Yes providing that the face of the door has 4 sides
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The face of a door is a quadrilateral but a door, itself, is a 3-dimensional object and so cannot be a quadrilateral.
a door, window, computer screen. -apex
It is an inscribed quadrilateral or cyclic quadrilateral.
It is simply a quadrilateral - there is no special name for such a quadrilateral.
It is an arrowhead quadrilateral
There is no specific name for such a quadrilateral.