Ltac convert_to_algebra :=
try setoid_rewrite characterization_of_betweenness_F;
try setoid_rewrite characterization_of_congruence_F;
try setoid_rewrite characterization_of_midpoint_F;
try setoid_rewrite characterization_of_collinearity_F;
try setoid_rewrite characterization_of_right_triangle_F;
try setoid_rewrite characterization_of_perpendicularity_F;
try setoid_rewrite characterization_of_parallelism_F;
try setoid_rewrite characterization_of_equality_F;
try setoid_rewrite characterization_of_neq_F.
Ltac express_disj_as_a_single_poly := repeat rewrite <- mulF__eq0.

Lemma centroid_theorem : forall A B C A1 B1 C1 G,
Midpoint A1 B C ->
Midpoint B1 A C ->
Midpoint C1 A B ->
Col A A1 G ->
Col B B1 G ->
Col C C1 G \/ Col A B C.