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# [讨论] 20多年了，我无力解出来的一道高中奥数题！

 根据前面的讨论我们知体积最大时\$P\$点为四面体\$ABCD\$的垂心： 又由于垂心四面体存在的充要条件为：三组对棱的平方和相等，我们可以设\$PA=a,PB=b,PC=c,PD=d,AB=z,AC=y,BC=x,BD=sqrt(t^2-y^2),AD=sqrt(t^2-x^2),CD=sqrt(t^2-z^2)\$ 可以得到： -a^8*z^4-2*a^6*b^2*x^2*z^2+2*a^6*b^2*y^2*z^2+2*a^6*b^2*z^4+2*a^6*c^2*x^2*z^2-2*a^6*c^2*y^2*z^2+2*a^6*c^2*z^4+2*a^6*x^2*z^4+2*a^6*y^2*z^4-2*a^6*z^6-a^4*b^4*x^4+2*a^4*b^4*x^2*y^2+2*a^4*b^4*x^2*z^2-a^4*b^4*y^4-4*a^4*b^4*y^2*z^2-a^4*b^4*z^4+2*a^4*b^2*c^2*x^4-4*a^4*b^2*c^2*x^2*y^2+2*a^4*b^2*c^2*x^2*z^2+2*a^4*b^2*c^2*y^4+2*a^4*b^2*c^2*y^2*z^2-4*a^4*b^2*c^2*z^4+2*a^4*b^2*x^4*z^2+2*a^4*b^2*x^2*y^2*z^2-4*a^4*b^2*x^2*z^4-4*a^4*b^2*y^4*z^2+2*a^4*b^2*y^2*z^4+2*a^4*b^2*z^6-a^4*c^4*x^4+2*a^4*c^4*x^2*y^2-4*a^4*c^4*x^2*z^2-a^4*c^4*y^4+2*a^4*c^4*y^2*z^2-a^4*c^4*z^4-4*a^4*c^2*x^4*z^2+2*a^4*c^2*x^2*y^2*z^2+2*a^4*c^2*x^2*z^4+ 2*a^4*c^2*y^4*z^2-4*a^4*c^2*y^2*z^4+2*a^4*c^2*z^6+2*a^4*d^2*x^4*z^2-4*a^4*d^2*x^2*y^2*z^2-4*a^4*d^2*x^2*z^4+2*a^4*d^2*y^4*z^2-4*a^4*d^2*y^2*z^4+2*a^4*d^2*z^6+2*a^4*t^2*x^4*z^2-4*a^4*t^2*x^2*y^2*z^2-4*a^4*t^2*x^2*z^4+2*a^4*t^2*y^4*z^2-4*a^4*t^2*y^2*z^4+2*a^4*t^2*z^6-a^4*x^4*z^4+4*a^4*x^2*y^2*z^4+2*a^4*x^2*z^6-a^4*y^4*z^4+2*a^4*y^2*z^6-a^4*z^8-2*a^2*b^6*x^2*y^2+2*a^2*b^6*y^4+2*a^2*b^6*y^2*z^2+2*a^2*b^4*c^2*x^4+2*a^2*b^4*c^2*x^2*y^2-4*a^2*b^4*c^2*x^2*z^2-4*a^2*b^4*c^2*y^4+2*a^2*b^4*c^2*y^2*z^2+2*a^2*b^4*c^2*z^4+2*a^2*b^4*x^4*y^2-4*a^2*b^4*x^2*y^4+2*a^2*b^4*x^2*y^2*z^2+2*a^2*b^4*y^6+2*a^2*b^4*y^4*z^2-4*a^2*b^4*y^2*z^4-4*a^2*b^2*c^4*x^4+2*a^2*b^2*c^4*x^2*y^2+2*a^2*b^2*c^4*x^2*z^2+2*a^2*b^2*c^4*y^4-4*a^2*b^2*c^4*y^2*z^2+ 