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 Math Help > Geometry > Polygons and Triangles > Cyclic Quadrilateral

A cyclic quadrilateral is a quadrilateral that can be inscribed in a circle.  In other words, a circle can be drawn that passes through all four vertices of the quadrilateral.  An equivalent condition is that both pairs of opposite angles must add up to 180 degrees.

### Why must both pairs of opposite angles add up to 180 degrees?

Let ABCD be a cyclic quadrilateral.  Since the quadrilateral is cyclic, all its vertices lie on a circle.  So inscribed angles A and angle C together subtend the entire circle.  Since the measure of an inscribed angle is half that of the central angle that is subtended by the same arc, the sum of the measures of angles A and C is 180 degrees.

### Related pages in this website:

Circles and Conic Sections

Circumscribed Triangle -- proves the formula for the circumradius of a triangle.

Ptolemy's Theorem

Brahmagupta's formula