  It is well known that a rigid motion of the Euclidean plane can be written as
the composition of at most three reflections. It is perhaps not so widely known
that a similar result holds for Euclidean space in any number of dimensions.
The purpose of the present article is, firstly, to present a natural proof of
this result in dimension 3 by explicitly constructing a suitable sequence of
reflections, and, secondly, to show how a careful analysis of this construction
provides a quick and pleasant geometric path to Euler's rotation theorem, and
to the complete classification of rigid motions of space, whether orientation
preserving or not. Finally, we present an example where we use the general
scheme of our proofs to classify the composition of two explicitly given
orientation preserving isometries. We believe that our presentation will
highlight the elementary nature of the results and hope that readers, perhaps
especially those more familiar with the usual linear algebra approach, will
appreciate the simplicity and geometric flavour of the arguments.
