the midpoints of any quadrilateral are joined in order, they form a parallelogram.
This theorem is due to a French mathematician named Pierre
Varignon (1654-1722). The resulting parallelogram is named in
The diagonals of the
Varignon parallelogram are the bimedians of the original
quadrilateral, the lines which connect the midpoints of the opposite sides.
The point where the bimedians
intersect is the barycenter, centroid,
or center of gravity, the balance point of equal weights placed at
the vertices of the quadrilateral. If the coordinates of the vertices
are known, the coordinates of the intersection of the bimedians would have
an x-coordinate that was the average of the x-coordinates of the vertices,
and similarly for the y-coordinate. Additional information about the centroid
of the general quadrilateral is found at the link to centroid.