Proof with try solve [ Geometry | congruence | field; Geometry ].
intros...
assert
(S P A B / S Q A B =
S P A B / S P A M * (S P A M / S Q A M * (S Q A M / S Q A B)))...
assert (A <> B); eauto with Geom...
assert (A <> M); eauto with Geom...
assert (A ** B / A ** M * (P ** M / Q ** M * (A ** M / A ** B)) = P ** M / Q ** M)...
assert (A ** B / A ** M = S P A B / S P A M)...
assert (P ** M / Q ** M = S P A M / S Q A M)...
assert (A ** M / A ** B = S Q A M / S Q A B)...
Qed.

Theorem co_side :
forall A B P Q M : Point,
~ Col Q A B ->
Q <> M -> Col A B M -> Col P Q M -> P ** M / Q ** M = S P A B / S Q A B.