Four points lying on one circle

Four points $A$, $B$, $C$, and $D$ are concyclic if and only if there exists a circle passing through all of them. This is equivalent to saying that the points all lie on the circumference of the same circle.

Important properties

• If four points are concyclic, the sum of the measures of the opposite angles in the quadrilateral they form is $180^ ext{o}$.

• Given any three non-collinear points, there is always a unique circle passing through them. For four points, this is not always possible.

• The condition for four points $A$, $B$, $C$, $D$ to be concyclic is that $ABCD$ is a cyclic quadrilateral.