Proof with try solve [ Geometry | congruence | field; Geometry ].
intros...
cases_equality C A...
assert (Col A B D)...
rewrite H4...
symmetry  in |- *...
assert (S P C D / S P A B = S P C D / S P C A * (S P C A / S P A B))...
assert (A <> C)...
assert (Col C D A)...
eauto with Geom...
assert (C ** D / C ** A = S P C D / S P C A)...
assert (A ** C / A ** B = S P A C / S P A B)...
assert (C ** A / A ** B = S P C A / S P A B)...
assert (C ** D / C ** A * (C ** A / A ** B) = C ** D / A ** B)...
Qed.

Theorem par_aux_1 : forall P Q U V : Point, ~ parallel P Q U V -> P <> Q.
Proof with Geometry.