■サマーヴィルの等面四面体(その395)
P1(0,0,0)
P2(1/√2,√3/√2,0)
P3(2/√2,0,0)
は△2
P1P2=P2P3=√2
P1P3=√2
を満たす.
P0(0,0,0)
P1(0,0,3h)
P2(m/√2,m√3/√2,h)
P3(2m/√2,0,2h)
とおくと
P0P1^2=9h^2
P0P2^2=2m^2+h^2
P0P3^2=2m^2+4h^2
P1P2^2=2m^2+4h^2
P1P3^2=2m^2+h^2
P2P3^2=2m^2+h^2
2m^2+h^2(3)<2m^2+4h^2(2)
4h^2(1)
ここで,
9h^2=2m^2+h^2,m^2=4h^2
ならば△3
P0P1^2=9h^2
P0P2^2=9h^2
P0P3^2=12h^2
P1P2^2=12h^2
P1P3^2=9h^2
P2P3^2=9h^2
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