2*a^2*b^2*c^4*z^4-2*a^2*b^2*c^2*x^6+2*a^2*b^2*c^2*x^4*y^2+2*a^2*b^2*c^2*x^4*z^2+2*a^2*b^2*c^2*x^2*y^4-12*a^2*b^2*c^2*x^2*y^2*z^2+2*a^2*b^2*c^2*x^2*z^4-2*a^2*b^2*c^2*y^6+2*a^2*b^2*c^2*y^4*z^2+ 2*a^2*b^2*c^2*y^2*z^4-2*a^2*b^2*c^2*z^6+2*a^2*b^2*d^2*x^6-6*a^2*b^2*d^2*x^4*y^2-6*a^2*b^2*d^2*x^4*z^2+6*a^2*b^2*d^2*x^2*y^4+4*a^2*b^2*d^2*x^2*y^2*z^2+6*a^2*b^2*d^2*x^2*z^4-2*a^2*b^2*d^2*y^6+2*a^2*b^2*d^2*y^4*z^2+2*a^2*b^2*d^2*y^2*z^4-2*a^2*b^2*d^2*z^6+2*a^2*b^2*t^2*x^6-6*a^2*b^2*t^2*x^4*y^2-6*a^2*b^2*t^2*x^4*z^2+6*a^2*b^2*t^2*x^2*y^4+4*a^2*b^2*t^2*x^2*y^2*z^2+ 6*a^2*b^2*t^2*x^2*z^4-2*a^2*b^2*t^2*y^6+2*a^2*b^2*t^2*y^4*z^2+2*a^2*b^2*t^2*y^2*z^4-2*a^2*b^2*t^2*z^6+4*a^2*b^2*x^4*y^2*z^2-6*a^2*b^2*x^2*y^4*z^2-6*a^2*b^2*x^2*y^2*z^4+2*a^2*b^2*y^6*z^2-4*a^2*b^2*y^4*z^4+2*a^2*b^2*y^2*z^6+2*a^2*c^6*x^4-2*a^2*c^6*x^2*y^2+2*a^2*c^6*x^2*z^2+2*a^2*c^4*x^6-4*a^2*c^4*x^4*y^2+2*a^2*c^4*x^4*z^2+2*a^2*c^4*x^2*y^4+2*a^2*c^4*x^2*y^2*z^2-4*a^2*c^4*x^2*z^4-2*a^2*c^2*d^2*x^6+6*a^2*c^2*d^2*x^4*y^2+2*a^2*c^2*d^2*x^4*z^2-6*a^2*c^2*d^2*x^2*y^4+4*a^2*c^2*d^2*x^2*y^2*z^2+2*a^2*c^2*d^2*x^2*z^4+2*a^2*c^2*d^2*y^6-6*a^2*c^2*d^2*y^4*z^2+6*a^2*c^2*d^2*y^2*z^4- 2*a^2*c^2*d^2*z^6-2*a^2*c^2*t^2*x^6+6*a^2*c^2*t^2*x^4*y^2+2*a^2*c^2*t^2*x^4*z^2-6*a^2*c^2*t^2*x^2*y^4+4*a^2*c^2*t^2*x^2*y^2*z^2+2*a^2*c^2*t^2*x^2*z^4+2*a^2*c^2*t^2*y^6-6*a^2*c^2*t^2*y^4*z^2+ 6*a^2*c^2*t^2*y^2*z^4-2*a^2*c^2*t^2*z^6+2*a^2*c^2*x^6*z^2-6*a^2*c^2*x^4*y^2*z^2-4*a^2*c^2*x^4*z^4+4*a^2*c^2*x^2*y^4*z^2-6*a^2*c^2*x^2*y^2*z^4+2*a^2*c^2*x^2*z^6-2*a^2*d^2*x^6*z^2+2*a^2*d^2*x^4*y^2*z^2+6*a^2*d^2*x^4*z^4+2*a^2*d^2*x^2*y^4*z^2+4*a^2*d^2*x^2*y^2*z^4-6*a^2*d^2*x^2*z^6-2*a^2*d^2*y^6*z^2+6*a^2*d^2*y^4*z^4-6*a^2*d^2*y^2*z^6+2*a^2*d^2*z^8-2*a^2*t^2*x^6*z^2+2*a^2*t^2*x^4*y^2*z^2+6*a^2*t^2*x^4*z^4+2*a^2*t^2*x^2*y^4*z^2+4*a^2*t^2*x^2*y^2*z^4-6*a^2*t^2*x^2*z^6-2*a^2*t^2*y^6*z^2+6*a^2*t^2*y^4*z^4-6*a^2*t^2*y^2*z^6+2*a^2*t^2*z^8- 