Collinear [P1_, P2_, P3_] :=
   Det [{P1, P2, P3}]


AA= {0,   0, 1};
BB= {1,   0, 1};
CC= {c1,  0, 1};
DD= {d1, d2, 1};
EE= {e1, e2, 1};
FF= {f1, f2, 1};
GG= {g1, g2, 1};
HH= {h1, h2, 1};
II= {i1, i2, 1};
JJ= {j1, j2, 1};


TheSystem = 
   {Collinear [AA,BB,CC],
    Collinear [DD,EE,FF],
    Collinear [AA,HH,EE],
    Collinear [DD,HH,BB],
    Collinear [DD,II,CC],
    Collinear [AA,II,FF],
    Collinear [EE,JJ,CC],
    Collinear [BB,JJ,FF],
    (* Non degenerate *)
    Collinear [AA,BB,DD] * z2 - 1,
    Collinear [AA,BB,EE] * z3 - 1,
    (* Conclusion *)
    Collinear [HH,II,JJ] * z1 - 1}

TheSystemGB = GroebnerBasis [TheSystem,  MonomialOrder -> DegreeReverseLexicographic]

    
