Proof with Geometry.
intros.
unfold weak_parallelogram in H.
DecompAndAll.
DecompExAnd H3 X.
unfold mid_point in *.
DecompAndAll.
assert (S A P Q + S C P Q = 2 * S X P Q).
apply mid_point_equation; Geometry.
assert (S B P Q + S D P Q = 2 * S X P Q).
apply mid_point_equation; Geometry.
congruence.
Qed.

Theorem l2_11a :
forall A B C D P Q : Point,
parallelogram A B C D -> S A P Q + S C P Q = S B P Q + S D P Q.