直接计算即可

2022-12-26 18:02 作者:数学老顽童 0人读过 | 我要投稿

（2022全国甲，16）已知 $%5Cbigtriangleup%20ABC$ 中，点 $D$ 在边 $BC$ 上， $%5Cangle%20ADB%3D120%5E%5Ccirc%20$ ， $AD%3D2$ ， $CD%3D2BD$ .当 $%5Cfrac%7BAC%7D%7BAB%7D$ 取得最小值时， $BD%3D$ ________.

解：设 $CD%3D2BD%3D2m$ ，

易知 $%5Cangle%20ADC%3D60%5E%5Ccirc%20$ ，所以

$%5Cbegin%7Baligned%7D%0A%09%5Cfrac%7BAC%7D%7BAB%7D%26%3D%5Cfrac%7B%5Csqrt%7B%5Cleft(%202m%20%5Cright)%20%5E2%2B2%5E2-2%5Ccdot%202m%5Ccdot%202%5Ccdot%20%5Ccos%2060%5E%5Ccirc%20%7D%7D%7B%5Csqrt%7Bm%5E2%2B2%5E2-2%5Ccdot%20m%5Ccdot%202%5Ccdot%20%5Ccos%20120%5E%5Ccirc%20%7D%7D%5C%5C%0A%09%26%3D%5Cfrac%7B%5Csqrt%7B4m%5E2%2B4-4m%7D%7D%7B%5Csqrt%7Bm%5E2%2B4%2B2m%7D%7D%5C%5C%0A%09%26%3D2%5Csqrt%7B1-%5Cfrac%7B3%7D%7Bm%2B1%2B%5Cfrac%7B3%7D%7Bm%2B1%7D%7D%7D%5C%5C%0A%09%26%5Cgeqslant%202%5Csqrt%7B1-%5Cfrac%7B3%7D%7B2%5Csqrt%7B3%7D%7D%7D%3D%5Csqrt%7B3%7D-1%5C%5C%0A%5Cend%7Baligned%7D$