Proof with Geometry.
intros...
assert (P <> Q)...
eapply par_aux_1; apply H...
cases_equality Q M...
assert (Q ** M = 0)...
rewrite H4...
rewrite <- H3 in H0...
assert (Col Q A B)...
unfold parallel in H...
rewrite H5; field...
cases_equality A Q...
assert (A = M)...
apply (inter_unicity A B P Q M H H0 H1 H4)...
rewrite H4...
assert (Q = M)...
congruence...
assert (Q ** M = 0)...
assert (S Q Q B = 0)...
rewrite H7...
rewrite H8...
field...
assert (~ Col Q A B)...
unfold not in |- *; intro...
unfold parallel in H...
unfold S4 in H...
assert (S P A B = S P A Q + S P Q B + S Q A B)...
rewrite H5 in H6...
IsoleVar (S P A Q) H6...
rewrite H6 in H...
assert (S P A B - 0 - S P Q B + S P Q B = S P A B)...
ring...
rewrite H7 in H...
clear H6 H7...
assert (Col A B Q)...
assert (A <> B)...
unfold not in |- *; intro...
assert (S P A B = 0)...
rewrite H7...
assert (Col A M Q); eauto with Geom...
assert (Col Q M P)...
assert (Col Q M A)...
assert (Col Q A P); eauto with Geom...
assert (Col A Q B)...
assert (Col A Q P)...
assert (Col A B P); eauto with Geom...
assert (Col P A B)...
assert (P ** M / Q ** M = S P A B / S Q A B)...
assert (Q ** M / P ** Q * (P ** Q / Q ** M) = 1)...
field...
assert (P ** M / Q ** M - 1 = P ** Q / Q ** M).
replace (P ** Q) with (P ** M - Q ** M).
field...
assert (Q ** M = - M ** Q)...
rewrite H8 in |- *.
ring_simplify...
rewrite <- H8 in H7.
clear H8.
rewrite H6 in H7.
assert (S4 P A Q B = S P A B - S Q A B)...
rewrite H8.
clear H8.
IsoleVar (Q ** M / P ** Q) H7...
rewrite H7.
field...
split...
assert (S4 P A Q B = S P A B - S Q A B)...
unfold parallel in H.
rewrite H9 in H...
assert (S P A B / S Q A B - 1 = (S P A B - S Q A B) / S Q A B).
field...
rewrite H8...
apply nonzerodiv...
assert (S4 P A Q B = S P A B - S Q A B)...
rewrite <- H9...
Qed.
Hint Resolve co_side_bis: Geom.

Theorem co_side_ter :
forall A B P Q M : Point,
~ parallel P Q A B ->
Col A B M -> Col P Q M -> P ** M / P ** Q = S P A B / S4 P A Q B.