Proof with Geometry.
intros.
assert (S A B C = S A B D + S A D C + S D B C)...
rewrite H0 in H2.
rewrite H1 in H2.
cases_col A D C...
cases_equality C A...
apply egalcol...
NormalizeRing H2...
assert (C ** B / C ** A = S D C B / S D C A)...
assert (C ** A + A ** B = C ** B)...
rewrite <- H6 in H5...
assert ((C ** A + A ** B) / C ** A = 1 + A ** B / C ** A)...
field...
rewrite H7 in H5.
clear H7 H6.
assert (S D B C = - S D C B)...
rewrite H6 in H2...
assert (S A D C = S D C A)...
rewrite H7 in H2...
IsoleVarRing (S D C B) H2...
rewrite H2 in H5...
assert (S D C A / S D C A = 1)...
field...
rewrite H8 in H5.
clear H6 H7 H8.
IsoleVarRing (A ** B) H5...
assert (A = B)...
intuition...
Qed.
Hint Resolve col_trans_1.

Lemma build_point_not_collinear_1 : forall A B, A<>B -> exists C, ~ Col A B C.
Proof with Geometry.