Prove that the midpoints of the sides of an arbitrary quadrilateral are the vertices of a parallelogram.

Let an arbitrary quadrilateral ABCD be given with the denoted midpoints of the sides M, N, K, E.

Prove that MNKE is a parallelogram.

Proof: Given that points M, N, K and E are the midpoints of the corresponding sides AB, BC, CD, AD, then MN, NK, KE, ME are the midlines of the corresponding triangles FBC, BCD. Following from this MN, KE are parallel to the AC and to each other.

We argue similarly with the pair of lines ME, NK. They are parallel to the side BD, and therefore parallel to each other.

Thus, in the resulting quadrilateral MNKE, the sides MN and KE, as well as KN and ME, are parallel to each other. And such a quadrilateral is called a parallelogram.