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

|AB| = |AC| + |CB|

by the third statement in Theorem 4.1.

But since   M   is an isometry

|M(A)M(B)| = |AB|,   |M(A)M(C)| = |AC|,   and |M(C)M(B)| = |CB|

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)