow us to conclude that a rotation is an invertible isometry.

At this point we know of three types of isometries: translations, reflections, and rotations. There is one other type of isometry: the glide reflection which is a composition of a translation and a reflection. However, all of the rigid motions of the plane which we will need in this development can be accomplished by composing a translation, rotation, and perhaps a reflection, so we will not need to discuss glide reflections here.

Since translations and rotations can be accomplished by composing two reflections, it follows that any isometry can be accomplished by composing at most three reflections.

Analytic Foundations of Geometry