6*a^2*x^4*y^2*z^4-6*a^2*x^2*y^4*z^4+6*a^2*x^2*y^2*z^6-b^8*y^4+2*b^6*c^2*x^2*y^2+2*b^6*c^2*y^4-2*b^6*c^2*y^2*z^2+2*b^6*x^2*y^4-2*b^6*y^6+2*b^6*y^4*z^2-b^4*c^4*x^4-4*b^4*c^4*x^2*y^2+2*b^4*c^4*x^2*z^2-b^4*c^4*y^4+2*b^4*c^4*y^2*z^2-b^4*c^4*z^4-4*b^4*c^2*x^4*y^2+2*b^4*c^2*x^2*y^4+2*b^4*c^2*x^2*y^2*z^2+2*b^4*c^2*y^6-4*b^4*c^2*y^4*z^2+2*b^4*c^2*y^2*z^4+2*b^4*d^2*x^4*y^2-4*b^4*d^2*x^2*y^4- 4*b^4*d^2*x^2*y^2*z^2+2*b^4*d^2*y^6-4*b^4*d^2*y^4*z^2+2*b^4*d^2*y^2*z^4+2*b^4*t^2*x^4*y^2-4*b^4*t^2*x^2*y^4-4*b^4*t^2*x^2*y^2*z^2+2*b^4*t^2*y^6-4*b^4*t^2*y^4*z^2+2*b^4*t^2*y^2*z^4-b^4*x^4*y^4+2*b^4*x^2*y^6+4*b^4*x^2*y^4*z^2-b^4*y^8+2*b^4*y^6*z^2-b^4*y^4*z^4+2*b^2*c^6*x^4+2*b^2*c^6*x^2*y^2-2*b^2*c^6*x^2*z^2+2*b^2*c^4*x^6+2*b^2*c^4*x^4*y^2-4*b^2*c^4*x^4*z^2-4*b^2*c^4*x^2*y^4+2*b^2*c^4*x^2*y^2*z^2+2*b^2*c^4*x^2*z^4-2*b^2*c^2*d^2*x^6+2*b^2*c^2*d^2*x^4*y^2+6*b^2*c^2*d^2*x^4*z^2+2*b^2*c^2*d^2*x^2*y^4+4*b^2*c^2*d^2*x^2*y^2*z^2-6*b^2*c^2*d^2*x^2*z^4-2*b^2*c^2*d^2*y^6+6*b^2*c^2*d^2*y^4*z^2-6*b^2*c^2*d^2*y^2*z^4+2*b^2*c^2*d^2*z^6-2*b^2*c^2*t^2*x^6+2*b^2*c^2*t^2*x^4*y^2+6*b^2*c^2*t^2*x^4*z^2+2*b^2*c^2*t^2*x^2*y^4+4*b^2*c^2*t^2*x^2*y^2*z^2- 6*b^2*c^2*t^2*x^2*z^4-2*b^2*c^2*t^2*y^6+6*b^2*c^2*t^2*y^4*z^2-6*b^2*c^2*t^2*y^2*z^4+2*b^2*c^2*t^2*z^6+2*b^2*c^2*x^6*y^2-4*b^2*c^2*x^4*y^4-6*b^2*c^2*x^4*y^2*z^2+2*b^2*c^2*x^2*y^6-6*b^2*c^2*x^2*y^4*z^2+ 