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Math Help - Circumcircle

  1. #1
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    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!
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  2. #2
    MHF Contributor Amer's Avatar
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    Quote Originally Posted by Aquafina View Post
    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

    Circumcircle-good2.jpg

    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
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