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Inscribed (cyclic) quadrilateral

Definition: A quadrilateral where all four vertices lie on a common circle.
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