■等面単体の体積(その251)

 (その246)の続き.(n=4)

  P1P2=P2P3=P3P4=2

  P1P3=P2P4=√6

  P1P4=√6

  P1(0,0,0,0)

  P2(2,0,0,0)

  P4(1,√5,0,0)

はこれを満たす.

  P3(x,y,z,0)

とおくと,

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

  (x−2)^2+y^2+z^2=4

  (x−1)^2+(y−√5)^2+z^2=4

  (x−2)^2+6−x^2=4

  −4x+4+6=4,x=3/2

  1/4+y^2+z^2=4

  1/4+(y−√5)^2+z^2=4,y=(√5)/2,z=(√10)/2

  P1(0,0,0,0)

  P2(2,0,0,0)

  P3(3/2,(√5)/2,(√10)/2,0)

  P4(1,√5,0,0)

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  P1を外す.

  P2(2,0,0,0)

  P3(3/2,(√5)/2,(√10)/2,0)

  P4(1,√5,0,0)

  P1(x,y,z,0)

とおいで,

  P1P2=P2P3=P3P4=2

  P1P3=P2P4=√6

  P1P4=√6

を満たすものを探す.

  (x−2)^2+y^2+z^2=4

  (x−3/2)^2+(y−(√5)/2)^2+(z−(√10)/2)^2=6

  (x−1)^2+(y−√5)^2+z^2=6

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