1643

在 ${\displaystyle \mathrm{\Delta }ABC}$ 中, ${\displaystyle D}$ 是 ${\displaystyle BC}$ 边上一点, 且 ${\displaystyle AD=BD}$, ${\displaystyle \mathrm{\angle }DAC=2\mathrm{\angle }ACB}$, ${\displaystyle AB+AC=5}$, 则 ${\displaystyle BC}$ 的最小值为_____.

设 ${\displaystyle \mathrm{\angle }BAD=\mathrm{\angle }ABD=\beta }$, ${\displaystyle \mathrm{\angle }BCA=\alpha }$, ${\displaystyle \mathrm{\angle }DAC=2\alpha }$. 则 ${\displaystyle 3\alpha +2\beta =\pi }$. 记 ${\displaystyle \alpha =2t}$, ${\displaystyle \beta =\frac{\pi }{2}-3t}$. 根据正弦定理 $$\frac{c}{\sin \alpha }=\frac{b}{\sin \beta }=\frac{a}{\sin (\alpha +\beta )},$$ 于是 $$\begin{array}{rl}\frac{5}{a}& =\frac{b+c}{a}=\frac{\sin \alpha +\sin \beta }{\sin (\alpha +\beta )}=\frac{\sin 2t+\cos 3t}{\cos t}\\ & =\frac{2\sin t\cos t+4{\cos }^{3}t-3\cos t}{\cos t}=-4{\sin }^{2}t+2\sin t+1\le \frac{5}{4},\end{array}$$ 取等条件为 ${\displaystyle \sin t=\frac{1}{2}}$ 也即 ${\displaystyle t=\frac{\pi }{6}}$, 所以 ${\displaystyle a\ge 4}$.