Translations and Halfturns

The methods of this chapter are from analytic geometry. To emphasize the use of numbers, we often call this approach coordinate geometry. In either case, we start with the translations as they are important and so easily studied by these methods. Later, we shall not hesitate to use the older synthetic methods from the geometry of Euclid when they are most convenient. Then, as we progress, we shall be using more and more group-algebraic methods. At times it may even be hard to say whether a particular part is essentially synthetic, analytic, or algebraic, but this will be completely unimportant.