4*b^2*c^2*x^2*y^2*z^4-2*b^2*d^2*x^6*y^2+6*b^2*d^2*x^4*y^4+2*b^2*d^2*x^4*y^2*z^2-6*b^2*d^2*x^2*y^6+4*b^2*d^2*x^2*y^4*z^2+2*b^2*d^2*x^2*y^2*z^4+2*b^2*d^2*y^8-6*b^2*d^2*y^6*z^2+6*b^2*d^2*y^4*z^4- 2*b^2*d^2*y^2*z^6-2*b^2*t^2*x^6*y^2+6*b^2*t^2*x^4*y^4+2*b^2*t^2*x^4*y^2*z^2-6*b^2*t^2*x^2*y^6+4*b^2*t^2*x^2*y^4*z^2+2*b^2*t^2*x^2*y^2*z^4+2*b^2*t^2*y^8-6*b^2*t^2*y^6*z^2+6*b^2*t^2*y^4*z^4-2*b^2*t^2*y^2*z^6-6*b^2*x^4*y^4*z^2+6*b^2*x^2*y^6*z^2-6*b^2*x^2*y^4*z^4-c^8*x^4-2*c^6*x^6+2*c^6*x^4*y^2+2*c^6*x^4*z^2+2*c^4*d^2*x^6-4*c^4*d^2*x^4*y^2-4*c^4*d^2*x^4*z^2+2*c^4*d^2*x^2*y^4-4*c^4*d^2*x^2*y^2*z^2+ 2*c^4*d^2*x^2*z^4+2*c^4*t^2*x^6-4*c^4*t^2*x^4*y^2-4*c^4*t^2*x^4*z^2+2*c^4*t^2*x^2*y^4-4*c^4*t^2*x^2*y^2*z^2+2*c^4*t^2*x^2*z^4-c^4*x^8+2*c^4*x^6*y^2+2*c^4*x^6*z^2-c^4*x^4*y^4+4*c^4*x^4*y^2*z^2-c^4*x^4*z^4+ 2*c^2*d^2*x^8-6*c^2*d^2*x^6*y^2-6*c^2*d^2*x^6*z^2+6*c^2*d^2*x^4*y^4+4*c^2*d^2*x^4*y^2*z^2+6*c^2*d^2*x^4*z^4-2*c^2*d^2*x^2*y^6+2*c^2*d^2*x^2*y^4*z^2+2*c^2*d^2*x^2*y^2*z^4-2*c^2*d^2*x^2*z^6+2*c^2*t^2*x^8- 6*c^2*t^2*x^6*y^2-6*c^2*t^2*x^6*z^2+6*c^2*t^2*x^4*y^4+4*c^2*t^2*x^4*y^2*z^2+6*c^2*t^2*x^4*z^4-2*c^2*t^2*x^2*y^6+2*c^2*t^2*x^2*y^4*z^2+2*c^2*t^2*x^2*y^2*z^4-2*c^2*t^2*x^2*z^6+6*c^2*x^6*y^2*z^2-6*c^2*x^4*y^4*z^2-6*c^2*x^4*y^2*z^4-d^4*x^8+4*d^4*x^6*y^2+4*d^4*x^6*z^2-6*d^4*x^4*y^4-4*d^4*x^4*y^2*z^2-6*d^4*x^4*z^4+4*d^4*x^2*y^6-4*d^4*x^2*y^4*z^2-4*d^4*x^2*y^2*z^4+4*d^4*x^2*z^6-d^4*y^8+4*d^4*y^6*z^2-6*d^4*y^4*z^4+ 4*d^4*y^2*z^6-d^4*z^8+2*d^2*t^2*x^8-8*d^2*t^2*x^6*y^2-8*d^2*t^2*x^6*z^2+12*d^2*t^2*x^4*y^4+8*d^2*t^2*x^4*y^2*z^2+12*d^2*t^2*x^4*z^4-8*d^2*t^2*x^2*y^6+8*d^2*t^2*x^2*y^4*z^2+8*d^2*t^2*x^2*y^2*z^4-8*d^2*t^2*x^2*z^6+2*d^2*t^2*y^8-8*d^2*t^2*y^6*z^2+12*d^2*t^2*y^4*z^4-8*d^2*t^2*y^2*z^6+2*d^2*t^2*z^8+10*d^2*x^6*y^2*z^2-20*d^2*x^4*y^4*z^2-20*d^2*x^4*y^2*z^4+10*d^2*x^2*y^6*z^2-20*d^2*x^2*y^4*z^4+ 