# The Cyclic Quadrilateral

A quadrilateral is said to be cyclic if its vertices all lie on a circle. In cyclic quadrilateral, the sum of two opposite angles is 180° (or π radian); in other words, the two opposite angles are supplementary.

$A + C = 180^\circ$

$B + D = 180^\circ$

The area of cyclic quadrilateral is given by

$A = \sqrt{(s - a)(s - b)(s - c)(s - d)}$

See the derivation of area of cyclic quadrilateral for profound details.

**Ptolemy's Theorem for Cyclic Quadrilateral**

For any cyclic quadrilateral, the product of the diagonals is equal to the sum of the products of non-adjacent sides. In other words

$d_1 d_2 = ac + bd$

See the proof of Ptolemy's theorem for cyclic quadrilateral.