Sine Rule (Proof)

Proof:
From the Trigonometric Formula for Areas of Triangles,
$$\frac{1}{2}bc\sin A = \frac{1}{2}ac\sin B = \frac{1}{2}ab\sin C$$
$$bc\sin A = ac\sin B = ab\sin C$$
$$\frac{{bc\sin A}}{{abc}} = \frac{{ac\sin B}}{{abc}} = \frac{{ab\sin C}}{{abc}}$$
$$\frac{{\sin A}}{a} = \frac{{\sin B}}{b} = \frac{{\sin C}}{c}$$
Thus,
$$\frac{a}{{\sin A}} = \frac{b}{{\sin B}} = \frac{c}{{\sin C}}$$
where $\angle A$ is the angle opposite the side $a$, $\angle B$ is the angle opposite the side $b$ and $\angle C$ is the angle opposite the side $c$.