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.

Contents of this section: Cyclic Quadrilateral Circumscribed Cyclic Q. Brahmagupta's Formula Ptolemy's Theorem Butterfly Theorem Brianchon's Hexagon Theorem Pascal's Hexagon Theorem Pappus Theorem

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:

Summary of geometrical theorems 

Circles and Conic Sections

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

Ptolemy's Theorem

Brahmagupta's formula

Triangle Home Page

The webmaster and author of this Math Help site is Graeme McRae.