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24 lines
910 B
24 lines
910 B
2 years ago
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- 定角:则做$\triangle BAC$的外接圆
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- 既然都有了外接圆,就需要有圆心有半径,设圆心$O$,连接$OA=OB=OC=r$
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- $\because \angle BAC=60^{\circ}$
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$\therefore \angle BOC=120 ^ \circ$
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- 从$O$引$BC$垂线$OE$交$BC$于$E$点
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则$\angle EOC=60 ^{\circ},\angle ECO=30 ^{\circ}$
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$\therefore OE=\frac{1}{2}r,EC=\frac{\sqrt{3}}{2}r$
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$\therefore BC=\sqrt{3}r$
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预求$\triangle ABC$的面积最小值,面积表示为
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$\frac{1}{2} BC \times AD$
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其中$AD$为定高,等于$3$,所以$BC$最小,则面积最小。
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也就是$\sqrt{3}r$最小即可,也就是$r$最小。
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$OA+OE>=AE>=AD$
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$\therefore r+\frac{1}{2}r>=3$
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$\therefore r>=2$
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$\therefore BC>=2\sqrt{3}$
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$S=\frac{1}{2}\times 2\sqrt{3}\times 3=3\sqrt{3}$
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