Proof with Geometry.
intros.
unfold weak_parallelogram in H0.
DecompAndAll.
DecompExAnd H4 X.
unfold parallelogram.
assert (parallel A B C D).
eapply diag_mid_point_parallel;eauto.
repeat split;try assumption.
cut (parallel C B A D).
Geometry.
eapply diag_mid_point_parallel.
2:apply H5.
unfold mid_point in *;DecompAndAll;Geometry.
Qed.

Theorem l2_11a_strong :
forall A B C D P Q : Point,
weak_parallelogram A B C D -> S A P Q + S C P Q = S B P Q + S D P Q.