Cyclic quadrilateral

A cyclic quadrilateral is a quadrilateral that can be inscribed in a circle, meaning all four of its vertices lie on the circle's circumference. The circle is called the circumscribed circle (circumcircle), and its existence imposes powerful angle and length relationships on the quadrilateral.

The signature property: opposite angles are supplementary. If a cyclic quadrilateral has angles A, B, C, D in order, then A + C = 180° and B + D = 180°. This follows from the inscribed angle theorem, since opposite angles subtend arcs that together cover the entire circle. The converse also holds — if a quadrilateral's opposite angles sum to 180°, it must be cyclic — which makes the property a two-way test in proofs.

Cyclic quadrilaterals unlock several classic results. Ptolemy's theorem says the product of the diagonals equals the sum of the products of opposite sides: AC · BD = AB · CD + AD · BC. Brahmagupta's formula gives the area from the side lengths alone: √((s − a)(s − b)(s − c)(s − d)), where s is the semiperimeter. Not every quadrilateral is cyclic — squares, rectangles, and isosceles trapezoids always are, but a general parallelogram is not.

Competition math leans on these facts heavily: the AMC 8, AMC 10, and AMC 12 regularly feature problems where recognizing a cyclic quadrilateral — and applying supplementary opposite angles or Ptolemy's theorem — is the key step.