Proof of the sine rule

The proof uses the chord theorem applied to the circle circumscribed about the triangle.

Proof — With the chord theorem

Consider the circle circumscribed about the triangle. The side $a$ is a chord of this circle, and the opposite angle $\widehat{A}$ is the inscribed angle subtending the arc $BC$. By the chord theorem, $a = 2R\sin\widehat{A},$ whence $a/\sin\widehat{A} = 2R$. Likewise for $b$ and $c$: all three ratios equal $2R$. $\blacksquare$

Links

Topics: Triangle trigonometry
Concepts: Circumscribed circle · Sine rule · Chord theorem
Functions: Sine
Skills: Proving