Cyclic Quadrilaterals

Opposite angles supplementary, the concyclicity test, and the exterior angle.

The idea

A quadrilateral is cyclic when all four of its vertices lie on a single circle. Four points chosen at random do not, so the condition is a genuine restriction, and it constrains the angles: in a cyclic quadrilateral $ABCD$ the opposite angles are supplementary,

$A + C = 180^{\circ}, \qquad B + D = 180^{\circ}.$

The reason is that each angle of the quadrilateral is an inscribed angle of the circle, so it measures half the arc it subtends. The angle at $A$ subtends the arc from $B$ through $C$ to $D$, and the angle at the opposite vertex $C$ subtends the remaining arc. Together the two arcs make up the whole circle, and half of $360^{\circ}$ is $180^{\circ}$.

The converse also holds: if a convex quadrilateral has one pair of opposite angles summing to $180^{\circ}$, then its four vertices lie on a circle. Two angles therefore certify that four points are concyclic, without producing the circle.

One consequence follows immediately. Extend a side past a vertex; the exterior angle there is $180^{\circ}$ minus the interior angle, which equals the interior angle at the opposite vertex.

Ways to work on it

• Walkthrough. Why opposite angles of an inscribed quadrilateral sum to 180^ .
• Practice. Recover a missing opposite or exterior angle.
• Hardest. Decide whether four points lie on a circle, then find a further angle.

Not sure where to start? Take the ten-question placement test.