Thread: Circumcircle

1. Circumcircle

Hi guys, i was doing a question and this part of it which I didn't get. They basically said that the Law of Sines:

a/sin A = b/sin B = c/sin C = 2R

where R is the radius of the circumcircle (circle around the triangle)

Why does the ratio between the law of sines = the diameter?

Thanks!

2. Originally Posted by Aquafina
Hi guys, i was doing a question and this part of it which I didn't get. They basically said that the Law of Sines:

a/sin A = b/sin B = c/sin C = 2R

where R is the radius of the circumcircle (circle around the triangle)

Why does the ratio between the law of sines = the diameter?

Thanks!

let O be a circumcenter of ABC let the angle BAC be $\alpha$ ok see the picture use law of cosines on the triangle BOC

you will have

$a^2 = R^2 + R^2 -2R(R) \cos 2\alpha$

$a^2 = 2R^2 ( 1-\cos 2\alpha )$

$a^2 = 2R^2 ( 1-(1-2\sin ^2\alpha))$

$a^2 = 4R^2 \sin ^2 \alpha \Rightarrow \frac{a}{\sin \alpha} = 2R$

do the same thing to the other sides you will get

$\frac{a}{\sin \alpha } = \frac{b}{\sin \beta } = \frac{c}{\sin \gamma } = 2R$