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

|A, B| = |A, C| + |C, B|

by the third statement in Theorem 4.1.

But since M is an isometry

|M(A), M(B)| = |A, B|, |M(A), M(C)| = |A, C|, and |M(C), M(B)| = |C, B|

so

|M(A), M(B)| = |M(A), M(C)| + |M(C), M(B)|

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

next theorem (4.3)