■等面単体の体積(その305)

135→(√10,√10,√12)に注目して

P1(0,0,0,0)

P3(√3,√7,0,0)

P5(√12,0,0,0)

P2(x,y,z,0)とおくと

  x^2+y^2+z^2=6

  (x-√3)^2+(y-√7)^2+z^2=6

  (x-√12)^2+y^2+z^2=12

  (x-√12)^2+6-x^2=12

-2x√12+6=0→x=(√3)/2

  y^2+z^2=6-3/4=21/4

  3/4+(y-√7)^2+21/4-y^2=6

-2y√7+7=0→y=(√7)/2

  z^2=21/4-7/4=14/4→z=(√14)/2

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P1(0,0,0,0)

P2((√3)/2,(√7)/2,(√14)/2,0)

P3(√3,√7,0,0)

P5(√12,0,0,0)

P4(x,y,z,w)とおく.

  P1P2=P2P3=P3P4=P4P5=P5P6=√6

  P1P3=P2P4=P3P5=P4P6=√10

  P1P4=P2P5=P3P6=√12

  P1P5=P2P6=√12

  P1P6=√10

より

  x^2+y^2+z^2+w^2=12

  (x-√3/2)^2+(y-√7/2)^2+(z-√14/2)^2+w^2=10

  (x-√3)^2+(y-√7)^2+z^2+w^2=6

  (x-√12)^2+y^2+z^2+w^2=6

  (x-√12)^2+12-x^2=6

  -2x√12=-18→x=9/√12

  y^2+z^2+w^2=12-81/12=63/12

  9/12+(y-√7)^2+z^2+w^2=6

  9/12+(y-√7)^2+63/12-y^2=6

  -2y√7+13=6→y=(√7)/2

  z^2+w^2=63/12-7/4=42/12

  3+(z-√14/2)^2+w^2=10

  3+(z-√14/2)^2+42/12-z^2=10

-z√14+14/4+3+14/4=10

-z√14+10=10→z=0,w=(√14)/2

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