■等面単体の体積(その400)

 (その399)の続き.Q4=Q5

  P1(0,0,0,0)

  P2(2m,0,0,h)

  P3(3m/2,m√5/2,m√10/2,2h)

  P4(m,m√5,0,3h)

  P5(m,m√5,0,−2h)

としてみると,

  P1P2^2=4m^2+h^2

  P1P3^2=6m^2+4h^2

  P1P4^2=6m^2+9h^2

  P1P5^2=6m^2+4h^2

  P2P3^2=4m^2+h^2

  P2P4^2=6m^2+4h^2

  P2P5^2=6m^2+9h^2

  P3P4^2=4m^2+h^2

  P3P5^2=4m^2+16h^2

  P4P5^2=25h^2

4m^2+h^2(3)<4m^2+16h^2(1)

6m^2+4h^2(3)<6m^2+9h^2(2)

25h^2(1)

  P1P2=P2P3=P3P4=P4P5=√5

  P1P3=P2P4=P3P5=√8

  P1P4=P2P5=3

  P1P5=√8

より,

  4m^2+h^2=25h^2=5→h^2=1/5,m^2=6/5

  6m^2+4h^2=4m^2+16h^2=8

  6m^2+9h^2=9

h^2=1/5,m^2=6/5はこれらを満たす.

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