10*d^2*x^2*y^2*z^6-t^4*x^8+4*t^4*x^6*y^2+4*t^4*x^6*z^2-6*t^4*x^4*y^4-4*t^4*x^4*y^2*z^2-6*t^4*x^4*z^4+4*t^4*x^2*y^6-4*t^4*x^2*y^4*z^2-4*t^4*x^2*y^2*z^4+4*t^4*x^2*z^6-t^4*y^8+4*t^4*y^6*z^2-6*t^4*y^4*z^4+ 4*t^4*y^2*z^6-t^4*z^8-6*t^2*x^6*y^2*z^2+12*t^2*x^4*y^4*z^2+12*t^2*x^4*y^2*z^4-6*t^2*x^2*y^6*z^2+12*t^2*x^2*y^4*z^4-6*t^2*x^2*y^2*z^6-9*x^4*y^4*z^4=0                                                                               (1) -a^8*t^4+2*a^8*t^2*x^2-a^8*x^4-2*a^6*b^2*t^2*x^2+2*a^6*b^2*t^2*y^2+2*a^6*b^2*t^2*z^2+2*a^6*b^2*x^4-2*a^6*b^2*x^2*y^2-2*a^6*b^2*x^2*z^2+4*a^6*d^2*t^4-6*a^6*d^2*t^2*x^2-2*a^6*d^2*t^2*y^2-2*a^6*d^2*t^2*z^2+2*a^6*d^2*x^4+2*a^6*d^2*x^2*y^2+2*a^6*d^2*x^2*z^2+2*a^6*t^4*x^2-2*a^6*t^4*y^2+2*a^6*t^4*z^2-4*a^6*t^2*x^4+4*a^6*t^2*x^2*y^2-4*a^6*t^2*x^2*z^2+2*a^6*x^6-2*a^6*x^4*y^2+2*a^6*x^4*z^2- 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2*t^6*y^4*z^2+2*t^6*y^2*z^4-t^4*x^8+2*t^4*x^6*y^2+4*t^4*x^6*z^2-3*t^4*x^4*y^4-2*t^4*x^4*y^2*z^2-6*t^4*x^4*z^4+2*t^4*x^2*y^6+8*t^4*x^2*y^4*z^2-2*t^4*x^2*y^2*z^4+4*t^4*x^2*z^6- t^4*y^8+2*t^4*y^6*z^2-3*t^4*y^4*z^4+2*t^4*y^2*z^6-t^4*z^8-6*t^2*x^6*y^2*z^2+6*t^2*x^4*y^4*z^2+12*t^2*x^4*y^2*z^4-6*t^2*x^2*y^6*z^2+6*t^2*x^2*y^4*z^4-6*t^2*x^2*y^2*z^6-9*x^4*y^4*z^4=0                 (4) \$-t^2*x^4+2*t^2*x^2*y^2+2*t^2*x^2*z^2-t^2*y^4+2*t^2*y^2*z^2-t^2*z^4-4*x^2*y^2*z^2-144*V^2=0 \$                                          (5) 显然通过求解(1),(2),(3),(4)可以得到\$x,y,z,t\$(即可以求出四面体六条棱边长），然后结合（5）即可以求出最大的体积\$V\$

