The applet below is a quick proof of the law of sines. The vertices [math]A, [/math] [math]B,[/math] and [math]C[/math] can all be moved to show that the proof works for any triangle. Click on any of the equations to show the right triangle that supports it. The explanation is below the applet.

From the applet above, note that [math]a \sin C = c \sin A = h_1[/math]. [br]Divide both sides of the equation [math]a \sin C = c \sin A [/math] by [math]ac.[/math][br][br][center][math]\begin{align} \frac{a \sin C}{ac}=& \frac{c \sin A}{ac} \\[br]\frac{\sin C}{c}=& \frac{ \sin A}{a} \end{align}[/math][/center][br]Similarly, note that [math]a \sin B = b \sin A = h_2[/math]. [br]Divide both sides of the equation [math]a \sin B = b \sin A [/math] by [math]ab.[/math][br][br][center][math]\begin{align} \frac{a \sin B}{ab}=& \frac{b \sin A}{ab} \\[br]\frac{\sin B}{b}=& \frac{ \sin A}{a} \end{align}[/math][/center][br]From the equations [math]\frac{\sin C}{c}=\frac{\sin A}{a}[/math] and [math]\frac{\sin b}{b}=\frac{\sin A}{a}[/math], we conclude that [br][br][center][math]\Large \frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}.[/math][/center]Note that this is true for obtuse as well as acute angles because the sine of an angle [math]\theta[/math] is equal to the sine of the supplement of [math]\theta,[/math] or [math]\sin \left(180^{\circ} - \theta \right) = \sin \theta.[/math]