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[几何] 难题求助!1立体几何求体积最大值

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word3000 posted 2017-1-19 07:51 |Read mode
三棱锥A-BCD中,AD=6,BC=2,AB+BD=AC+CD=7,求VA-BCD的最大值

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realnumber posted 2017-1-19 09:12
取AD中点M,可以用解析几何中办法证明BM垂直AD时最大,
同样在三角形CDB中也是一样,此时$BM=CM=\frac{\sqrt{13}}{2}$.
此时$V_{ABCD}=\frac{1}{3}AD\times S_{ΔMBC}=3$.
以下说明此时ΔMBC是最大的,以B,C为焦点,BC=2,满足$MB+MC=\sqrt{13}$的M点在平面内是椭圆C,而当MB=CM时,M为短轴端点,ΔMBC面积最大.
当$MB+MC=2a_{0}<\sqrt{13}$,$a_0$为某正常数时,其轨迹是椭圆C内的某个椭圆.
三角形MBC面积不会大于之前的情景.

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original poster word3000 posted 2017-1-19 20:02
谢谢

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