 对于42#的表达式： 代入\$a=b=c=d=sqrt(6)/4\$,可以得到\$x=y=z=1,t=sqrt(2),V=sqrt(2)/12\$ 但是当\$a=b=c\$时，我并没能通过消元的方法得到\$V\$的表达式，即我怀疑12#给出的\$V\$的表达式有误： 即\$V=(9*a^2*d-d^3+(3*a^2+d^2)*sqrt(3*a^2+d^2))/(18*sqrt(3))\$ 当然\$V=(3*(a*b*c+a*b*d+a*c*d+b*c*d)-(a^3+b^3+c^3+d^3)+(a^2+b^2+c^2+d^2)*sqrt(a^2+b^2+c^2+d^2))/(18*sqrt(3))\$也是值得怀疑的。 关于垂心四面体，我还没有在网上找到任何有关求棱长及体积的公式，因此我怀疑根本不存在根式表达式，或许只能给出楼上的代数方程。

### 评分

Lwins_G 0 0 + 2 + 2 保留意见

 根据42#方程可以得到： 例1. \$AB = sqrt(2), AC = sqrt(2), AD = sqrt(5), BC = sqrt(2), BD = sqrt(5), CD = sqrt(5), AP = 1.179132857, BP= 1.590562125, CP= 1.179132857, DP=0 .7075924853\$ \$V=0.4166666666\$而根据12#结果 \$V_2=(3*(a*b*c+a*b*d+a*c*d+b*c*d)-a^3-b^3-c^3-d^3+(a^2+b^2+c^2+d^2)*sqrt(a^2+b^2+c^2+d^2))/(18*sqrt(3))=0.7665992111\$ 例2.\$AB = sqrt(5), AC = sqrt(10), AD = sqrt(17), BC = sqrt(13), BD = sqrt(20), CD = sqrt(25), AP = 1.882105750, BP = 3.572281328, CP= 2.786510772, DP = 1.955215158,V=4.149966532\$， 而根据12#结果 \$V_2=(3*(a*b*c+a*b*d+a*c*d+b*c*d)-a^3-b^3-c^3-d^3+(a^2+b^2+c^2+d^2)*sqrt(a^2+b^2+c^2+d^2))/(18*sqrt(3))=8.042024794\$ 显然公式是有误的，有兴趣的可以试着验算一下？

 这个题目应该用向量做

 第一问由于已经确定是垂心，还是比较容易的，就是已有四向量两两内积相等。

### 评分

 我们记四个向量两两内积为T 底面面积2倍可以写成\$|(a-b) xx (a-c)|\$由于向量积平方和内积平方之和等于两向量长度平方的积。所以底面面积平方的4倍可以写成 \$(a-b)^2(a-c)^2-((a-b)(a-c))^2=(a^2+b^2-2T)(a^2+c^2-2T)-(a^2-T)^2=(a^2b^2+b^2c^2+c^2a^2)-2T(a^2+b^2+c^2)+3T^2\$ 而对应的高的长度可以写成\${d(d-a)}/{|d|}\$所以高的平方为\${(d^2-T)^2}/{d^2}\$ 于是\$36V^2 = {((a^2b^2+b^2c^2+c^2a^2)-2T(a^2+b^2+c^2)+3T^2)(d^4-2Td^2+T^2)}/{d^2}\$ 同样\$36V^2={((d^2b^2+b^2c^2+c^2a^2)-2T(d^2+b^2+c^2)+3T^2)(a^4-2Ta^2+T^2)}/{a^2}\$ 我们得出 \$a^2((a^2b^2+b^2c^2+c^2a^2)-2T(a^2+b^2+c^2)+3T^2)(d^4-2Td^2+T^2)=d^2((d^2b^2+b^2c^2+c^2a^2)-2T(d^2+b^2+c^2)+3T^2)(a^4-2Ta^2+T^2)\$ 得出: \$3T^4-2(a^2+b^2+c^2+d^2)T^3+(a^2b^2+a^2c^2+a^2d^2+b^2c^2+b^2c^2+c^2d^2)T^2-a^2b^2c^2d^2=0\$ 由于这里T是一个四次方程的解，感觉表达式V应该没有简单的形式,最终结果写成对称形式为： \$36V^2=-4T^3+3(a^2+b^2+c^2+d^2)T^2-2T(a^2b^2+a^2c^2+a^2d^2+b^2c^2+b^2d^2+c^2d^2)+d^2b^2a^2+d^2c^2a^2+d^2c^2b^2+a^2b^2c^2\$

