A cyclic quadrilateral is inscribed into a circle whose vertices all lie on a circle. The properties of cyclic quadrilateral in circle geometry are;
- Opposite angles in a cyclic quadrilateral supplementary.
- The exterior angle and its opposite angle are equal.
(a) Prove that \( FADG \) is a cyclic quadrilateral.
\( \begin{aligned} \require{AMSsymbols} \require{color}
\text{Let } \angle BCD &=\theta \\
\angle FAD &= \theta &\color{red} \text{opposite angle’s of a cyclic quadrilateral} \\
\angle FGD &= 180^{\circ}-\theta &\color{red} \text{co-interior angles are supplementary} \\
\angle FAD + \angle FGD &= 180^{\circ} \\
\therefore FADG \text{ is a } &\text{cyclic quadrilateral} &\color{red} \text{opposite angles are supplementary}
\end{aligned} \)
(b) Prove that \( GA \) is a tangent to the circle through \(A\), \(B\), \(C\) and \(D\).
\( \begin{aligned} \require{AMSsymbols} \require{color}
\angle DAG &= \angle DFG &\color{red} \text{angles in the same segment} \\
\angle DFG &= \angle AEF &\color{red} \text{alternate angles in parallel lines} \\
\angle DAG &= \angle AED \\
\therefore GA \text{ is a } &\text{tangent} &\color{red} \text{alternate segment}
\end{aligned} \)
Algebra Algebraic Fractions Arc Binomial Expansion Capacity Common Difference Common Ratio Differentiation Double-Angle Formula Equation Exponent Exponential Function Factorials Factorise Functions Geometric Sequence Geometric Series Index Laws Inequality Integration Kinematics Length Conversion Logarithm Logarithmic Functions Mass Conversion Mathematical Induction Measurement Perfect Square Perimeter Prime Factorisation Probability Product Rule Proof Quadratic Quadratic Factorise Ratio Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume
