四面体ABCD各顶点到对面的距离分别为a,b,c,d体内一点到各个面的距离为a1,b1,c1,d1求证a1\a+b1\b+c1\c+d1\d=1
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![四面体ABCD各顶点到对面的距离分别为a,b,c,d体内一点到各个面的距离为a1,b1,c1,d1求证a1\a+b1\b+c1\c+d1\d=1](/uploads/image/z/8838838-46-8.jpg?t=%E5%9B%9B%E9%9D%A2%E4%BD%93ABCD%E5%90%84%E9%A1%B6%E7%82%B9%E5%88%B0%E5%AF%B9%E9%9D%A2%E7%9A%84%E8%B7%9D%E7%A6%BB%E5%88%86%E5%88%AB%E4%B8%BAa%2Cb%2Cc%2Cd%E4%BD%93%E5%86%85%E4%B8%80%E7%82%B9%E5%88%B0%E5%90%84%E4%B8%AA%E9%9D%A2%E7%9A%84%E8%B7%9D%E7%A6%BB%E4%B8%BAa1%2Cb1%2Cc1%2Cd1%E6%B1%82%E8%AF%81a1%5Ca%2Bb1%5Cb%2Bc1%5Cc%2Bd1%5Cd%3D1)
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四面体ABCD各顶点到对面的距离分别为a,b,c,d体内一点到各个面的距离为a1,b1,c1,d1求证a1\a+b1\b+c1\c+d1\d=1
四面体ABCD各顶点到对面的距离分别为a,b,c,d体内一点到各个面的距离为a1,b1,c1,d1
求证a1\a+b1\b+c1\c+d1\d=1
四面体ABCD各顶点到对面的距离分别为a,b,c,d体内一点到各个面的距离为a1,b1,c1,d1求证a1\a+b1\b+c1\c+d1\d=1
令满足条件的四面体ABCD的内部一点为O,则:
V(A-BCD)=(1/3)S(△BCD)×a、V(O-BCD)=(1/3)S(△BCD)a1,
∴V(O-BCD)/V(A-BCD)=a1/a.······①
同理,有:
V(O-ACD)/V(B-ACD)=b1/b,······②
V(O-ABD)/V(C-ABD)=c1/c,······③
V(O-ABC)/V(D-ABC)=d1/d.······④
显然有:
V(A-BCD)=V(B-ACD)=V(C-ABD)=V(D-ABC),
V(O-BCD)+V(O-ACD)+V(O-ABD)+V(O-ABC)=V(A-BCD),
∴①+②+③+④,得:a1/a+b1/b+c1/c+d1/d=1.