### 点评

 而T的方程有多个根，取唯一的负数根即可

 一些特例：当a=1,b=2,c=3,d=4时，要使V最大，应有 AB=2.7476962050544724834734263418124774737034288451755985578428886640455026666604774750979049046764082286504741354625063066347486667558786455652610559850348782690204113624378542655762711758742180267837632235621105672807500392977494645832739524025144973883321544472807070936937186140015603111961790282602145186506228951857452651064354771293188488927166207971992826261804956376461549284472522606755707833385870819079346644960697288040759954539491134597566179888193634111586783494561117252452672756297839900 AC=3.5425745490068024392387512934969883332361771287647493257729451481644603418460321245736492574545515015443479094337787184637038485914688025789492035175052576647482730671632392266803030107043816281302516536530756680756843869829944736872220623439045174979491258625899196050913756816713462383461092644982631706703536675218899200244749548000563959963334593449874576234459702093084843615975571755978113347164366582386075810101292976476120794234037777580699783938993773235247692118343857740925113837910375349 AD=4.4215194713210016233957282017726779042599620077866405969125048346967280274691239140255764611904873655350650774697263544564996148513797452948927286429280005461035558700408075673838993602630607881581946721835620659089334826898615600074707689226011126034005662796674238475489784088628459309853574667436693382274245689217836342228593046140184455964731844612895947142751957397463903136040244436307707753314436946804899159161617981580454852805242447686820131087056035010417894692422691306039475082404376201 CD=5.2487936171343934541119359091041567271540322138977020234130608958333614338583421912010859985227584175609516245172023389824017952906751380752064339923672132564293986130084694908881650258807735967055450192724418358304727781887156786682843550148675790966328718881331256884066855456004679681455251179808448517533416407573743835798040220849241507458937122041669218533763350282734220899966632695305043131501918393373157010767386385580388164124884103800501815020878350501082385286602597324021475537390484385 BD=4.748666595505600422341406032094185052625841972083707490284089617377306041326791845603567333501547047865491508349121810894715043552355712159423672884047377361317037504069675625806970642635051631980681320520020082601908082712971800781043011767916077705622383499164304962287601066949779715735075349736617114980152005521195719740798018576662106632189904147748563537566378990391605815249385918700224053414940164542814962872382782161988132324411368758208641048157720226618783104293986321598470017814313234 BC=3.943327837660818900079699763020689070691214988946079387809078835376225123742988201816966005618843656163423463348831617789665253168392012024861375786381978605098306243987890118723172266038172852600374737966315449606553513766551523275977039295233931103294265957938336100783398024622778470403435758873563658040965912862582299140245290843808426206963266945580486001717784302341918716220231195390559380259775432346570849567065248889446156479255132820133448073142597638694149944067250254371019092534017177 此时 V=6.81232772589693021059932086228599847714137786486277311255752185391411632379372953906802625571578365343983849042018281936011949627964014414508549978802254100040372748595596941405310099460781314511198346352383097588104464671443062012649710342462667963212878248205710479127285140526428815870082839400780832007221062507638684116179928336336556991817768767468335321649919463984081536416988393961663548377194831672699630945463355301068009853609059961335470115846690859662280273917752459540235097794389189

