Proof with Geometry.
intros...
assert (Q ** M / P ** Q = S Q A B / S4 P A Q B)...
assert (P ** Q + Q ** M = P ** M)...
rewrite <- H3...
RewriteVar (Q ** M) H2...
assert (S P A B = S P A Q + S P Q B + S Q A B)...
rewrite H5...
unfold S4 in |- *...
field...
assert (P <> Q)...
eapply par_aux_1; apply H...
Qed.
Hint Resolve co_side_ter: Geom.

Theorem l2_9_weak :
forall A B P Q R : Point,
P <> Q ->
~ Col Q A P ->
~ Col Q P B ->
Col R P Q -> S R A B = P ** R / P ** Q * S Q A B + R ** Q / P ** Q * S P A B.