给定棱长的四面体体积 2个公式

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AOPS - High School Math Formula for Volume of Tetrahedron with given lengths

验证给定棱长的四面体体积 2个公式相等，需要证明以下恒等式\[\det\left(\begin{array}{ccccc}0 & u^{2} & v^{2} & w^{2} & 1 \\ u^{2} & 0 & W^{2} & V^{2} & 1 \\ v^{2} & W^{2} & 0 & U^{2} & 1 \\ w^{2} & V^{2} & U^{2} & 0 & 1 \\ 1 & 1 & 1 & 1 & 0\end{array}\right) =\frac {(-a+b+c+d)(a-b+c+d)(a+b-c+d)(a+b+c-d)}{128u^2v^2w^2}\]其中\begin{aligned}a&={\sqrt {xYZ}}\\b&={\sqrt {yZX}}\\c&={\sqrt {zXY}}\\d&={\sqrt {xyz}}\\X&=(w-U+v)\,(U+v+w)\\x&=(U-v+w)\,(v-w+U)\\Y&=(u-V+w)\,(V+w+u)\\y&=(V-w+u)\,(w-u+V)\\Z&=(v-W+u)\,(W+u+v)\\z&=(W-u+v)\,(u-v+W).\end{aligned}

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1. u, v, w, U, V, W = var('u v w U V W')
2. X = (w-U+v)*(U+v+w)
3. x = (U-v+w)*(v-w+U)
4. Y = (u-V+w)*(V+w+u)
5. y = (V-w+u)*(w-u+V)
6. Z = (v-W+u)*(W+u+v)
7. z = (W-u+v)*(u-v+W)
8. a = sqrt(x*Y*Z)
9. b = sqrt(y*Z*X)
10. c = sqrt(z*X*Y)
11. d = sqrt(x*y*z)
12. M = Matrix([[0, u^2, v^2, w^2, 1],
13.             [u^2, 0, W^2, V^2, 1],
14.             [v^2, W^2, 0, U^2, 1],
15.             [w^2, V^2, U^2, 0, 1],
16.             [1, 1, 1, 1, 0]])
17. expand((-a+b+c+d)*(a-b+c+d)*(a+b-c+d)*(a+b+c-d)-(128*u**2*v**2*w**2)*M.determinant())

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expand的结果不是0，不知哪里输入错了