Proof with Geometry.
intros.
unfold parallelogram in H.
DecompAndAll.
unfold parallel,S4 in *.
assert (S A B D = S A B C).
RewriteVar (S A B D) H2...
replace (-(1) * S A C B) with (- S A C B) by ring...
assert (S B C D = S B C A).
RewriteVar (S B C D) H4...
replace (-(1) * S B A C) with (- S B A C) by ring...
assert (S C B D = - S B C A).
rewrite <- H3...
assert (~ Col C B D).
unfold not in |- *; intro.
rewrite H7 in H6...
assert (A ** O / C ** O = S A B D / S C B D).
apply co_side...
unfold not in |- *; intro.
subst O...
rewrite H in H8...
rewrite H6 in H8...
assert (S A B C = S B C A)...
rewrite H9 in H8...
assert (C ** O = - O ** C)...
rewrite H10 in H8...
RewriteVar (A ** O) H8...
field...
rewrite <- H10.
assert (O <> C).
unfold not; intro;subst O...
Geometry.
Qed.

Theorem diago_par_intersect :
forall A B C D : Point,
parallelogram A B C D ->
~ parallel A C B D.