Proof with Geometry.
intros...
assert (S R A B = S R A P + S R P B + S P A B)...
assert (P ** R / P ** Q = S R A P / S Q A P)...
RewriteVar (S R A P) H4...
clear H4...
assert (P ** R / P ** Q = S B P R / S B P Q)...
assert (- S B P R = S R P B)...
rewrite <- H6 in H3...
RewriteVar (S B P R) H4...
clear H4 H6...
assert (S R A B = P ** R / P ** Q * (S Q A P - S B P Q) + S P A B)...
rewrite H3...
field...
clear H3...
assert (S A P Q - S B P Q = S4 A P B Q)...
unfold S4 in H3...
assert (S Q A P = S A P Q)...
rewrite <- H6 in H3...
rewrite H3 in H4...
clear H3 H6...
assert (S A B Q = S Q A B)...
rewrite H3 in H4...
clear H3...
assert (S A P B = - S P A B)...
rewrite H3 in H4...
clear H3...
assert
(P ** R / P ** Q * (- S P A B + S Q A B) + S P A B =
P ** R / P ** Q * S Q A B + (1 - P ** R / P ** Q) * S P A B)...
field...
rewrite H3 in H4...
clear H3...
assert (1 - P ** R / P ** Q = R ** Q / P ** Q)...
assert (R ** P + P ** Q = R ** Q)...
rewrite <- H3...
assert (R ** P = - P ** R)...
rewrite H6.
field...
congruence...
Qed.

Theorem l2_9aux :
forall A B P Q R : Point,
P <> Q ->
Col R P Q ->
Col Q A P ->
~ Col Q P B -> S R A B = P ** R / P ** Q * S Q A B + R ** Q / P ** Q * S P A B.