■サマーヴィルの等面四面体(その250)

P1(0,0,0,0,0,0)

P2(5/√10,(√14)/2,0,0,0,0)

P3(10/√10,0,0,0,0,0)

P4(8/√10,0,56/√560,0,0,0)

P5(6/√10,0,42/√560,21/2√21,0,0)

P6(4/√10,0,28/√560,14/2√21,√(14/3),0)

P0(a,b,c,d,e,f)とおくと

  a^2+b^2+c^2+d^2+e^2+f^2=6

  (a-5/√10)^2+(b-√14/2)^2+c^2+d^2+e^2+f^2=10

  (a-10/√10)^2+b^2+c^2+d^2+e^2+f^2=12

  (a-8/√10)^2+b^2+(c-56/√560)^2+d^2+e^2+f^2=12

  (a-6/√10)^2+b^2+(c-42/√560)^2+(d-21/2√21)^2+e^2+f^2=10

  (a-4/√10)^2+b^2+(c-28/√560)^2+(d-14/2√21)^2+(e-√(14/3))^2+f^2=6

  (a-√10)^2+6-a^2=12

  -2√10a+16=12,a=2/√10

  b^2+c^2+d^2+e^2+f^2=6-4/10=56/10

  (a-5/√10)^2+(b-√14/2)^2+c^2+d^2+e^2+f^2=10

  9/10+(b-√14/2)^2+56/10-b^2=10

  -2・√14/2・b+70/20+130/20=10,b=0

  c^2+d^2+e^2+f^2=56/10

  (a-8/√10)^2+b^2+(c-56/√560)^2+d^2+e^2+f^2=12

  36/10+(c-56/√560)^2+56/10-c^2=12

  -2・56/√560・c+56/10+92/10=12

  -56/√560・c=-7/5

  c=√560/40=14/√560

  d^2+e^2+f^2=56/10-196/560=2940/560=21/4

  (a-6/√10)^2+b^2+(c-42/√560)^2+(d-21/2√21)^2+e^2+f^2=10

  16/10+14/10+(d-21/2√21)^2+21/4-d^2=10

  -2・21/2√21・d+21/4+3+21/4=10

  -2・21/2√21・d=7-42/4=-14/4

  d=7/2√21

  e^2+f^2=441/84-49/84=392/84=98/21

  (a-4/√10)^2+b^2+(c-28/√560)^2+(d-14/2√21)^2+(e-√(14/3))^2+f^2=6

  4/10+196/560+49/84+(e-√(14/3))^2+98/21-e^2=6

  -2√(14/3)・e+14/3+3/4+441/84=6

  -2√(14/3)・e+455/84+441/84=6

  -2√(14/3)・e+896/84=6

  -2√(14/3)・e=(504-896)/84=-392/84

  -√(14/3)・e=-7/3

  e=7/3・√(3/14)=√(7/6)

  f^2=196/42-49/42=147/42=7/2

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