Cyclic quadrilateral

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. The circle through them is its circumcircle.

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.