Now assume A is an isometry on L. What could A be?
There are two cases:
- A has a fixed point.
- A does not have a fixed point.
There are only two such things! Either B is the identity, or B is v↦−v. In the first case, A is the identity on L. In the second case, A is reflection in the point P on L.
Now consider the case where A has no fixed point. Take any point P on L, and let S be reflection in the point midway between P and A(P). The map S∘A clearly has P as a fixed point. Call it A′. According to the reasoning above, A′ is either the identity or a reflection in some point. But A′ cannot be the identity, because then A would be S, and P would be a fixed point, but A had no fixed points. So A′ must be a reflection in some other point. Call it S′. We have S∘A=S′A=S∘S′ So A is the composition of two reflections. Some simple considerations will show that A is a translation on L.
In conclusion, the kinds of isometries on a euclidean line are:
- The identity
- Reflections
- Compositions of two reflections (these are translations)