Quadrilaterals with Varignon Parallelogram

The applet below demonstrates the quadrilaterals that can be formed given a specific Varignon parallelogram. Since three of the vertices of WXYZ depend on the fourth, the sliders represent the independent vertex. Move the sliders to see the possible quadrilaterals that give the Varignon parallelogram ABCD.

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1. If a quadrilateral is not convex, can it still have a Varignon parallelogram?
2. How many different quadrilaterals can be formed that have the same Varignon parallelogram?

Mindy Burcham, Created with GeoGebra