Proof with Geometry.
intros...
assert (A ** C / B ** A = S P A C / S P B A)...
assert (A ** C / B ** A = C ** A / A ** B)...
assert (S P C A / S P A B = S P A C / S P B A)...
congruence...
Qed.
Hint Resolve A6_4: Geom.

Theorem A6_5 :
forall A B O P : Point,
O <> A -> ~ Col O A B -> Col P A O -> P ** A / O ** A = S P A B / S O A B.