 当a=5,b=6,c=7,d=8时，要使V最大，应有 AB=9.354504601822144040036543928965253501623835173117051944744003395641372587052066809984653977696087123149527356815116031915720354658667666039596884276556323710293885443492151984235860089638380431281748194394625271340597237675647843126334097300121921652320398182448409934198446431178354816575454106832784787692362631986311004172136043349317302732071009354694341395901851908607368875618466942677159439089813058346938301198214661136586860020357204275690405856031574959819567916492085066055289528047492921493997520028314431156025670492267281357587760961694776710612564192964318529473564009211276489837295501884029618654923787366765206973760158544075041587817647558051419032758981916743624842628740494658563259069984133234340366906612671556321874665772834933665427156585903289938292766480151901819229649165335923707667510161620590460245255538676836619886625779329633806886305933358563293717683559044953093989830690656178013026917216846819123899871640670116932174906908179188067135732058653991154897400433258 AC=10.02530579810469769453301398430895245498826727177334991031435993305407866206255749992016300069371696578540497584110933933714013785624242975974563928530166772920284185469788313987009813184110642080735009024957959018504642407707480979566434958857518272215068591435068928280191741450736440512643342708462792005029878122841044306024665164513769216717726543336218061017045351154327587232047696162094506048095355220585213732975976767349264281335139439186357484425966761839015291284913579637178490108322558326758819454602207419730235950470347035337629178410712142335360107172459214898799703911955079081808753605701366704455684463361230757171147310097395780759408523273096343552821096319380083918925469793144141286924948760627842819915743303240080065545810763673440933529113381549189344730962759090022075439167059391489251156680632826338529276736175365369987605088061579478928452524232912180867746601050895165163247144254591032310116378192215821363031794741513321578593198010973660733759137594451549584537680 AD=10.74740695914654892771273168914539197681757882984966087419823011627142041367160658944487863887466625120536077033774713314963546976073861662791583043601228376783217831128532961262194864140961671613936173940005768223394801258401707397497543583823761450811106253910734537924013111232085664143413066651921210270118829690253024255906700994874105490304107278487007334262775622534014276965232683297101611542661217644879287195664731066323767428503347396549892979120016668182320903255790533068247175957153360023482696195355737734958219297819056686244287768961159308959383224609896877951696935554969869718390532947794516448398783207200428496231697213120665502806713501735064503527954374620853644748780287841975989238386659793874798514890212784581475051159404341932437203709683387353873771028238626630557984105795197961264464718233805873740352647066284377784883400606315552574046693164087672095212918284388288096299095990753843372820782788165111537746491526380587026207496161755038416887209980012685064832616906 CD=11.81129782646732135191655137423755362987318240475250424704426877767851061780208349846799753539156030201240359038394297419416033289650718979409095619226457044609880116216460149193784766486948625379921987812973808749736198097706800576185545535122163660067842413066321664277508541667700394626783688635455691323646980668968232704489295059503872700436796169951172903540578787640247210143670262197281355236284101560121005647207975133730550977094755594759141325519179765773007580812642921402625190298101451232438787708761298784542374910771517678819123324127642936412503444011512720269969446853417806806181995085504568325388970999881239184477763262746472354688672741152611556096830028632095905092700622064789676865848325390671836061146215041383212239165709198650881765094769105544866458122876864629033238760510218160770837299266065546068283624890494263990786481626693776603865481710197814511045184084833093919616089107754566996262487714137230020567650712261491170497198579667340719040866202515949815311985120 另外两条棱长满足对棱平方和相等条件，这里略去不写 V=137.809311953656022823001066731083630863882637366997282296033200511905081123449143177463033317067263007421026443813580021685447021723694331428234216224739700379392945643093392232016273083213502846575501255100277824331031845825076940541846656160582725570140283106062549894290106387601586381942615538371128744980286863108493728835919753083851731480228983902454055122504708626870959273193018227670869981267278704088788214327862883078710317596222205315370737410994970445207341219159942396612333842452810704134022190325861009690371831110059673748437079638426515808966654326142526049874520130354967350189251707798346720323000401469137427509564494265443320202228183463954481338826127302591526367725185792797477916789007219378918441711537946823892128247925962607265533963933838171869164967270504264215155443027058425607077731754557458427276975179454559224916748490427728476250987744927054818061741797369199744033702845893639126057380208665819355068730799300307556607829907982679627583332212277270815297700233

### 点评

a=5,b=6,c=7,d=8,得出T满足3*x^4 - 348*x^3 + 10929*x^2 - 2822400=0 得出T=-13.25337817275583480604453880, V=137.8093119536560228230010667  发表于 2013-11-25 18:12