Cao Yi

正弦定理 Law of Sines

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设$\mathrm{△}ABC$的三个角$\mathrm{\angle }A$, $\mathrm{\angle }B$和$\mathrm{\angle }C$对应的三条边是a, b和c, R是三角形的外接圆的半径，则

${\displaystyle \frac{a}{\sin A}}={\displaystyle \frac{b}{\sin B}}={\displaystyle \frac{c}{\sin C}}=2R$

证明

如下图，直线AO过圆心O交圆周于D，

根据圆周角的性质，

$\mathrm{\angle }ACB=\mathrm{\angle }ADB$, $\mathrm{\angle }ABD={\displaystyle \frac{\pi }{2}}$

$\therefore |AB|=|AD|\sin \mathrm{\angle }ADB=|AD|\sin \mathrm{\angle }ACB=2R\sin \mathrm{\angle }ACB$

即 $c=2R\sin C$

$\therefore {\displaystyle \frac{c}{\sin C}}=2R$

同理可证

$\therefore {\displaystyle \frac{a}{\sin A}}=2R$

$\therefore {\displaystyle \frac{b}{\sin B}}=2R$

$\therefore {\displaystyle \frac{a}{\sin A}}={\displaystyle \frac{b}{\sin B}}={\displaystyle \frac{c}{\sin C}}=2R$

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参考

• Law of Sines, Wikipedia

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