**Theorem 4.2**: If *M *is an isometry and *C *is a point on the
line segment between *A *and *B*, then *M*(*C*) is a point on the
line segment between *M*(*A*) and *M*(*B*).

**Proof**: If *C *is a point on the
line segment between *A *and *B*, then

by the third statement in Theorem 4.1.

But since *M *is an isometry

so

and, again, by the third statement in Theorem
4.1, *M*(*C*) is on the line segment between *M*(*A*) and *M*(*B*).