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April 23[edit]

Varignon parallelogram and triangles[edit]

Given any quadrilateral, the midpoints of its four sides always form a parallelogram, called the Varignon parallelogram. Does the Varignon parallelogram exist even for "quadrilaterals" that look like triangles in either of the following two ways?

Two coincident vertices forming a side of length zero
Three collinear vertices

For the first case, one has that the midpoints of the three sides together with the three vertices of the triangle should form three parallelograms. For the second case, one has a side of a triangle divided into two parts, and the midpoints of those two parts together with the midpoints of the other two sides should then also form a parallelogram. GeoffreyT2000 (talk) 19:31, 23 April 2021 (UTC)[reply]

If you form a crossing quadrilateral from a trapezoid by taking the vertices which alternate between the parallel sides, then the Varignon parallelogram will be a line segment. If you form a degenerate quadrilateral by alternating between two points, then the Varignon parallelogram will be a single point. Other than these degenerate cases you get a parallelogram. I think it's a matter of how strict you are with the definitions whether these degenerate cases would still count anyaway. In any case, the configurations you gave would still produce Varignon parallelograms; there's nothing that would break the proof given in the article. --RDBury (talk) 21:26, 23 April 2021 (UTC)[reply]