Quadrilateral - Bimedians

Bimedians

The midpoints of the sides of a quadrilateral are the vertices of a parallelogram called the Varignon parallelogram. The sides in this parallelogram are half the lengths of the diagonals of the original quadrilateral, the area of the Varignon parallelogram equals half the area of the original quadrilateral, and the perimeter of the Varignon parallelogram equals the sum of the diagonals of the original quadrilateral. The diagonals of the Varignon parallelogram are the bimedians of the original quadrilateral.

The two bimedians in a quadrilateral and the line segment joining the midpoints of the diagonals in that quadrilateral are concurrent and are all bisected by their point of intersection.

In a convex quadrilateral with sides a, b, c and d, the length of the bimedian that connects the midpoints of the sides a and c is

where p and q are the length of the diagonals. The length of the bimedian that connects the midpoints of the sides b and d is

Hence

This is also a corollary to the parallelogram law applied in the Varignon parallelogram.

The length of the bimedians can also be expressed in terms of two opposite sides and the distance x between the midpoints of the diagonals. This is possible when using Euler's quadrilateral theorem in the above formulas. Whence

and

Note that the two opposite sides in these formulas are not the two that the bimedian connects.

In a convex quadrilateral, there are the following dual connection between the bimedians and the diagonals:

• The two bimedians have equal length if and only if the two diagonals are perpendicular
• The two bimedians are perpendicular if and only if the two diagonals have equal length