■等面単体の体積(その180)

 結局,

  b^2=4−16λ^2/(4−b^2)

とおくのが最速だったようだ.

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  (4−b^2)^2=16λ^2

  1/λ^2=4/(4−a^2)+4/(4−b^2)+4/(4−c^2)

=4{(4−a^2)(4−b^2)+(4−b^2)(4−c^2)+(4−c^2)(4−a^2)}/(4−a^2)(4−b^2)(4−c^2)

  (4−b^2)=4(4−a^2)(4−c^2)/{(4−a^2)(4−b^2)+(4−b^2)(4−c^2)+(4−c^2)(4−a^2)}

(4−b^2){(4−a^2)(4−b^2)+(4−b^2)(4−c^2)+(4−c^2)(4−a^2)}=4(4−a^2)(4−c^2)

(4−b^2){(8−a^2−c^2)(4−b^2)+(4−c^2)(4−a^2)}=4(4−a^2)(4−c^2)

(4−b^2){b^2(4−b^2)+(4−c^2)(4−a^2)}=4(4−a^2)(4−c^2)

(4−b^2)b^2(4−b^2)=b^2(4−a^2)(4−c^2)

(4−c^2)(4−a^2)=(4−b^2)^2

(4−c^2)(−4+c^2+b^2)=(4−b^2)^2

(4−c^2)^2−b^2(4−c^2)+(4−b^2)^2=0

4−c^2=1/2{b^2−(b^4−4(4−b^2)^2)^1/2}

4−c^2=1/2{b^2−(−64+32b^2−3b^4)^1/2}

4−c^2=1/2{b^2−{(3b^2−8)(8−b^2))^1/2}

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