MT【290】内外圆求三角最值

求$\sqrt{\dfrac{5}{4}-\sin x}+2\sqrt{\dfrac{9}{4}+\cos x-\sin x}$的最小值.

提示:
$\sqrt{\dfrac{5}{4}-\sin x}+2\sqrt{\dfrac{9}{4}+\cos x-\sin x}$

$=\sqrt{(\dfrac{1}{2}\cos x)^2+(1-\dfrac{1}{2}\sin x)^2}+2\sqrt{(\dfrac{1}{2}\cos x+1)^2+(\dfrac{1}{2}\sin x-1)^2}$
令$A(-1,1),B(0,1),D(0,\dfrac{1}{4}),C(\dfrac{1}{2}cos x,\dfrac{1}{2}sin x)$
则由内外圆知识$|BC|+2|AC|=2(|DC|+|AC|)\ge 2AD=\dfrac{5}{2}$

练习:MT 【191】阿波罗尼乌兹圆