

Inscribed (cyclic) quadrilateral
Try this Drag any orange dot.
Note how the four
vertices
of the quadrilateral always lie on the circle.
An inscribed, or cyclic,
quadrilateral
is one where all the four
vertices
lie on a common circle.
Another way to say it is that the quadrilateral is 'inscribed' in the circle. Here, inscribed means to 'draw inside'.
In the figure above, as you drag any of the vertices around the circle the quadrilateral will change.
Note that if you drag a vertex past an adjacent one, the quadrilateral will be 'crossed'.
It will have one side that crosses over another.
As with all polygons, this is not regarded as a valid quadrilateral,
and most theorems and properties described below do not hold for them.
Interior angles
In a cyclic quadrilateral, opposite pairs of interior angles are always
supplementary  that is, they always add to 180°.
For more on this see
Interior angles of inscribed quadrilaterals.
Area
If you know the four sides lengths,
you can calculate the area of an inscribed quadrilateral using a formula very similar to
Heron's Formula.
For more see
Area of an inscribed quadrilateral.
Diagonals
It turns out there is a relationship between the side lengths and the diagonals of a cyclic quadrilateral.
For more see
Diagonals of an inscribed quadrilateral.
Other properties
There are numerous other properties of cyclic figures.
Wikipedia has a large selection.
Other polygon topics
General
Types of polygon
Area of various polygon types
Perimeter of various polygon types
Angles associated with polygons
Named polygons
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