Proposition: The segment joining the midpoints of two sides of a triangle is parallel to the third side and is 1/2 of the measure of the third side.

A vector proof:

Suppose the two side correspond to the position vectors, v and w. so the midpoints correspond to the vectors 1/2 v and 1/2 w.

Since z is a scalar multiple of w-v, it is parallel to w-v, the vector determined by the third side of the triangle.

The scalar is 1/2 , so z has a length 1/2 of the vector w-v. IRMC.