# Cyclic Quadrilateral in Circle Geometry

A cyclic quadrilateral is inscribed into a circle, whose vertices all lie on a circle. The properties of cyclic quadrilateral in circle geometry are;

1. Opposite angles in a cyclic quadrilateral supplementary.
2. Exterior angle and its opposite angle are equal.

(a)    Prove that $FADG$ is a cyclic quadrilateral.

\begin{aligned} \require{color} \text{Let } \angle BCD &=\theta \\ \angle FAD &= \theta &\color{red} \text{opposite angle’s of a cyclic quadrilateral} \\ \angle FGD &= 180^{\circ} – \theta &\color{red} \text{co-interior angles are supplementary} \\ \angle FAD + \angle FGD &= 180^{\circ} \\ \therefore FADG \text{ is a } &\text{cyclic quadrilateral} &\color{red} \text{opposite angles are supplementary} \\ \end{aligned}

(b)    Prove that $GA$ is a tangent to the circle through $A$, $B$, $C$ and $D$.

\begin{aligned} \require{color} \angle DAG &= \angle DFG &\color{red} \text{angles in the same segment} \\ \angle DFG &= \angle AEF &\color{red} \text{alternate angles in parallel lines} \\ \angle DAG &= \angle AED \\ \therefore GA \text{ is a } &\text{tangent} &\color{red} \text{alternate segment} \\ \end{aligned}