弦外与圆内角的测量

Blurb/stub article

If a chord $AB$ "intercepts" an (inscribed) angle $\angle{ACB}$ of measure $\theta$, then minor arc $\widehat{AB} = \min(2\theta,2\pi-2\theta)$ (by this in the "normal chord case" and this in the "tangent case"), so $\frac{AB/2}{R} = \sin(\widehat{AB}/2) = \sin\theta$.

In other words, $AB = 2R\sin\theta$.

So, you can find chord length from the central, inscribed, or circumscribed angle. You can also go in reverse from the angle to the chord length. Either way, this angle-chord relationship can be used to prove the extended law of sines.