A cyclic quadrilateral is one where the sum of measures of opposite angles is 180 degrees.
I t mostly is formed with the vertices as part of the circumference of a circle.
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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