Inside a Tetrahedron

It was mentioned in another article that, given four points
(0,0), (x1,y1), (x2,y2), and (x3,y3) in the plane, one of the
points lies inside the triangle formed by the other three iff

                         abc(a-b+c)  <  0

where a,b,c are the determinants

          | x1  y1 |           | x1  y1 |           | x2  y2 |
    a  =  |        |     b  =  |        |     c  =  |        |
          | x2  y2 |           | x3  y3 |           | x3  y3 |


I'm wondering about the analagous problem in three dimensions,
i.e., given five points 

    (0,0,0)   (x1,y1,z1)   (x2,y2,z2)   (x3,y3,z3)   (x4,y4,z4)

in space, can we express a necessary and sufficient condition that
any one of these points is inside the tetrahedron formed by the
other three in terms of the four determinants

                                  | xi  yi  zi |
                   D_(i,j,k)  =   | xj  yj  zj |
                                  | xk  yk  zk |

with (i,j,k) = (1,2,3), (1,2,4), (1,3,4), and (2,3,4)?