Varignon's theorem: Difference between revisions

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The Varignon parallelogram is a [[rectangle]] if and only if the diagonals of the quadrilateral are [[perpendicular]], that is, if the quadrilateral is an [[orthodiagonal quadrilateral]].<ref name=Josefsson/>{{rp|p. 14}} <ref name=deV />{{rp|p. 169}}

For a [[list of self-intersecting polygons|self-crossing]] quadrilateral, the Varignon parallelogram can degenerate to four collinear points, forming a line segment traversed twice. This happens whenever the polygon is formed by replacing two parallel sides of a [[trapezoid]] by the two diagonals of the trapezoid, such as in the [[antiparallelogram]].<ref>{{citation

If a crossing quadrilateral is formed from either pair of opposite parallel sides and the diagonals of a parallelogram, the Varignon parallelogram is a line segment traversed twice.

| last = Muirhead | first = R. F. | author-link = Robert Franklin Muirhead

| date = February 1901

| doi = 10.1017/s0013091500032892

| journal = Proceedings of the Edinburgh Mathematical Society

| pages = 70–72

| title = Geometry of the isosceles trapezium and the contra-parallelogram, with applications to the geometry of the ellipse

| volume = 20}}</ref>

==See also==