Yes
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No, the circle is inscribed in the quadrilateral.
All geometric shapes can be inscribed in a circle, since the circle is bigger than the other geometric shape inscribed in it. (Obviously)
There are different formula for: Height, Area, Perimeter, Angle, Length of Median Radius of inscribed circle Perimeter of inscribed circle Area of inscribed circle etc.
Yes all inscribed angles in a circle have their vertex on the circumference of the circle. Central angles have their vertex at the center of the circle.
An inscribed circle, possibly.