向量法

如下的"AB","CD"等都表示向量。x表示叉乘

  S_EFHG = S_EGH + S_EFH
      = 1/2(GHxGE) + 1/2(FExFH)
      = 1/2(1/3*ABx(GA+AD+DE)) + 1/2(1/3CDx(FC+CB+BH)
      = 1/6*ABx(1/3*BA + AD + 1/3*DC)) + 1/6*CDx(1/3DC + CB + 1/3BA)
      = 1/6*ABx(AD + 1/3*DC) + 1/6*CDx(CB + 1/3BA)
      = 1/6*(ABxAD + CDxCB) + 1/18(ABxDC + CDxBA)
      = 1/6*(ABxAD + CDxCB) + 1/18(ABxDC - BAxCD)
      = 1/6*(ABxAD + CDxCB)
      = 1/3*1/2(ABxAD + CDxCB)
      = 1/3*S_ABCD
      = 1/3*3
      = 1