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python/数学课程/【辅助圆问题】定高定角.md

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910 B

2 years ago
![](http://dsideal.obs.cn-north-1.myhuaweicloud.com/HuangHai/BlogImages/2023/03/df8cadfe004d592f4dacbce784b24027.png)
- 定角:则做$\triangle BAC$的外接圆
- 既然都有了外接圆,就需要有圆心有半径,设圆心$O$,连接$OA=OB=OC=r$
- $\because \angle BAC=60^{\circ}$
$\therefore \angle BOC=120 ^ \circ$
- 从$O$引$BC$垂线$OE$交$BC$于$E$点
则$\angle EOC=60 ^{\circ},\angle ECO=30 ^{\circ}$
$\therefore OE=\frac{1}{2}r,EC=\frac{\sqrt{3}}{2}r$
$\therefore BC=\sqrt{3}r$
预求$\triangle ABC$的面积最小值,面积表示为
$\frac{1}{2} BC \times AD$
其中$AD$为定高,等于$3$,所以$BC$最小,则面积最小。
也就是$\sqrt{3}r$最小即可,也就是$r$最小。
$OA+OE>=AE>=AD$
$\therefore r+\frac{1}{2}r>=3$
$\therefore r>=2$
$\therefore BC>=2\sqrt{3}$
$S=\frac{1}{2}\times 2\sqrt{3}\times 3=3\sqrt{3}$