---
tag: 0CMC2KAXFS
kind: definition
name: Cyclic quadrilateral
msc: [51M04]
defines: 0RCP929Y1M
uses: [049DCJZ448, 03VZJE6C8C]
sources:
  - url: https://en.wikipedia.org/wiki/Cyclic_quadrilateral
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Cyclic_quadrilateral
    locator: Supplementary angles
---

## Definition

Let $ABCD$ be a quadrilateral in the Euclidean plane $\mathbb{R}^2$, with the vertices $A$, $B$, $C$, $D$ in this order.
The quadrilateral $ABCD$ is cyclic if its four vertices are [[concyclic|049DCJZ448]]. The circle through them is its [[circumcircle|03VZJE6C8C]].

## Remarks

A convex quadrilateral $ABCD$ is cyclic if and only if its opposite angles add up to $\pi$: $\angle DAB + \angle BCD = \pi$. The direct part is Proposition 22 of Book III of Euclid's Elements.
Each rectangle is cyclic. A rhombus that is not a square is not cyclic.
Usually the quadrilateral is taken to be convex. A crossed quadrilateral with concyclic vertices is a crossed cyclic quadrilateral.

## Explanations

### L1 human

Each triangle fits in a circle, but a quadrilateral does not always. A quadrilateral is cyclic when one circle passes through all four corners. A simple test: the angles at two opposite corners add up to a straight angle.