first you must define "quadrilateral" then you must see the word plain then the defined explanation of "quadrilateral" or if you're lazy then the answer is a plain good old fashioned square. :3
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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.
No, a square is an equilateral quadrilateral. http://en.wikipedia.org/wiki/Quadrilateral