A bisect-diagonal quadrilateral is a convex quadrilateral in which at least one polygon diagonal is bisected by the other (Josefsson 2017, p. 214). Every kite and every parallelogram is a bisect-diagonal quadrilateral. In a parallelogram, both diagonals bisect each other.
A polygon diagonal of a convex quadrilateral bisects the other polygon
diagonal iff it divides the area
into two equal parts (Josefsson 2017, p. 215). Equivalently, for such a quadrilateral 
,
let 
be the intersection of its diagonals,
and let 
,
,
,
and 
be the areas of the triangles 
,
,
,
and 
,
respectively. Then 
is bisect-diagonal iff
 |
The bimedians of a convex quadrilateral intersect on one of its diagonals iff it is bisect-diagonal (Josefsson 2017, p. 216). Since the bimedians are the diagonals of the Varignon parallelogram, this is equivalent to the center of the Varignon parallelogram lying on a polygon diagonal of the original quadrilateral.
