Circle Theorems: Cyclic Quadrilaterals

Definition of Cyclic Quadrilateral

A cyclic quadrilateral is a four-sided polygon where all four vertices lie on the circumference of a circle. This circle is called the circumcircle of the quadrilateral.

Key property: The opposite angles of a cyclic quadrilateral always add up to $180^\circ$ (they are supplementary).

Properties of Cyclic Quadrilaterals

Sum of opposite angles: If a quadrilateral is cyclic, then the sum of each pair of opposite angles is $180^\circ$. For example, if the quadrilateral has angles $A$, $B$, $C$, and $D$ in order around the circle, then:

$$A + C = 180^\circ \quad \text{and} \quad B + D = 180^\circ$$

Exterior angle property: The exterior angle of a cyclic quadrilateral is equal to the opposite interior angle. For example, if you extend one side of the quadrilateral, the angle formed outside the quadrilateral equals the interior angle opposite it.

Diagonals and the circle: The diagonals of a cyclic quadrilateral intersect inside the quadrilateral (which lies inside the circle). There is no fixed length relationship between the diagonals, but their intersection helps define the shape inside the circumcircle.

Proofs and Applications

You can use the supplementary angle property to prove whether a quadrilateral is cyclic. If you can show that one pair of opposite angles adds to $180^\circ$, then the quadrilateral must be cyclic.

In problem-solving, identifying cyclic quadrilaterals allows you to apply these angle properties to find unknown angles or prove other geometric facts.

When looking at diagrams, check if all vertices of the quadrilateral lie on the circumference of the same circle. If so, use the cyclic quadrilateral properties to solve angle problems.

For instance, if you know three angles of a cyclic quadrilateral, you can find the fourth by using the fact that opposite angles sum to $180^\circ$. For example, if angles $A = 70^\circ$, $B = 110^\circ$, and $C = 110^\circ$, then since $A + C = 180^\circ$, the quadrilateral is cyclic and $B + D = 180^\circ$, so $D = 70^\circ$.