**Theorem 5.3**:
Arc length is preserved by
invertible
isometries.

**Proof**: Let *A *be an arc,
and let {*P*_{0}, *P*_{1}, *P*_{2}, . . . ,
*P*_{n}} be a partition
of the arc* A*. Let *O* be the center of the
circle which contains *A*, and let *M *be an invertible
isometry. Then, *P*_{i }is in the
angle between *OP*_{i-1 }and *OP*_{i+1} which is to say that* P*_{i} is on the
same side of *OP*_{i-1 }as *P*_{i+1} and on the same side of
* OP*_{i+1} as *P*_{i-1}. By
Theorem 4.3 *M*(*P _{i}*) will
be on the same
side of the line
determined by

Since the lengths of all the partitions will be preserved, the limits will be the same.